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THEORY  OF  FUNCTIONS 

OF  A 

COMPLEX  VARIABLE 


BY 

A.  R.  FORSYTH,    1 

Sc.D.,  LL.D.,  Math.D.,  F.R.S., 

CHIEF  PROFESSOR  OF  MATHEMATICS  IN  THE  IMPERIAL  ioLLEGE  OF  SCIENCE 

AND  TECHNOLOGY,  LONDON  :  AND  SOMETIME  SADLERIAN  PROFESSOR  OF  PURE 

MATHEMATICS  IN  THE  UNIVERSITY  OF  CAMBRIDGE 


THIRD  EDITION 


BOSTON    COLLEGE    LIBRARY 
CHESTNUT  HILL,  MASS. 

MATH.   " 

CAMBRIDGE 

AT  THE  UNIVERSITY  PRESS 

1918 


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PEEFACE. 

AMONG  the  many  advances  in  the  progress  of  mathematical 
-^^^  science  during  the  last  forty  years,  not  the  least  remarkable 
are  those  in  the  theory  of  functions.  The  contributions  that  are 
still  being  made  to  it  testify  to  its  vitality  :  all  the  evidence  points 
to  the  continuance  of  its  growth.  And,  indeed,  this  need  cause  no 
surprise.  Few  subjects  can  boast  such  varied  processes,  based 
upon  methods  so  distinct'  from  one  another  as  are  those  originated 
by  Cauchy,  by  Weierstrass,  and  by  Kiemann.  Each  of  these 
methods  is  sufficient  in  itself  to  provide  a  complete  development ; 
combined,  they  exhibit  an  unusual  wealth  of  ideas  and  furnish 
unsurpassed  resources  in  attacking  new  problems. 

It  is  difficult  to  keep  pace  with  the  rapid  growth  of  the 
literature  which  is  due  to  the  activity  of  mathematicians, 
especially  of  continental  mathematicians  :  and  there  is,  in  con- 
sequence, sufficient  reason  for  considering  that  some  marshalhng 
of  the  main  results  is  at  least  desirable  and  is,  perhaps,  necessary. 
Not  that  there  is  any  dearth  of  treatises  in  French  and  in 
German  :  but,  for  the  most  part,  they  either  expound  the  pro- 
cesses based  upon  some  single  method  or  they  deal  with  the 
discussion  of  some  particular  branch  of  the  theory. 

The  present  treatise  is  an  attempt  to  give  a  consecutive 
account  of  what  may  fairly  be  deemed  the  principal  branches  of 
the  whole  subject.  It  may  be  that  the  next  few  years  will  see 
additions  as  important  as  those  of  the  last  few  years  :  this  account 
would  then  be  insufficient  for  its  purpose,  notwithstanding  the 
breadth  of  range  over  which  it  may  seem  at  present  to  extend. 
My  hope  is  that  the  book,  so  far  as  it  goes,  may  assist  mathe- 
maticians, by  lessening  the  labour  of  acquiring  a  proper  knowledge 
of  the  subject,  and  by  indicating  the  main  lines  on  which  recent 
progress  has  been  achieved. 

No  apology  is  offered  for  the  size  of  the  book.  Indeed,  if 
there  were  to  be  an  apology,  it  would  rather  be  on  the  ground 
of  the  too  brief  treatment  of  some  portions  and  the  omissions 
of  others.     The  detail  in  the  exposition  of  the  elements  of  several 


VI 


PREFACE 


important  branches  has  prevented  a  completeness  of  treatment 
of  those  branches:  but  this  fuhiess  of  initial  explanations  is 
deliberate,  my  opinion  being  that  students  will  thereby  become 
better  qualified  to  read  the  great  classical  memoirs,  by  the  study 
of  which  effective  progress  can  best  be  made.  And  limitations  of 
space  have  compelled  me  to  exclude  some  branches  which  other- 
wise would  have  found  a  place.  Thus  the  theory  of  functions  of 
a  real  variable  is  left  undiscussed  :  happily,  the  treatises  of  Dini, 
Stolz,  Tannery,  and  Chrystal  are  sufficient  to  supply  the  omission. 
Again,  the  theory  of  functions  of  more  than  one  complex  variable 
receives  only  a  passing  mention  ;  but  in  this  case,  as  in  most 
cases,  where  the  consideration  is  brief,  references  are  given 
which  will  enable  the  student  to  follow  the  development  to 
such  extent  as  he  may  desire.  Limitation  in  one  other  direction 
has  been  imposed  :  the  treatise  aims  at  dealing  with  the  general 
theory  of  functions  and  it  does  not  profess  to  deal  with  special 
classes  of  functions.  I  have  not  hesitated  to  use  examples  of 
special  classes  :  but  they  are  used  merely  as  illustrations  of  the 
general  theory,  and  references  are  given  to  other  treatises  for 
the  detailed  exposition  of  their  properties. 

The  general  method  which  is  adopted  is  not  limited  so  that 
it  may  conform  to  any  single  one  of  the  three  principal  inde- 
pendent methods,  due  to  Cauchy,  to  Weierstrass  and  to  Riemann 
respectively :  where  it  has  been  convenient  to  do  so,  I  have 
combined  ideas  and  processes  derived  from  different  methods. 

The  book  may  be  considered  as  composed  of  five  parts. 

The  first  part,  consisting  of  Chapters  I — VII,  contains  the 
theory  of  uniform  functions  :  the  discussion  is  based  upon  power- 
series,  initially  connected  with  Cauchy's  theorems  in  integration, 
and  the  properties  established  are  chiefly  those  which  are  con- 
tained in  the  memoirs  of  Weierstrass  and  Mittag-Leffler. 

The  second  part,  consisting  of  Chapters  VIII — XIII,  contains 
the  theory  of  multiform  functions,  and  of  uniform  periodic 
functions  which  are  derived  through  the  inversion  of  integrals 
of  algebraic  functions.  The  method  adopted  in  this  part  is 
Cauchy's,  as  used  by  Briot  and  Bouquet  in  their  three  memoirs 
and  in  their  treatise  on  elliptic  functions  ;  it  is  the  method  that 


PREFACE  Vll 

has  been  followed  by  Hermite  and  others  to  obtain  the  properties 
of  various  kinds  of  periodic  functions.  A  chapter  has  been 
devoted  to  the  proof  of  Weierstrass's  results  relating  to  functions 
that  possess  an  addition-theorem. 

The  third  part,  consisting  of  Chapters  XIY — XVIII,  contains 
the  development  of  the  theory  of  functions  according  to  the 
method  initiated  by  Riemann  in  his  memoirs.  The  proof  which 
is  given  of  the  existence-theorem  is  substantially  due  to  Schwarz  ; 
in  the  rest  of  this  part  of  the  book,  I  have  derived  great  assist- 
ance from  Neumann's  treatise  on  Abelian  functions,  from  Fricke's 
treatise  on  Klein's  theory  of  modular  functions,  and  from  many 
memoirs  by  Klein. 

The  fourth  part,  consisting  of  Chapters  XIX  and  XX,  treats 
of  conformal  representation.  The  fundamental  theorem,  as  to  the 
possibility  of  the  conformal  representation  of  surfaces  upon  one 
another,  is  derived  from  the  existence-theorem  :  it  is  a  curious  fact 
that  the  actual  solution,  which  has  been  proved  to  exist  in  general, 
has  been  obtained  only  for  cases  in  which  there  is  distinct 
limitation. 

The  fifth  part,  consisting  of  Chapters  XXI  and  XXII,  contains 
an  introduction  to  the  theory  of  Fuchsian  or  automorphic  functions, 
based  upon  the  researches  of  Poincare  and  Klein  :  the  discussion  is 
restricted  to  the  elements  of  this  newly-developed  theory. 

The  arrangement  of  the  subject-matter,  as  indicated  in  this 
abstract  of  the  contents,  has  been  adopted  as  being  the  most 
convenient  for  the  continuous  exposition  of  the  theory.  But  the 
arrangement  does  not  provide  an  order  best  adapted  to  one  who  is 
reading  the  subject  for  the  first  time.  I  have  therefore  ventured 
to  prefix  to  the  Table  of  Contents  a  selection  of  Chapters  that 
will  probably  form  a  more  suitable  introduction  to  the  subject  for 
such  a  reader ;  the  remaining  Chapters  can  then  be  taken  in  an 
order  determined  by  the  branch  of  the  subject  which  he  wishes 
to  follow  out. 

In  the  course  of  the  preparation  of  this  book,  I  have  consulted 
many  treatises  and  memoirs.  References  to  them,  both  general 
and  particular,  are  freely  made  :  without  making  precise  reserva- 
tions as  to  independent  contributions  of  my  own,  I  wish  in  this 


Vlll  PREFACE 

place  to  make  a  comprehensive  acknowledgement  of  my  obligations 
to  such  works.  A  number  of  examples  occur  in  the  book  :  most  of 
them  are  extracted  from  memoirs,  which  do  not  lie  close  to  the 
direct  line  of  development  of  the  general  theory  but  contain 
results  that  provide  interesting  special  illustrations.  My  inten- 
tion has  been  to  give  the  author's  name  in  every  case  where  a 
result  has  been  extracted  from  a  memoir  :  any  omission  to  do  so 
is  due  to  inadvertence. 

Substantial  as  has  been  the  aid  provided  by  the  treatises  and 
memoirs  to  which  reference  has  just  been  made,  the  completion  of 
the  book  in  the  correction  of  the  proof-sheets  has  been  rendered 
easier  to  me  by  the  unstinted  and  untiring  help  rendered  by 
two  friends.  To  Mr  William  Burnside,  M.A./formerly  Fellow  of 
Pembroke  College,  Cambridge,  and  now  Professor  of  Mathematics 
at  the  Royal  Naval  College,  Greenwich,  I  am  under  a  deep  debt 
of  gratitude  :  he  has  used  his  great  knowledge  of  the  subject  in 
the  most  generous  manner,  making  suggestions  and  criticisms  that 
have  enabled  me  to  correct  errors  and  to  improve  the  book  in 
many  respects.  Mr  H,  M.  Taylor,  M.  A.,  Fellow  of  Trinity  College, 
Cambridge,  has  read  the  proofs  with  great  care  :  the  kind  assist- 
ance that  he  has  given  me  in  this  way  has  proved  of  substantial 
service  and  usefulness  in  correcting  the  sheets.  I  desire  to 
recognise  most  gratefully  my  sense  of  the  value  of  the  work  which 
these  gentlemen  have  done. 

It  is  but  just  on  my  part  to  state  that  the  willing  and  active 
co-operation  of  the  StafP  of  the  University  Press  during  the 
progress  of  printing  has  done  much  to  lighten  my  labour. 

It  is,  perhaps,  too  ambitious  to  hope  that,  on  ground  which 
is  relatively  new  to  English  mathematics,  there  will  be  freedom 
from  error  or  obscurity  and  that  the  mode  of  presentation  in  this 
treatise  will  command  general  approbation.  In  any  case,  my  aim 
has  been  to  produce  a  book  that  will  assist  mathematicians  in 
acquiring  a  knowledge  of  the  theory  of  functions  :  in  proportion 
as  it  may  prove  of  real  service  to  them,  will  be  my  reward. 

A.  R  FOUSYTH. 

Trinity  College,  Cambridge, 
25  February,  1893. 


PEEFACE  TO  THE  SECOND  EDITION. 

IN  issuing  the  second  edition  of  this  treatise,  I  desire  to  express 
my  grateful  sense  of  the  reception  which  has  already  been 
accorded  to  the  book.  When  it  was  first  published,  I  could  not 
but  fear  that,  if  from  no  other  reason  than  the  breadth  of  range 
which  it  covers,  it  would  contain  blemishes  in  the  way  of  inaccuracy 
and  obscurity.  During  the  preparation  of  the  second  edition,  I 
have  had  the  advantage  of  suggestions  and  criticisms  sent  to  me 
by  friends  and  correspondents,  to  whom  my  thanks  are  willingly 
returned  for  the  help  they  thus  have  afforded  me  ;  my  hope  is  that 
improvement  has  been  secured  in  several  respects.  The  principal 
changes  may  be  indicated  briefly. 

Some  moditications  have  been  made  in  the  portion  that  is 
devoted  to  the  theory  of  uniform  functions :  no  substantial 
additions  have  been  made  to  this  part  of  the  book,  but  new 
references  are  given  for  the  sake  of  readers  who  may  wish  to 
acquaint  themselves  with  the  most  recent  developments. 

The  exposition  of  Schwarz's  proof  of  the  existence  of  various 
classes  of  functions  upon  a  Niemann's  surface  has  been  considerably 
changed.  The  new  form  seems  to  me  to  be  free  from  some  of  the 
difficulties  to  which  exception  has  been  taken  from  time  to  time  : 
the  general  features  of  the  proof  have  been  retained. 

Several  sections  have  been  inserted  in  Chapter  XVIII,  which 
are  intended  to  serve  as  a  simple  introduction  to  the  theory  of 
birational  transformation  of  algebraic  equations  and  curves  and  of 
Riemann's  surfaces.  Moreover,  as  that  part  of  the  book  is  occupied 
with  integrals  of  algebraic  functions  and  with  Abelian  functions,  it 
seems  not  unnatural  that  a  proof  of  Abel's  Theorem  should  be 
given,  as  well  as  some  illustrations  :  this  has  been  effected  in  some 
supplementary  notes  appended  to  Chapter  XVIII.  With  minor 
exceptions,  these  additions  constitute  the  whole  of  the  new  matter 
relating  to  algebraic  functions  and  their  integrals. 


X  PREFACE  TO  THE  SECOND  EDITION 

The  chief  omission  from  the  contents  of  the  former  edition  is 
caused  by  the  transference,  to  the  second  volume  of  my  Theory  of 
Differential  Equations,  of  the  sections  that  discussed  the  properties 
of  certain  binomial  differential  equations  of  the  first  order.  The 
space  thus  placed  at  my  disposal  has  been  assigned  to  the  theory 
of  birational  tranvsformation  ;  and  I  have  been  enabled  to  keep  the 
numbering  of  the  paragraphs  the  same  as  in  the  former  edition 
with  only  very  few  exceptions. 

The  increased  size  of  the  book  has  prevented  me,  even  more 
definitely  than  before,  from  attempting  to  discuss  some  of  the 
subjects  left  undiscussed  in  the  first  edition.  The  volume  will 
probably  be  regarded  as  sufficiently  large  in  its  present  form  : 
I  hope  that  it  may  continue  to  be  found  a  useful  introduction  to 
one  of  the  most  important  subjects  in  modern  pure  mathematics. 

A.  R.  F. 


Trinity  College,  Cambridge, 
31  October,  1900. 


PEEFACE  TO  THE  THIED  EDITION. 

THE  differences  between  the  present  edition  and  the  second 
edition  are  not  substantial. 

The  general  plan  of  the  book  is  unaltered ;  and  no  change  has 
been  made  in  the  numbering  of  the  paragraphs.  Not  a  few 
detailed  changes  have  been  made  in  places  as,  for  instance,  in  the 
establishment  of  the  fundamental  functions  in  the  Weierstrass 
theory  of  elliptic  functions ;  but  some  chapters  remain  entirely 
unaltered. 

The  theory  of  conformal  representation  is  important  in  particular 
ranges  of  subjects  such  as  hydrodynamics  and  electrostatics  ;  so 
I  have  included  a  note  giving  some  applications  of  that  theory  to 
some  branches  of  mathematical  physics.  It  is  intended  only  as 
an  introduction  ;  but  it  may  suffice  to  shew  that  many  analytical 
results  are  common  to  these  selected  ranges,  though  they  are  ex- 
pressed in  the  various  vocabularies  appropriate  to  the  respective 
subjects. 

In  passing  from  the  first  edition  to  the  second,  I  omitted  certain 
sections  which  discussed  the  properties  of  certain  differential  equa- 
tions of  the  first  order.  These  sections  are  now  contained  in  the 
second  volume  of  my  Theory  of  Differential  Equations.  Owing 
to  their  importance  as  illustrations  of  the  theory  of  functions,  I 
have  included  a  note  stating  the  results. 

Here  and  there,  throughout  the  book,  some  further  examples 
have  been  added.  At  the  end  of  the  book,  I  have  given  a  set  of 
some  two  hundred  miscellaneous  examples,  which  have  been 
collected  from  Cambridge  examination  papers.  For  making  the 
collection,  I  am  indebted  to  Mr  C.  H.  Kebby,  B.Sc,  A.RC.S.,  a 
demonstrator  in  the  department  of  mathematics  and  mechanics  in 
the  Imperial  College  of  Science  and  Technology,  London. 

The  Staff  of  the  University  Press  have  shewn  to  me  the  same 
courteous  consideration  that  I  have  experienced  for  many  years  ; 

62 


Xll  PREFACE  TO  THE  THIRD  EDITION 

and  they  have  achieved  the  task  of  printing  the  volume  within  a 
brief  period  in  spite  of  their  grave  depletion  by  the  demands  of 
this  world-wide  war.  To  all  of  them,  who  have  been  concerned 
with  the  book,  I  tender  my  most  cordial  and"  appreciative  thanks. 

A.  R.  F. 


Imperial  College  op  Science  and  Technology, 
London,  S.W. 

11   October,  1917.       ' 


CONTENTS. 


The    following    course    is    recom mended,    in    the    order    specified,    to    those    who    are 

,  reading  the  subject  for  the  first  time  :    The  theory  of  uniform  functions,   Chapters 

I — V  ;   Gonformal  representation,  Chapter  XIX  ;   Multiform  functions  and  uniform 

periodic  fimctions,  Chapters  VIII — XI ;    Rieinann^s  surfaces,   and  Riemann)s  theory 

of  algebraic  functions  and  their  integrals,  Chapters  XIV — XVI,  XVIII. 


CHAPTER   I. 

GENERAL   INTRODUCTION. 

§§  PAGE 

1 — 3.      The  complex  variable  and  the  representation  of  its  variation  by  points 

in  a  plane      .............  1 

4.  Neumann's  representation  by  points  on  a  sphere 4 

5.  Propei'ties  of  functions  assumed  known     .......  6 

6,  7.      The  idea  of  complex  functionality  adopted,  with  the  conditions  necessary 

and  sufficient  to  ensure  functional  dependence  .....  6 

8.  Riemann's  definition  of  functionality  .......  8 

9.  A  functional    relation   between   two   complex   variables  establishes   the 

geometrical  property  of  conformal  representation  of  their  planes  .  10 
10,  11.     Relations  between  the  real  and  the  imaginary  parts  of  a  function  of  z; 

with  examples       ...........         12 

12,  13.     Definitions  and  illustrations  of  the  terms  monogenic,  uniform,  multiform, 

branch,  branch-point,  holomorphic,  zero,  pole,  meromorphic  ...         15 


CHAPTER   II. 

INTEGRATION    OF   UNIFORM   FUNCTIONS. 

14,  15.     Definition  of  an  integral  with  complex  variables;  inferences.    Definitions 

as  to  convergence  of  series .         .         .         .         .         ...         j         .         20 

16.      Proof  of  the   lemma    I  /(a    —  ^)  dxdy  =  \{pdx  +  qdy),   under  assigned 

conditions 24 


XIV  CONTENTS 


PAGE 


17,  18.     The  integral   \f{z)dz    round   any   simple   curve   is   zero,   when  f{z)  is 

holomorphic   within   the   curve ;    and    1    f{z)  dz    is    a  holomorphic 

J  a 

function  when  the  path  of  integration  lies  within  the  curve    .         .         27 
19.       The  path  of  integration  of  a  holomorphic   function   can   be  deformed 

without  changing  the  value  of  the  integral 30 

20—22.     The  integral  — A-L^dz,   round  a   curve   enclosing  a,   is    f{a)   when 
°        2TnJz-a 

f{z)  is  a  holomorphic  function  within  the  curve;    and  the  integral 

I  — L\2 —  dz  is 4-^^ .     Superior  limit  for  the  modulus  of 

27ril{z-a)^  +  ^  n\     da^^  ^ 

the  nth  derivative  of /(a)  in  terms  of  the  modulus  of  f{a)     .         .         31 

23.  The  path  of  integration  of  a  meromorphic  function  cannot  be  deformed 

across  a  pole  without  changing  the  value  of  the  integral         .         .         39 

24.  The  integral  of  any  function  (i)  round  a  very  small  circle,  (ii)  round  a 

very  large  circle,  (iii)  round  a  circle  which  encloses  all  its  infinities 

and  all  its  branch -points 40 

25.  Examples 43 

CHAPTER   III. 

EXPANSION   OF   FUNCTIONS   IN   SERIES    OF   POWERS. 

26  27.  Cauchy's  expansion  of  a  function  in  positive  powers  of  z-a;  with  re- 
marks and  inferences 50 

28 30.     Laurent's  expansion  of  a  function  in  positive  and  negative  powers  of 

z-a;  with  corollary 54 

31.  Application  of  Cauchy's  expansion  to  the  derivatives  of  a  function        .         59 

32.  Definition  of  an  ordinary/  point  of  a  function,    of  the  domain  of  an 

ordinary  point,  of  (a  jooZe)  or  an  accidental  singularity^,  and  of  an 
essential  singularity/.  Behaviour  of  a  uniform  function  at  and  near 
an  essential  singularity -  .         .         .         .         60 

33.  Weierstrass's  theorem  on  the  values  of  a  uniform  function  in  the  imme- 

diate vicinity  of  an  essential  singularity 64 

34    35.     Continuation   of  a  function   by    means   of  elements   over  its   region   of 

continuity      ............         66 

36.  Schwarz's  theorem  on  symmetric  continuation  across  the  axis  of  real 

quantities 70 

CHAPTER  IV. 

UNIFORM   FUNCTIONS,    PARTICULARLY   THOSE   WITHOUT   ESSENTIAL 

SINGULARITIES. 

37.  A  function,   constant  over   a  continuous  series   of  points,  is  constant 

everywhere  in  its  region  of  continuity 72 

38,  39.     The  multiplicity  of  a  zero,  which  is  an  ordinary  point,  is  finite;   and 

a  multiple  zero  of  a  function  is  a  zero  of  its  first  derivative  .         .         75 


CONTENTS 


XV 


§§ 
40.  ■ 

41,  42. 

43,  44. 
45. 

46. 
47. 

48. 


A  function,  that  is  not  a  constant,  must  have  infinite  values 

Form  of  a  function  near  an  accidental  singularity  . 

Poles  of  a  function  are  poles  of  its  derivatives 

A  function,  vsrhich  has  infinity  for  its  only   pole  and  has  no  essential 

singularity,  is  a  polynomial  ..... 

Polynomial  and  transcendental  functions  .... 
A  function,  all  the  singularities  of  which   are  accidental,  is  a  rational 

meromorphic  function  ....... 

Some  properties  of  polynomials  and  rational  functions  . 


PAGE 

77 
78 
80 

83 
84 

85 
87 


CHAPTER   V. 


TRANSCENDENTAL   INTEGRAL    FUNCTIONS. 


49,  50. 


51. 
52,  53. 

54. 

55—57. 

58. 

59,  60. 
61. 


Construction  of  a   transcendental  integral  function  with  assigned  ■  zeros 
ax,  a^i  ^3,  ...,  when  an  integer  s  can  be  found  such  that  2|a„|~' 
is  a  converging  series.     Definitions  as  to  convergence  of  products    .         90 
Weierstrass's  construction  of  a  function  with  any  assigned  zeros    .         .         95 

The  most  general  form  of  function  with  assigned  zeros  and  having  its 

single  essential  singularity  at  2=00      .         .         .         .         .         .         .         99 

Functions  with  the  singly-infinite  system  of  zeros  given  by  0=mco,  for 

integral  values  of  ?«      .         .         .         .         .         .         .         .         .         .101 

Weierstrass's  cr-function  with  the  doubly-infinite  system  of  zeros  given 

by  z  =  ma-{-m'a>',  for  integral  values  of  m  and  oi  m'   .         .  ,       .         .       104 

A  uniform  function  cannot  exist  with  a  triply-infinite  arithmetical  pro- 
gression of  zeros   ...........       108 

Class  (genre)  of  a  function 109 

Laguerre's  criterion  of  the  class  of  a  function;  with  examples       .         .       Ill 


CHAPTER   VI. 


FUNCTIONS   WITH    A   LIMITED   NUMBER   OF   ESSENTIAL   SINGULARITIES. 


62.  Indefiniteness  of  value  of  a  function  at  and  near  an  essential  singularity       115 

63.  A  function  is  of  the  form  O  y  — j)  +P{z-b)m  the  vicinity  of  an  essential 

singularity  at  b,  a  point  in  the  finite  part  of  the  plane  .  .  .117 
64,  65.     Expression  of  a  function  with  n  essential  singularities  as  a  sum  of  n 

functions,  each  with  only  one  essential  singularity  .  .  .  .120 
66,  67.     Product-expression  of  a  function  with  n  essential  singularities  and  no 

zeros  or  accidental  singularities  .         . 122 

68 — 71.     Product-expression  of  a  function  with  n  essential  singularities  and  with 

assigned  zeros  and  assigned  accidental  singularities ;  with  a  note  on 

the  region  of  continuity  of  such  a  function 126 


XVI  CONTENTS 


CHAPTER   VII. 


FUNCTIONS   WITH   UNLIMITED   ESSENTIAL   SINGULAEITIES, 
AND   EXPANSION   IN    SERIES   OF    FUNCTIONS. 


PAGE 


72.  Mittag-Leffler's   theorem    on   functions  with  unlimited   essential   singu- 

larities, distributed  over  the  whole  plane 134 

73.  Construction  of  subsidiary  functions,  to  be  terms  of  an  infinite  sum     .       135 
74_76.     Weierstrass's  proof  of  Mittag-Leffler's  theorem,  with  the  generalisation 

of  the  form  of  the  theorem 136 

77,  78.     Mittag-Leffler's   theorem   on  functions    with  unlimited  essential   singu- 
larities, distributed  over  a  finite  circle 140 

79.  Expression  of  a  given  function  in  Mittag-Leflfler's  form     ....       146 

80.  General  remarks  on  infinite  series,  v»rhether  of  powers  or  of  functions     .       150 

81.  A  series  of  powers,  in  a  rfegion  of  continuity,  represents  one  and  only 

one  function;   it  cannot  be  continued  beyond  a  natural  limit  .         .       152 

82.  Also  a  series  of  functions  :    but  its  region  of  continuity  may  consist  of 

distinct  parts         ...........       153 

83.  A  series  of  functions  does  not  necessarily  possess  a  derivative  at  points 

on  the  boundary  of  any  one  of  the  distinct  portions  of  its  region 

of  continuity  ...........       155 

84.  A  series  of  functions  may  represent  different  functions  in  distinct  parts 

of  its  region  of  continuity ;  Tannery's  series 161 

85.  Construction  of  a  function  which  represents  different  assigned  functions 

in  distinct  assigned  parts  of  the  plane       ......       163 

86.  Functions  with  a  line  of  essential  singularity 164 

87.  Functions  with  an  area  of  essential  singularity  or  lacunary  spaces ;  with 

examples        ............       166 

88.  Arrangement  of  singularities  of  functions  into  classes  and  species    .         .       175 


CHAPTER   VIII. 

MULTIFORM   FUNCTIONS. 

89.  Branch-points  and  branches  of  functions .178 

90.  Branches  obtained  by  continuation  :   path  of  variation  of  independent 

variable  between  two   points  can  be  deformed  without  affecting  a 
branch  of  a  function  if  it  be  not  made  to  cross  a  branch-point       .       179 

91.  If  the  path  be  deformed  across  a  branch-point  which  affects  the  branch, 

then  the  branch  is  changed  .         .         .         .         .         .         .         .184 

92.  The  interchange  of  branches  for  circuits  round  a  branch-point .  is  cyclical       185 

93.  Analytical  form  of  a  function  near  a  branch-point 186 

94.  Branch-points  of  a  function  defined  by  an  algebraic  equation  in  their 

relation  to  the  branches  :    definition  of  algebraic  function         .         .       190 

95.  Infinities  of  an  algebraic  function      .         .         .         .         .         .         .         .192 

96.  Determination  of  the  branch-points  of  an  algebraic  function,  and  of  the 

cyclical  systems  of  the  branches  of  the  function        .  .         .197 


CONTENTS  XVll 

? 

§§  PAGE 

97.  The  analytic  character  of  a  function  defined  by  an  algebraic  equation  203 

98.  Special  case,  when  the  branch-points  are  simple  :    their  number   .         .  208 

99.  A  function,  with  n  branches  and  a  limited  number  of  branch-points  and 

singularities,  is  a  root  of  an  algebraic  equation  of  degree  n    .         .  210 


CHAPTER   IX. 

PERIODS   OF    DEFINITE   INTEGRALS,   AND   PERIODIC    FUNCTIONS 
IN   GENERAL. 

100.  Conditions  under  which  the  path  of  variation   of  the   integral   of  a 

multiform  function  can  be  deformed  without  changing  the  value 

of  the  integral 214 

101.  Integral   of  a   multiform  function   round   a  small   curve   enclosing  a 

branch-point 217 

102.  Indefinite  integrals  of  uniform  functions  with  accidental  singularities 
dz 


fdz       f  dz 


218 


103.  Hermite's  method  of  obtaining  the  multiplicity  in  value  of  an  integral 

sections  in  the  plane,  made  to  avoid  the  multiplicity    .         .         .       219 

104.  Examples  of  indefinite  integrals  of  multiform  functions  ;   \wdz  round 

any  loop,  the  general  value  of  J(l  -  z^)~^dz,  of  J{(1  -  z^)  (I  -  kH'^)]'^dz, 

and  of  J{(z  -  e^  {z  —  e^)  (z  -  es)}~^  dz 224 

105.  Graphical    representation    of    simply-periodic   and   of  dcmbly-periodic 

functions 235 

106.  The  ratio  of  the  periods  of  a  uniform  doubly-periodic  function  is  not 

real 238 

107,  108.     Triply-periodic  uniform  functions  of  a  single  variable  do  not  exist     .       239 

109.  Construction  of  a  fundamental  parallelogram  for  a  uniform  doubly- 

periodic  function       . 243 

110.  An  integral,  with  more  periods  than  two,  can  be  made  to  assume  any 

value  by  a  modification  of  the  path  of  integration  between  the 
limits 246 


CHAPTER   X. 

UNIFORM   SIMPLY-PERIODIC   AND   DOUBLY-PERIODIC    FUNCTIONS, 

2Trzi 

111.  Simply -periodic  functions,  and  the  transformation  Z=e  '»    .         .         .       250 

112.  Fourier's  series  and  simply -periodic  functions 252 

113,  114.     Properties  of  simply -periodic  functions  without  essential  singularities 

in  the  finite  part  of  the  plane 253 

115.  Uniform   doubly-periodic  functions,  without  essential  singularities  in 

the  finite  part  of  the  plane      . 257 

116.  Properties  of  uniform  doubly-periodic  functions    .....       268 


XVlll 


CONTENTS 


117.         The  zeros  and  the  singularities  of  the  derivative  of  a  doubly-periodic 

function  of  the  second  order .271 

118,  119.     Relations  between  homoperiodic  functions     ......       273 

Note  on  differential  equations  of  the  first  order  having  uniform  integrals       283 


CHAPTER   XL 


DOUBLY-PERIODIC    FUNCTIONS   OF   THE   SECOND   ORDER. 


120,  121.     Formation   of  an   uneven   function  with  two  distinct  irreducible  in- 
finities ;   its  addition-theorem  .         .         .         .         .         .         .         •  286 

122,  123.     Properties  of  Weierstrass's  o--function    .......  291 

124.         Introduction  of  ^{z)  and  of  ^(2) 295 

125,  126.     Periodicity  of  the  function  ^{z\  with  a  single  irreducible  infinity  of 

degree  two  ;  the  differential  equation  satisfied  by  the  function  ^  {£)  296 

127.  Pseudo-periodicity  of  {"(2)        .         .         .         .         .         .         .         .         .  300 

128.  Construction  of  a  doubly-periodic  function  in  terms  of  ^{z)  and  its 

derivatives          ...........  301 

129.  On  the  relation  7;&)'  — ?;'a)= +^7ri      ........  302 

130.  Pseudo-periodicity  of  a- (2) 304 

131.  Construction  of  a  doubly-periodic  function  as  a  product  of  tr-functions  ; 

with  examples  ...........  305 

132.  On   derivatives   of  periodic   functions   with   regard  to  the   invariants 

gi  and  g^ 309 

133 — 135.    Formation  of  an  even  function  of  either  class      .....  312 


CHAPTER   XIL 


PSEUDO-PERIODIC   FUNCTIONS. 


136.         Three  kinds  of  pseudo-periodic  functions,  with  the  characteristic  equa 
tions  ............ 

137,  138.     Hermite's  and  Mittag-Leffler's  expression  for  doubly-periodic  functions 
of  the  second  kind    ....... 

139.         The  zeros  and  the  infinities  of  a  secondary  function    . 
140,  141.     Solution  of  Lame's  differential  equation 

142.  The  zeros  and  the  infinities  of  a  tertiary  function 

143.  Product-expression  for  a  tertiary  function 
144 — 146.    Two  classes  of  tertiary  functions  ;  Appell's  expressions  for  a  function 

of  each  class  as  a  sum  of  elements 

147.  Expansion  in  trigonometrical  series         .... 

148.  Examples  of  other  classes  of  pseudo-periodic  functions 


320 

322 
327 
328 
333 
334 

335 

340 
342 


CONTENTS  XIX 

CHAPTER   XIII. 

FUNCTIONS   POSSESSING   AN   ALGEBRAICAL   ADDITION-THEOREM. 

§§  PAGE 

149.  Definition  of  an  algebraical  addition -theorem         .....       344 

150.  A   function   defined    by   an    algebraical    equation,   the   coefficients   of 

which  are  algebraical  functions,  or  simply-periodic  functions,  or 
doubly-periodic  functions,  has  an  algebraical  addition-theorem     .       344 

151 — 154.  A  function  possessing  an  algebraical  addition-theorem  is  either 
algebraic,  simply-periodic  or  doubly-periodic,  having  in  each  in- 
stance only  a  finite  number  of  values  for  an  argument         .         .       347 

155,  156.  A  function  with  an  algebraical  addition-theorem  can  be  defined  by  a 
differential  equation  of  the  first  order,  into  which  the  independent 
variable  does  not  explicitly  enter 356 

CHAPTER  XIV. 

CONNECTIVITY   OF   SURFACES. 

157 — 159.  Definitions  of  connection^  simple  connection,  multiple  connection,  cross- 
cut, loop-cut       ...........  359 

160.  Relations  between  cross-cuts  and  connectivity 362 

161.  Relations  between  loop-cuts  and  connectivity        .....  367 

162.  Effect  of  a  sht 368 

163,  164.     Relations  between  boundaries  and  connectivity     .....  369 

165.  Lhuilier's    theorem    on    the    division    of    a    connected    surface    into 

curvilinear  polygons 372 

166.  Definitions  of  circuit,  reducible,  irreducible,  simple,  multiple,  compound, 

recondleable        ...........  374 

167,  168.     Properties  of  a   complete  system  of  irreducible  simple  circuits  on  a 

surface,  in  its  relation  to  the  connectivity     .         .         .         .         .  375 

169.  Deformation  of  surfaces 379 

170.  Conditions  of  equivalence  for  representation  of  the  variable        .         .  380 


CHAPTER   XV. 

riemann's  surfaces. 

171.  Character  and  general  description  of  a  Riemann's  surface  .         .         .  382 

172.  Riemann's  surface  associated  with  an  algebraic  equation     .         .         .  384 

173.  Sheets  of  the  surface  are  connected  along  lines,  called  branch-lines    .  384 

174.  Properties  of  branch-lines 386 

175,  176.     Formation  of  system  of  branch-lines  for  a  surface  ;   with   examples  .  387 

177.         Spherical  form  of  Riemann's  surface 393 


XX  CONTENTS 


§§ 


PAGE 


178.  The  connectivity  of  a  Riemann's  surface 393 

179.  Irreducible   circuits  :    examples   of  resolution   of   Riemann's   surfaces 

into  surfaces  that  are  simply  connected  .... 

180,  181.     General  resolution  of  a  Riemann's  surface 

182.         A  Riemann's  ?i-sheeted  surface  when  all  the  branch -points  are  simple 

183,  184.     On  loops,  and  their  deformation 

185.         Simple  cycles  of  Clebsch  and  Gordan 

186 189.    Canonical  form  of  Riemann's  surface  when  all  the  branch-points  are 

simple,  deduced  from  theorems  of  Lliroth  and  Clebsch 

190.  Deformation  of  the  surface 

191.  Remark  on  rational  transformations 


397 
400 
403 

404 
407 

408 
412 
415 


CHAPTER   XVI. 

ALGEBRAIC   FUNCTIONS   AND   THEIR   INTEGRALS. 

192.         Two  subjects  of  investigation  .'        .         .         .         .         .         .         .416 

193,  194.     Determination  of  the  most  general  uniform  function  of  position  on  a 

Riemann's  surface     .         .         .         .         .         .         .         •         •         .417 

195.         Preliminary  lemmas  in  integration  on  a  Riemann's  surface  .         .       422 

196,  197.     Moduli  of  periodicity  for  cross-cuts  in  the  resolved  siu-face  .         .       423 

198.  The  number  of  linearly  independent  moduli  of  periodicity  is  equal  to 

the  number  of  cross-cuts,  which  are  necessary  for  the  resolution 

of  the  surface  into  one  that  is  simply  connected  ....       427 

199.  Periodic  functions  on  a  Riemann's  surface  ;  with  examples .         .         .       428 

200.  Integral    of    the    most   general    uniform    function    of    position   on    a 

Riemann's  surface 436 

201.  Integrals,  everywhere  finite  on  the  surface,  connected  with  the  equa- 

tion ^v^-S{z)  =  0 438 

202 — 204.  Infinities  of  integrals  on  the  surface  connected  with  the  algebraic 
equation  /{w,  s)  =  0,  when  the  equation  is  geometrically  interpret- 
able  as  the  equation  of  a  (generalised)  curve  of  the  nth  order     .       438 

205,  206.  Integrals  of  the  Jirst  kind  connected  with  f{w,  z)  =  0,  being  functions 
that  are  everywhere  finite  :  the  number  of  such  integrals,  linearly 
independent  of  one  another  :  they  are  multiform  functions  .         .       444 

207,  208.     Integrals   of  the   second  kind  connected  with  f{w,  z)  =  0,  being  func- 
tions  that  have  only  algebraic   infinities  ;   elementary   integral  of 
the  second  kind         ..........       446 

209.         Integrals  of  the  third  kind  connected  with  /(w,  s)=0,  being  functions 

that  have  logarithmic  infinities 450 

210,  211.     An  integral  of  the  third  kind  cannot  have  less  than  two  logarithmic 

infinities  ;   elementary  integral  of  the  third  kind     .         .         .         .451 


CONTENTS 


XXI 


CHAPTER   XVII. 


SCHWAEZS    PROOF   OF   THE   EXISTENCE-THEOREM. 
§§  PAGE 

212,  213.     Existence  of  functions  on  a  Riemann's  surface ;    initial  limitation  of 

the  problem  to  the  real  parts  u  of  the  functions  ....       455 

Conditions  to  which  u,  the  potential  function,  is  subject     .         .         .       457 

Methods  of  proof :    summary  of  Schwarz's  investigation       .         .         .       458 

The   potential-function  u  is  uniquely  determined   for  a  circle   by  the 

general  conditions  and  by  the  assignment  of  finite  boundary  values  .       460 

Also  for  any  plane  area,  on  which  the  area  of  a  circle  can  be  con- 
formally  represented  ...         .         .         .         .         .         .         .477 

Also  for  any  plane  area  which  can  be  obtained  by  a  topological  com- 
bination of  areas,  having  a  common  part  and  each  conformally 
representable  on  the  area  of  a  circle       ......       480 

Also  for   any  area   on   a   Riemann's  surface  in  which  a  branch-point 

occurs  ;  and  for  any  simply  connected  surface       ....       485 
224 — 227.    Real  functions  exist   on   a   Riemann's  surface,  everywhere  finite,  and 
having  arbitrarily  assigned  real  moduli  of  periodicity,  whether  the 
surface  has  a  boundary  or  not         .......       487 

And  the  number  of  the  linearly  independent  real  functions  thus  ob- 
tained is  2jo       ..........         .       495 

Real  fimctions  exist  with  assigned  infinities  on  the  surface  and 
assigned  real  moduli  of  periodicity.  Classes  of  functions  of  the 
complex  variable  proved  to  exist  on  the  Riemann's  surface  .         .       495 


214. 

215. 

216—220 

221. 

222. 


223. 


228. 


229. 


CHAPTER   XVIII. 


APPLICATIONS   OF   THE   EXISTENCE-THEOREM. 

230.         Three  special  kinds  of  functions  on  a  Riemann's  surface     . 
231 — 233.    Relations  between  moduli  of  functions  of  the  first  kind  and  those  of 
functions  of  the  second  kind 

234.  The  number  of  linearly  independent  functions  of  the  first  kind  on  a 

Riemann's  surface  of  connectivity  2p  +  l  is  p 

235.  Normal  functions  of  the  first  kind ;  properties  of  their  moduli   . 

236.  Normal  elementary  functions  of  the  second  kind :  their  moduli  . 
237,  238.     Normal  elementary  functions  of  the  third  kind :  their  moduli :  inter 

change  of  arguments  and  parametric  points  .... 

239.  The  inversion--^voh\em.  for  functions  of  the  first  kind   . 

240.  Algebraic  functions  on  a  Riemann's  surface  without  infinities  at  the 

branch-points  but  only  at  isolated  ordinary  points  on  the  surface 
Riemann-Roch's  theorem  :    the   smallest  number   of  singularities 
that  such  functions  may  possess     ...... 

241.  A  class  of  algebraic  functions  infinite  only  at  branch-points 

242.  The  Brill-Nother  law  of  reciprocity        ...... 

243.  Fundamental  equation  associated  with  an  assigned  Riemann's  surface 


498 

500 

504 
506 
509 

511 

515 


519 
524 
526 

528 


XXll  CONTENTS 


PAGE 


531 

537 

542 
548 
554 
562 
566 
567 


244.  Appell's  factoHal  functions  on  a  RiemaQn's  surface :    their  multipliers 

at  the  cross-cuts ;  expression  for  a  factorial  function  with  assigned 
zeros  and  assigned  infinities ;  relations  between  zeros  and  infinities 
of  a  factorial  function       ....... 

245.  Birational  transformation  of  equations  and  Riemann's  surfaces 

246.  Conservation  of  genus  under  birational  transformation  :  moduli 

247.  Equations  of  genus  0 

248.  Equations  of  genus  1 -    . 

249.  Equations  of  genus  2        ......         . 

250.  Equations  of  genus  jo  (^  3)       . 

251.  Normal  equivalents  of  equations  for  birational  transformation 

252.  Birational  transformation  of  any  algebraic  plane  curve  into  an  algebraic 

plane  curve  having  no  singularities  except  simple  nodes       .         .       569 

SUPPLEMENTARY  NOTES :  ABEL's  THEOEEM. 

I.  Proof  of  Abel's  Theorem  in  general;  with  examples    .         .         .       579 

II.  Application  of  Abel's  Theorem  to  the  normal  elementary  integrals 

of  three  kinds  on  a  Riemann's  surface     ......       590 

III.  Proof  that  the  sum  of  any  number  of  integrals  is  expressible  as 

a  sum  of  p  integrals  together  with  an  additive  function         .       598 


CHAPTER   XIX. 

CONFORMAL   REPRESENTATION  :     INTRODUCTORY. 

253.  A  relation  between  complex  variables  is  the  most  general  relation  that 

secures  conformal  similarity  between  two  surfaces  ....       602 

254.  One  of  the  surfaces  for  conformal  representation  may,  without  loss  of 

generality,  be  taken  to  be  a  plane  .......       606 

255,  256.     Application  to  surfaces  of  revolution;   in  particular,  to  a  sphere,  so 

as  to  obtain  maps     ..........       607 

257.  Some  examples  of  conformal   representation  of  plane  areas,   in  par- 

ticular, of  areas  that  can  be  conformally  represented  on  the  area 

of  a  circle  ............       614 

258.  Linear   homographic    transformations    (or    substitutions)    of    the    form 

w= -, ;   their  fundamental  properties 625 

cz  +  d  ^     ^ 

259.  Parabolic,  elliptic,  hyperbolic,  and  loxodromic  substitutions  .         ."        .       631 

260.  An  elliptic  substitution  is  either  periodic  or  infinitesimal:  substitutions 

of  the  other  classes  are  neither  periodic  nor  infinitesimal      .         .       635 

261.  A  linear  substitution  can  be  regarded  geometrically  as  the  result  of 

an  even  number  of  successive  inversions  of  a  point  with  regard 

to  circles   ............       637 

NOTE:    SOME    APPLICATIONS   OF    CONFORMAL   REPRESENTATION 
TO    MATHEMATICAL   PHYSICS. 

I.  Applications  to  hydrodynamics       . 639 

II.  Applications  to  electrostatics  ........       646 

III.  Applications  to  conduction  of  heat 649 


CONTENTS  xxiii 

CHAPTER  XX. 

CONFORMAL  REPRESENTATION :  GENERAL  THEORY. 

§§  PAGE 

262.         Riemann's  theorem  on  the  conformal  representation   of  a  given   area 

upon  the  area  of  a  circle  with  unique  correspondence  .  .  .  653 
263,  264.     Proof    of    Riemann's  theorem :    how    far    the    functional  equation   is 

algebraically  determinate  .........       654 

265,  266.     The    method    of    Beltrami   and   Cayley   for    the   construction   of  the 

functional  equation  for  an  analytical  curve  .....  658 
267,  268.     Conformal  representation  of  a   convex   rectilinear  polygon   upon  the 

half-plane  of  the  variable  ........       665 

269.  The  triangle,  and  the  quadrilateral,  conformally  represented         .         .       671 

270.  A  convex  curve,  as  a  limiting  case  of  a  polygon  .....       678 
271,  272.     Conformal  representation  of  a  convex  figure,  bounded  by  circular  arcs : 

the    functional    relation    is    connected   with   a   linear   differential 

equation  of  the  second  order 679 

273.         Conformal  representation  of  a  crescent  .......       684 

274 — 276.    Conformal  representation  of  a  triangle,  bounded  by  circular  arcs        .       685 
277 — 279.    Relation  between  the  triangle,  bounded  by  circles,  and  the  stereographic 

projection  of  regular  solids  inscribed  in  a  sphere  ....       694 

280.  On  families  of  plane  algebraic  curves,  determined  as  potential-curves 

by  a  single  potential-parameter  u  :  the  forms  of  functional  relation 
z=(f)(u  +  iv),  which  give  rise  to  such  curves    .....       706 
Supplementary  note ;  surfaces  of  constant  negative  curvature,  and  their 

representation  on  a  plane,  in  connection  with  §  275      .         .         .       712 

CHAPTER   XXI. 

GROUPS   OF   LINEAR   SUBSTITUTIONS. 

281.  The  algebra  of  group-symbols 715 

282.  Groups,   which   are   considered,  are   discontinuous   and    have   a   finite 

number  of  fundamental  substitutions       .         .         .         .         .         .717 

283,  284.     Anharmonic  group :   gsoup  for  the  modular -functions,  and  division  of 

the  plane  of  the  variable  to  represent  the  group  .  .  .  .719 
285,  286.     Fuchsian  groups :    division  of  plane  into  convex  curvilinear  polygons : 

polygon  of  reference  ..........       724 

287.         Cycles  of  angular  points  in  a  curvilinear  polygon         ....       729 

288,  289.     Character  of  the  division  of  the  plane:    example 732 

290.  Fuchsian  groups  which  conserve  a  fundamental  circle  ....       736 

291.  Essential  singularities  of  a  group,  and    of  the  automorphic  functions 

determined  by  the  group 739 

292,  293.     Families  of  groups:    and  their  genus 740 

294.  Kleinian  groups :    the  generalised  equations  connecting  two  points  in 

space  ............       743 

295.  Division  of  plane  and  division  of  space,  in  connection  with  Kleinian 

groups 747 

296.  Example  of  improperly  discontinuous  group 749 


XXIV  CONTENTS 


CHAPTER  XXII.      - 

AUTOMORPHIC    FUNCTIONS. 
§§  PAGE 

297.  Definition  of  automorphic  functions         .......       753 

298.  Examples  of  functions,  automorphic  for  finite  discrete  groups  of  sub- 

stitutions  ............       754 

^99.         Cayley's  analytical  relation  between  stereographic  projections  of  posi- 
tions of  a  point  on  a  rotated  sphere       ......       754 

300.         Polyhedral  groups ;   in  particular,  the  dihedral  group,  and  the  tetra- 

hedral  group 757 

301,  302.     The  tetrahedral  functions,  and  the  dihedral  functions  ....       762 

303.  Special    illustrations    of    infinite    discrete    groups,    from    the    elliptic 

modular-functions 767 

304.  Division  of  the  plane,  and  properties  of  the  fundamental  polygon  of 

reference,  for  any  infinite  discrete  group  that  conserves  a  funda- 
mental circle      ...........       771 

305,  306.     Construction  of   Thetafuchsian  functions,    pseudo-automorphic  for  an 

infinite  group  of  substitutions  .         .         .         .         .         .         .775 

307.  Relations  between  the  number  of  irreducible  zeros  and  the  number 

of  irreducible  poles  of  a  pseudo-automorphic  function,  constructed 

with  a  rational  meromorphic  function  as  element  ....       779 

308.  Construction  of  automorphic  functions  .......       784 

309.  The  number  of  irreducible  points,  for  which  an  automorphic  function 

acquires  an  assigned  value,  is  independent  of  the  value        .         .       786 

310.  Algebraic    relations    between    functions,    automorphic   for    a    group  : 

application  of  Riemann's  theory  of  functions  ....       788 

311.  Connection    between    automorphic    functions    and    linear    differential 

equations;  with  illustrations  from  elliptic  modular-functions         .       789 


Miscellaneous  examples 794 

Glossary  of  technical  terms 829 

Index 833 


CHAPTER   T. 

Genekal  Introduction. 

1.  Algebraical  operations  are  either  direct  or  inverse.  Withoiit 
entering  into  a  general  discussion  of  the  nature  of  rational,  irrational,  and 
imaginary  quantities,  it  will  be  sufficient  to  point  out  that  direct  algebraical 
operations  on  numbers  that  are  positive  and  integral  lead  to  numbers  of  the 
same  character ;  and  that  inverse  algebraical  operations  on  numbers  that  are 
positive  and  integral  lead  to  numbers,  which  may  be  negative  or  fractional 
or  irrational,  or  to  numbers  which  may  not  even  fall  within  the  class  of  real 
quantities.  The  simplest  case  of  occurrence  of  a  quantity,  which  is  not 
real,  is  that  which  arises  when  the  square  root  of  a  negative  quantity  is 
required. 

Combinations  of  the  various  kinds  of  quantities  that  may  occur  are  of 
the  form  x  +  iy,  where  x  and  y  are  real,  and  i,  the  non-real  element  of  the 
quantity,  denotes  the  square  root  of  —  1.  It  is  found  that,  when  quantities 
of  this  character  are  subjected  to  algebraical  operations,  they  always  lead 
to  quantities  of  the  same  formal  character ;  and  it  is  therefore  inferred  that 
the  most  general  form  of  algebraical  quantity  is  ^  +  iy. 

Such  a  quantity  x  4-  iy,  for  brevity  denoted  by  z,  is  usually  called  a 
complex  variable  * ;  it  therefore  appears  that  the  complex  variable  is  the 
most  general  form  of  algebraical  quantity  which  obeys  the  fundamental  laws 
of  ordinary  algebra. 

2.  The  most  general  complex  variable  is  that,  in  which  the  constituents 
X  and  y  are  independent  of  one  another  and  (being  real  quantities)  are 
separately  capable  of  assuming  all  values  from  —  oo  to  +  cc  ;  thus  a  doubly- 
infinite  variation  is  possible  for  the  variable.  In  the  case  of  a  real  variable, 
it  is  convenient  to  use  the  customary  geometrical  representation  by  measure- 
ment of  distance  along  a  straight  line;  so  also  in  the  case  of  a  complex 
variable,  it  is  convenient  to  associate  a  geometrical  representation  with 
the   algebraical   expression ;   and    this   is  the  well-known  representation  of 

*  The  conjugate  complex,  viz.  x  -  iy,  is  frequently  denoted  by  z^. 
F.  F.  1 


2  GEOMETRICAL   REPRESENTATION    OF  [2. 

the  variable  x  +  iy  by  means  of  a  point  with  coordinates  x  and  y  referred 
to  rectangular  axes*.  The  complete  variation  of  the  complex  variable  z 
is  represented  by  the  aggregate  of  all  possible  positions  of  the  associated 
point,  which  is  often  called  the  point  z;  the  special  case  of  real  variables 
being  evidently  included  in  it  because,  when  y  =  0,  the  aggregate  of 
possible  points  is  the  line  which  is  the  range  of  geometrical  variation  of  the 
real  variable. 

The  variation  of  z  is  said  to  be  continuous  when  the  variations  of  x  and  y 
are  continuous.  Continuous  variation  of  z  between  two  given  values  will 
thus  be  represented  by  continuous  variation  in  the  position  of  the  point  z, 
that  is,  by  a  continuous  curve  (not  necessarily  of  continuous  curvature) 
between  the  points  corresponding  to  the  two  values.  But  since  an  infinite 
number  of  curves  can  be  drawn  between  two  points  in  a  plane,  continuity  of 
line  is  not  sufficient  to  specify  the  variation  of  the  complex  variable ;  and, 
in  order  to  indicate  any  special  mode  of  variation,  it  is  necessary  to  assign, 
either  explicitly  or  implicitly,  some  determinate  law  connecting  the  variations 
of  a;  and  y  or,  what  is  the  same  thing,  some  determinate  law  connecting 
a?  and  y.  The  analytical  expression  of  this  law  is  the  equation  of  the  curve 
which  represents  the  aggregate  of  values  assumed  by  the  variable  between 
the  two  given  values. 

'  In  such  a  case  the  variable  is  often  said  to  describe  the  part  of  the  curve 
between  the  two  points.  In  particular,  if  the  variable  resume  its  initial 
value,  the  representative  point  must  return  to  its  initial  position ;  and  then 
the  variable  is  said  to  describe  the  whole  curve  f. 

When  a  given  closed  curve  is  continuously  described  by  the  variable, 
there  are  two  directions  in  which  the  description  can  take  place.  From 
the  analogy  of  the  description  of  a  straight  line  by  a  point  representing  a 
real  variable,  one  of  these  directions  is  considered  as  positive  and  the  other 
as  negative.  The  usual  convention  under  which  one  of  the  directions  is 
selected  as  the  positive  direction  depends  upon  the  conception  that  the  curve 

*  This  method  of  geometrical  representation  of  imaghiary  quantities,  ordinarily  assigned  to 
Gauss,  was  originally  developed  by  Argand  who,  in  18U6,  published  his  Essai  sur  une  maniere 
de  representer  les  quantites  imaginaires  dans  les  constructions  geometriques.  This  tract  was 
republished  in  1874  as  a  second  edition  (Gauthier-Villars) ;  an  interesting  preface  is  added 
to  it  by  Hoiiel,  who  gives  an  account  of  the  earlier  history  of  the  publications  associated  with 
the  theory. 

.Other  references  to  the  historical  development  are  given  in  Chrystal's  Text-hook  of  Algebra, 
vol.  i,  pp.  248,  249 ;  in  Holzmiiller's  Einfuhrung  in  die  Theorie  d'r  isogonalen  Verwandschaftcn 
und  der  confo)-men  Abbildungen,  verbunden  mit  Amoendungen  auf  mathematische  Physik,  pp.  1 — 10, 
21^-23;  in  Schlomilch's  Compendium  der  hoheren  Analysis,  vol.  ii,  p.  38  (note) ;  and  in  Casorati, 
^Teorica  delle  funzioni  di  variabili  complesse,  only  one  volume  of  which  was  published.  In  this 
connection,  an  article  by  Cayley  {Quart.  Journ.  of  Math.,  vol.  xxii,  pp.  270 — 308;  Coll.  Math. 
Papers,  t.  xii,  pp.  459 — 489)  may  be  consulted  with  advantage. 

t  In  these  elementary  explanations,  it  is  unnecessary  to  enter  into  any  discussion  of 
the  effects  caused  by  the  occurrence  of  singularities  in  the  curve. 


2.] 


THE    COMPLEX    VARIABLE 


Fig.  1. 


is  the  boundary,  partial  or  complete,  of  some  area ;  under  it,  that  direction  is 
taken  to  be  positive  which  is  such  that  the  bounded  area  lies  to  the  left  of 
the  direction  of  description.  It  is  easy  to  see  that  the  same  direction  is  taken 
to  be  positive  under  an  equivalent  convention 
which  makes  it  related  to  the  normal  drawn 
outwards  from  the  bounded  area  in  the  same 
way  as  the  positive  direction  of  the  axis  of  y 
is  usually  related  to  the  positive  direction  of 
the  axis  of  x  in  plane  coordinate  geometry. 

Thus  in  the  figure  (fig.  1),  the  positive 
direction  of  description  of  the  outer  curve 
for  the  area  included  by  it  is  DEF\  the 
positive  direction  of  description  of  the  inner 
curve  for  the  area  without  it  (say,  the  area 
excluded  by  it)  is  ACB:  and  for  the  area 
between  the  curves  the  positive  direction  of  description  of  the  boundary, 
which  consists  of  two  parts,  is  DEF,  ACB. 

3.  Since  the  position  of  a  point  in  a  plane  can  be  determined  by  means 
of  polar  coordinates,  it  is  convenient  in  the  discussion  of  complex  variables 
to  introduce  two  quantities  corresponding  to  polar  coordinates. 

In  the  case  of  the  variable  z,  one  of  these  quantities  is  {x-  +  y^)^,  the 
positive  sign  being  always  associated  with  it;  it  is  called  the  modulus^ 
(sometimes  the  absolute  value)  of  the  variable  and  it  is  denoted,  sometimes 
by  mod. ^'j  sometimes  by  \z\.  The  modulus  of  a  complex  variable  is  quite 
definite,  and  it  has  only  one  value. 

The  other  is  6,  the  angular  coordinate  of  the  point  z;  it  is  called  the 
argument  (and.  less  frequently,  the  amplitude)  of  the  variable.  It  is 
measured  in  the  trigonometrically  positive  sense,  and  is  determined  by 
the  equations 


x=  \z\  cos 


Sim 


so  that  z  =  \z\  e^\  The  actual  value  depends  upon  the  way  in  which  the 
variable  has  acquired  its  value  ;  when  variation 
of  the  argument  is  considered,  its  initial  value 
is  usually  taken  to  lie  between  0  and  27r  or,  less 
frequently,  between  —  ir  and  +  it.  The  argu- 
ment of  a  variable  is  not  definite ;  it  has  an 
unlimited  number  of  values  differing  from  one 
another  by  integer  multiples  of  27r.  This 
characteristic  property  will  be  found  to  be  of 
essential  importance. 


Fig. 


*  Der  absolute  Betrag  is  often  used  by  German  writers. 


1—2 


4 


GREAT   VALUES   OF 


[3. 


As  z  varies  in  position,  the  values  of  |  ^  |  and  Q  vary.  When  z  has  com- 
pleted a  positive  description  of  a  closed  curve,  the  modulus  of  z  returns  to 
the  initial  value  whether  the  origin  be  without,  within,  or  on,  the  curve. 
The  argument  of  z  resumes  its  initial  value,  if  the  origin  0'  (fig.  2)  be  with- 
out the  curve ;  but,  if  the  origin  0  be  within  the  curve,  the  value  of  the 
argument  is  increased  by  27r  when  z  returns  to  its  initial  position. 

If  the  origin  be  on  the  curve,  the  argument  of  z  undergoes  an  abrupt 
change  by  tt  as  ^^  passes  through  the  origin ;  and  the  change  is  an  increase 
or  a  decrease  according  as  the  variable  approaches  its  limiting  position  on  the 
curve  from  without  or  from  within.  No  choice  need  be  made  between  these 
alternatives ;  for  care  is  always  exercised  to  choose  curves  which  do  not 
introduce  this  element  of  doubt. 

Later  on,  it  will  appear  that,  for  the  discussion  of  particular  types  of 
functions  of  z,  a  knowledge  of  the  actual  value  of  z  or  the  actual  position 
of  z  is  not  sufficient ;  account  has  to  be  taken  of  the  fact  that  the  argument 
of  z  is  not  uniquely  determinate. 

4.  Representation  on  a  plane  is  obviously  more  effective  for  points  at  a 
finite  distance  from  the  origin  than  for  points  at  a  very  great  distance. 

One  method  of  meeting  the  difficulty  of  representing  great  values  is  to 
introduce  a  new  variable  z'  given  by  z'z-=\:  the  part  of  the  new  plane  for 
z  which  lies  quite  near  the  origin  corresponds  to  the  part  of  the  old  plane 
for  z  which  is  very  distant.  The  two  planes  combined  give  a  complete 
representation  of  variation  of  the  complex  variable. 

Another  method,  in  many  ways  more  advantageous,  is  as  follows.  Draw 
a  sphere  of  unit  diameter,  touching  the  ^-plane  at  the  origin  0  (fig.  3)  on 
the  under  side:  join  a  point  z  in  the  plane  to  0',  the  other  extremity  of 


Pig-  3. 


the  diameter  through  0,  by  a  straight  line  cutting  the  sphere  in  Z. 
Then  Z  is  a  unique  representative  of  z,  that  is,  a  single  point  on  the 
sphere  corresponds  to  a  single  point  on  the  plane :  and  therefore  the  variable 


4.]  THE    COMPLEX   VARIABLE  5 

can  be  represented  on  the  surface  of  the  sphere.  With  this  mode  of 
representation,  0'  evidently  corresponds  to  an  infinite  value  of  z ;  and  points 
at  a  very  great  distance  in  the  ^•-plane  are  represented  by  points  in  the 
immediate  vicinity  of  0'  on  the  sphere.  The  sphere  thus  has  the  advantage 
of  putting  in  evidence  a  part  of  the  surface  on  which  the  variations  of 
great  values  of  2  can  be  traced*,  and  of  exhibiting  the  uniqueness  of 
2  =  X  as  a  value  of  the  variable,  a  fact  that  is  obscured  in  the  represent- 
ation on  a  plane. 

The  former  method  of  representation  can  be  deduced  by  means  of  the 
sphere.  At  0'  draw  a  plane  touching  the  sphere :  and  let  the  straight  line 
OZ  cut  this  plane  in  /.  Then  z'  is  a  point  uniquely  determined  by  Z 
and  therefore  uniquely  determined  by  z.  In  this  new  /-plane  take  axes 
parallel  to  the  axes  in  the  2^-plane. 

The  points  z  and  z'  move  in  the  same  direction  in  space  round  00' 
as  an  axis.  If  we  make  the  upper  side  of  the  ^•-plane  correspond  to  the 
lower  side  of  the  /-plane,  and  take  the  usual  positive  directions  in  the 
planes,  being  the  positive  trigonometrical  directions  for  a  spectator  looking 
at  the  surface  of  the  plane  in  which  the  description  takes  place,  we  have 
these  directions  indicated  by  the  arrows  at  0  and  at  0'  respectively,  so 
that  the  senses  of  positive  rotations  in  the  two  planes  are  opposite  in 
space.  Now  it  is  evident  from  the  geometry  that  Oz  and  O'z'  are 
parallel;  hence,  if  0  be  the  argument  of  the  point  z  and  6'  that  of  the 
point  /,  so  that  6  is  the  angle  from  Ox  to  Oz  and  6'  the  angle  from  OV 
to  O'z',  we  have 

6  +  6'=  27r. 

Further,  by  similar  triangles,        ^yy  =  -y^, , 

that  is,  Oz.O'z'=00'-  =  l. 

Now,  if  z  and  z'  be  the  variables,  we  have 

z=Oz.  e^',     z  =  O'z  .  e^'\ 
so  that  5/=0^.0V.e'»+«''^ 

which  is  the  former  relation. 

The  /-plane  can  therefore  be  taken  as  the  lower  side  of  a  plane  touching 

the  sphere  at  0'  when  the  ^r-plane  is  the  upper  side  of  a  plane  touching 

it  at  0.     The  part  of  the  ^-plane  at  a  very  great  distance  is  represented  on 

the  sphere  by  the  part  in  the  immediate  vicinity  of  0'.     Conversely,  this 

part  of  the  sphere  is  represented  on  the  very  distant  part  of  the  ^r-plane. 

Consequently,  the  portion  of  the  sphere  in  the  immediate  vicinity  of  0'  is  a 

space  wherein  the  variations  of  infinitely  great  values  of  z  can  be  traced, 

*  This  sphere  is  sometimes  called  Neumann's  sphere;  it  is  used  by  him  for  the  representation 
of  the  complex  variable  throughout  his  treatise  Vorlesungen  ilher  Riemann's  Theorie  der  Abel'schen 
Integrale  (Leipzig,  Teubner,  2nd  edition,  1884). 


6  CONDITIONS   OF  [4. 

But  it  need  hardly  be  pointed  out  that  any  special  method  of  represent- 
ation of  the  variable  is  not  essential  to  the  development  of  the  theory  of 
functions;  and,  in  particular,  the  foregoing  representation  of  the  variable, 
when  it  has  very 'great  values,  merely  provides  a  convenient  method  of 
dealing  with  quantities  that  tend  to  become  infinite  in  magnitude. 

5.  The  simplest  propositions  relating  to  complex  variables  will  be 
assumed  known.  Among  these  are,  the  geometrical  interpretation  of  opera- 
tions such  as  addition,  multiplication,  root-extraction ;  some  of  the  relations 
of  complex  variables  occurring  as  roots  of  algebraical  equations  with  real 
coefficients ;  the  elementary  properties  of  functions  of  complex  variables 
which  are  polynomial,  or  exponential,  or  circular,  functions ;  and  simple 
tests  of  convergence  of  infinite  series  and  of  infinite  products*. 

6.  All  ordinary  operations  effected  on  a  complex  variable  lead,  as 
already  remarked,  to  other  complex  variables;  and  any  definite  quantity, 
thus  obtained  by  operations  on  z,  is  necessarily  a  function  of  z. 

But  if  a  complex  variable  w  be  given  as  a  complex  function  of  x 
and  y  without  any  indication  of  its  source,  the  question  as  to  whether 
w  is  or  is  not  a  function  of  z  requires  a  consideration  of  the  general  idea 
of  functionality. 

It  is  convenient  to  postulate  w  -\-  iv  as  a  form  of  the  complex  variable  w, 
where  u  and  v  are  real.  Since  iv  is  initially  unrestricted  in  variation,  we 
may  so  far  regard  the  quantities  u  and  v  as  independent  and  therefore  as 
any  functions  of  x  and  y,  the  elements  involved  in  z.  But  more  explicit 
expressions  for  these  functions  are  neither  assigned  nor  supposed. 

The  earliest  occurrence  of  the  idea  of  functionality  is  in  connection  with 
functions  of  real  variables ;  and  then  it  is  coextensive  with  the  idea  of 
dependence.  Thus,  if  the  value  of  X  depends  on  that  of  x  and  on  no  other 
variable  magnitude,  it  is  customary  to  regard  X  as  a  function  of  x ;  and 
there  is  usually  an  implication  that  X  is  derived  from  x  by  some  series  of 
operations  j". 

A  detailed  knowledge  of  z  determines  x  and  y  uniquely ;  hence  the  values 
of  u  and  v  may  be  considered  as  known  and  therefore  also  w.  Thus  the 
value  of  IV  is  dependent  on  that  of  z,  and  is  independent  of  the  values 

*  These  and  other  introductory  parts  of  the  subject  are  discussed  in  Chrystal's  Text-book  of 
Algebra,  Hobson's  Treatise  on  Plane  Trigonometry,  Bromwich's  Theory  of  infinite  series,  and 
Hardy's  Course  of  pure  mathematics. 

They  are  also  discussed  at  some  length  in  the  translation,  by  G.  L.  Cathcart,  of  Harnack's 
Elements  of  the  differential  and  integral  calculus  (Williams  and  Norgate,  1891),  the  second  and 
the  fourth  books  of  which  contain  developments  that  should  be  consulted  in  special  relation 
with  the  first  few  chapters  of  the  present  treatise. 

These  books,  together  with  Neumann's  treatise  cited  in  the  note  on  p.  5,  will  hereafter  be  cited 
by  the  names  of  their  respective  authors. 

t  It  is  not  important  for  the  present  purpose  to  keep  in  view  such  mathematical  expressions 
as  have  intelligible  meanings  only  when  the  independent  variable  is  confined  within  limits. 


6.]  FUNCTIONAL    DEPENDENCE  7 

of  variables  unconnected  with  z ;  therefore,  with  the  foregoing  view  of 
functionality,  w  is  a  function  oi  z. 

It  is,  however,  equally  consistent  with  that  view  to  regard  -?/;  as  a  complex 
function  of  the  two  independent  elements  from  which  z  is  constituted ;  and 
we  are  then  led  merely  to  the  consideration  of  functions  of  two  real 
independent  variables  with  (possibly)  imaginary  coefficients. 

Both  of  these  aspects  of  the  dependence  of  lu  on  z  require  that  z  be 
regarded  as  a  composite  quantity  involving  two  independent  elements  which 
can  be  considered  separately.  Our  purpose,  however,  is  to  regard  z  as  the 
most  general  form  of  algebraical  variable  and  therefore  as  an  irresoluble 
entity ;  so  that,  as  this  preliminary  requirement  in  regard  to  z  is  unsatisfied^ 
neither  of  the  aspects  can  be  adopted. 

7.  Suppose  that  w  is  regarded  as  a  function  of  z  in  the  sense  that  it 
can  be  constructed  by  definite  operations  on  z  regarded  as  an  irresoluble 
magnitude,  the  quantities  u  and  v  arising  subsequently  to  these  operations 
by  the  separation  of  the  real  and  the  imaginary  parts  when  z  is  replaced  by 
X  +  iy.  It  is  thereby  assumed  that  one  series  of  operations  is  sufficient  for 
the  simultaneous  construction  of  u  and  v,  instead  of  one  series  for  u  and 
another  series  for  v  as  in  the  general  case  of  a  complex  function  in  §  6. 
If  this  assumption  be  justified  by  the  same  forms  resulting  from  the  two 
different  methods  of  construction,  it  follows  that  the  two  series  of  opera- 
tions, which  lead  in  the  general  case  to  ?(.  and  to  v,  must  be  equivalent  to 
the  single  series  and  must  therefore  be  connected  by  conditions ;  that  is, 
u  and  V  as  functions  of  x  and  y  must  have  their  functional  forms  related. 

We  thus  take 

li  +  iv  =  IV  =f{z)  =f{co  +  iy) 

without  any  specification  of  the  form  of/".     When  this  postulated  equation 

is  valid,  we  have 

dti)      dw  dz  _  V /.  _dw 

dx      dz  dx     -      '       dz  ' 

dw  _  dw  dz  _  . ,,      __  .  dw 

dy      dz  dy  d,z' 

,  ^,        „                                      dw      I  div      dw  ^ 

and  thereiore  ^r- =  -  ^:;- = -y-   (1), 

dx      t  dy      dz  ^  ^ 

equations  from  which  the  functional  form  has  disappeared.     Inserting  the 

value  of  IV,  we  have 

.  9  ,        .  ,        d   ,        .  . 
dx  dy 

whence,  after  equating  real  and  imaginary  parts, 

dv  _  du       du  _  dv 

dx~  dy'     dx      dy ^ 

These  are  necessary  relations  between  the  functional  forms  of  u  and  v. 


8  riemann's  [7. 

These  relations  are  easily  seen  to  be  sufficient  to  ensure  the  required 
functionality.     For,  on  taking  w  =  u+  iv,  the  equations  (2)  at  once  lead  to 

dw  _ldw 

doc      i   dy ' 

.    ,  ■     ,  dw      .dw     ,. 

that  IS,  to  7^  +  ^  t;—  =  0, 

dw         01/ 

a  linear  partial  differential  equation  of  the  first  order.     To  obtain  the  most 

general  solution,  we  form  a  subsidiary  system 

dx     dy     dw 

T  ""  T  ^  "0"  • 

It  possesses  the  integrals  tv,  x  +  iy ;  then  from  the  known  theory  of  such 
equations  we  infer  that  every  quantity  w  satisfying  the  equation  can  be 
expressed  as  a  function  of  a;  +  iy,  that  is,  of  z.  The  conditions  (2)  are  thus 
proved  to  be  sufficient,  as  well  as  necessary. 

8.  The  preceding  determination  of  the  necessary  and  sufficient  conditions 
of  functional  dependence  is  based  upon  the  existence  of  a  functional  form : 
and  yet  that  form  is  not  essential,  for,  as  already  remarked,  it  disappears 
from  the  equations  of  condition.  Now  the  postulation  of  such  a  form  is 
equivalent  to  an  assumption  that  the  function  can  be  numerically  calculated 
for  each  particular  value  of  the  independent  variable,  though  the  immediate 
expression  of  the  assumption  has  disappeared  in  the  present  case.  Experience 
of  functions  of  real  variables  shews  that  it  is  often  more  convenient  to  use 
their  properties  than  to  possess  their  numerical  values.  This  experience  is 
confirmed  by  what  has  preceded.  The  essential  conditions  of  functional 
dependence  are  the  equations  (1),  and  they  express  a  property  of  the  function  w, 

viz.,  that  the  value  of  the  ratio  -7—  is  the  same  as  that  of  ^r-  ,  or,  in  other 

dz  ox 

Avords,  it  is  independent  of  the  manner  in  which  dz  ultimately  vanishes  by 
the  approach  of  the  point  z  ^  dz  to  coincidence  with  the  point  z.  We  are 
thus  led  to  an  entirely  different  definition  of  functionality,  viz. : — 

A  cotwplex  quantity  w  is  a  function  of  another  complex  quantity  z,  ivhen 

dw 
they  change  together  in  such  a  manner  tliat  the  value  of  -^  is  independent  of 

the  value  of  the  differential  element  dz. 

This  is  Riemann's  definition*  ;  we  proceed  to  consider  its  significance. 
We  have 

dw     du  +  idv 
dz      dx  +  idy 

_  (du      .dv\       dx  fdu      .dv\       dy 

\dx        dxj  dx  +  idy      \dy        dyj  dx+idy' 

*  Ges.  Werke,  p.  5  ;  a  modified  definition  is  adojDted  by  him,  ib.,  p.  81. 


8.]  DEFINITION   OF   A   FUNCTION 

Let  <^  be  the  argument  of  dz ;  then 

dec  cos  <f) 


and  therefore 


dx  +  idy      cos  ^  +  i  sin  ^ 

idy 
dx  +  idy 


c?w     1  f9?t      .  3'y      .8w      3w)      ,     „a- (9^      •9^'      -^^      9^1 


rf^r      ^  [9a;        'bx       9j/      9.y]      ^         |9.r        "bx        dy     dy\ 

Since  -^  is  to  be  independent  of  the  value  of  the  differential  element  dz, 

it  must  be  independent  of  <^  which  is  the  argument  of  dz ;  hence  the  coefficient 
of  e~2*''  in  the  preceding  expression  must  vanish,  which  can  happen  only  if 

du  _dv         dv          du  ,^. 

dx     dy '      dx         dy 

These  are  necessary  conditions ;  the}^  are  evidently  also  sufficient  to  make 
-T-  independent  of  the  value  of  dz  and  therefore,  by  the  definition,  to  secure 
that  tu  is  a  function  of  2'. 

By  means  of  the  conditions  (2),  we  have 

div  _du  .    .dv  _  dw 
dz      dx        dx      dx' 

1     1  dw  .du     dv      1  dw 

and  also  3-=-*5-+^=^^' 

dz  dy     dy      t  oy 

agreeing  with  the  former  equations  (1).  They  are  immediately  derivable  from 
the  present  definition  by  noticing  that  dx  and  idy  are  possible  forms  of  dz. 

It  should  be  remarked  that  equations  (2)  are  the  conditions  necessary 
and  sufficient  to  ensure  that  each  of  the  expressions 

adx  —  vdy   and    vdx  -\-  udy 

is  a  perfect  differential — a  result  of  great  importance  in  many  investigations 
ill  the  region  of  mathematical  physics.  Within  that  region,  the  quantities 
u  and  V  are  frequently  called  conjugate  functions.  Sometimes  they  are 
called  harmonic  functions ;  but  the  latter  term  usually  has  a  wider  signi- 
ficance associated  with  classes  of.  functions  that  satisfy  the  equation  of  the 
potential  in  ordinary  three-dimensional  space. 

When  the  conditions  (2)  are  expressed,  as  is  sometimes  convenient,  in 
terms  of  derivatives  with  regard  to  the  modulus  of  z,  say  r,  and  the 
argument  of  z,  say  6,  they  take  the  new  forms 

du  _1  dv        dv 1  du  ., 

dr^rdd'      di^~~rd9 ^^' 


10  CONFORM  AL  [8. 

We  have  so  far  assumed  that  the  function  has  a  differential  coefficient — 
an  assumption  justified  in  the  case  of  functions  which  ordinarily  occur.  But 
functions  do  occur  which  have  different  values  in  different  regions  of  the 

2^-plane,  and  there  is  then  a  difficulty  in  regard  to  the  quantity  -^  at  the 

boundaries  of  such  regions  ;  and  functions  do  occur  which,  though  themselves 
definite  in  value  in  a  given  region,  do  not  possess  a  differential  coefficient  at 
all  points  in  that  region.  The  consideration  of  such  functions  is  not  of 
substantial  importance  at  present :    it  belongs  to  another  part  of  our  subject. 

It  must  not  be  inferred  that,  because  -j-  is  independent  of  the  direction 

in  which  dz  vanishes  when  lu  is  a  function  of  z,  therefore  -j-  has  only  one 

value.  The  number  of  its  values  is  dependent  on  the  number  of  values  of  w; 
no  one  of  its  values  is  dependent  on  dz. 

A  quantity,  defined  as  a  function  by  Riemann  on  the  basis  of  this 
property,  is  sometimes*  called  an  analytic  function;  but  it  seems  pre- 
ferable to  reserve  the  term  analytic  in  order  that  it  may  be  associated 
hereafter  (§  34)  with  an  additional  quality  of  the  functions. 

9.  In  the  same  way  as  the  complex  variable  z  is  represented  upon 
a  plane,  which  is  often  called  the  ^-plane,  so  the  complex  variable  w  is 
also  represented  upon  a  plane,  which  is  often  called  the  w-plane.  The 
two  variables  can  obviously  be  represented  upon  different  parts  of  the 
2^- plane.  The  relations  of  the  two  planes  to  one  another,  or  of  the 
different  parts  of  the  same  plane,  w^hen  there  is  a  functional  connection 
between  z  and  w,  will  be  the  subject  of  later  investigations;  one  important 
property  will,  however,  be  established  at  once. 

Let  P  and  p  be  two  points  in  different  planes,  or  in  different  parts  of 

the  same  plane,  representing  w  and  z  respectively :  and  suppose  that  P  and 

p  are  at  a  finite  distance  from  the  points  (if  any)  which  cause  discontinuity 

in   the  functional  connection  between  the  two  variables.     Let  q  and  r  be 

any  two  other  points,  z -{-dz  and  z  +  hz,  in   the  immediate   vicinity  of  p ; 

and  let  Q  and  R  be  the  corresponding  points,  %u  +  dw  and  iv  +  hw,  in  the 

immediate  vicinity  of  P.     Then 

,        dw  .-,        ^        dw  - 
dw  =  -i-  dz,     mv  =  -r-  oz, 
dz  dz 

dw 
the  value  of  -^  being  the  same  for  both  equations,  because,  as  w  is  a  function 

of  z,  that  quantity  is  independent  of  the  differential  element  of  z.     Hence 

hw  _  hz 
dw     dz' 

on  the  ground  that  ~r-  is  neither  zero  nor  infinite  at  z,  which  is  assumed  not 

*  Harnack,  §  84. 


9.]  REPRESENTATION   OF    PLANES  11 

to  be  a  point  of  discontinuity  in  the  functional  connection.  Expressing  all 
the  differential  elements  in  terms  of  their  moduli  and  arguments,  let 

dz  =  ae^\       diu  =  T/e"^'", 

hz  =  o-'e^'^,      8w  =  7]'e'^'^, 

and  let  these  values  be  substituted  in  the  foregoing  relation ;  then 

7)'  _  a' 
Tj       a  ' 

Hence  the  triangles  QPR  and  qpr  are  similar  to  one  another,  though 
not  necessarily  similarly  situated.  Moreover,  the  directions  originally  chosen 
for  pq  and  pr  are  quite  arbitrary.  Thus  it  appears  that  afunctional  connection 
between  two  complex  variables  establishes  the  similarity  of  the  corresponding 
infinitesimal  elements  of  those  parts  of  two  planes  which  are  in  the  immediate 
vicinity  of  the  points  representing  the  two  variables. 

The  magnification  of  the  w-plane  relative  to  the  ^-plane  at  the  corre- 
sponding points  P  and  p  is   the  ratio  of  two  corresponding  infinitesimal 

lengths,  say  of  QP  and  qp.     This  is  the  modulus  of  ^  ;  if  it  be  denoted  by 

m,  we  have 

dw  '^      fdu\-      /dvV      fduY      f^vV 

"'-=  dz  =u)+y  ==y +y 

du  dv      du  dv 
doc  dy     dy  dx  ' 

Evidently  the  quantity  m,  in  general,  depends  on  the  variables  and 
therefore  it  changes  fi-om  one  point  to  another;  hence  a  functional  relation 
between  w  and  z  does  not,  in  general,  establish  similarity  of  finite  parts  of 
the  two  planes  corresponding  to  one  another  through  the  relation. 

It  is  easy  to  prove  that  w=  az  +  b,  where  a  and  b  are  constants,  is  the 
only  relation  which  establishes  similarity  of  finite  parts ;  and  that,  with  this 
relation,  a  must  be  a  real  constant  in  order  that  the  similar  parts  may  be 
similarly  situated. 

If  u  +  iv  =  tu  =  (j)  (z),  the  curves  u  =  constant  and  v  =  constant  cut  at 
right  angles ;  a  special  case  of  the  proposition  that,  if  ^{x  +  iy)  =  u  +  ve^'^, 
where  X,  is  a  real  constant  and  «,  v  are  real,  then  u  =  constant  and  v  —  constant 
cut  at  an  angle  \. 

The  process,  which  establishes  the  infinitesimal  similarity  of  two  planes 
by  means  of  a  functional  relation  between  the  variables  of  the  planes,  may  be 
called  the  conformal  representation  of  one  plane  on  another*, 

*  By  Gauss  {Ges.  Werke,  t.  iv,  p.  262)  it  was  styled  conforme  Abbildung,  the  name 
universally  adopted  by  German  mathematicians.  The  French  title  is  representation  conforme ; 
and,  in  England,  Cayley  has  used  orthomorphosis  and  orthomorphic  transformation. 


12  CONDITIONS    OF    FUNCTIONAL   DEPENDENCE  [9. 

The  discussion  of  detailed  questions  connected  with  the  conformal  representation  is 
deferred  until  the  later  part  of  the  treatise,  principally  in  order  to  group  all  such 
investigations  together;  but  the  first  of  the  two  chapters,  devoted  to  it,  need  not  be 
deferred  so  late,  and  an  immediate  reading  of  some  portion  of  Chapter  XIX.  will  tend 
to  simplify  many  of  the  explanations  relative  to  functional  relations  as  they  occur  in 
the  early  chapters  of  this  treatise. 

10.     The    analytical    conditions    of    functionality,    under   either   of   the 
adopted  definitions,  are  the  equations  (2).    From  them  it  at  once  follows  that 

d^u      dhi  _ 

dx-     dy'^ 
so  that  neither  the  real  nor  the  imaginary  part  of  a  complex  function  can  be 
arbitrarily  assumed. 

If  either  part  be  given,  the  other  can  be  deduced.     For  example,  let  u  be 
given ;  then  we  have 

cii)  — —-  dx  + -;:-  ay 

dx  dy    ^ 

du  ^         du  -, 
=  —7:-  dx  +  ^  dy, 
dy  ox 

and  therefore,  except  as  to  an  additive  constant,  the  value  of  v  is 

In  particular,  when  u  is  an  integral  function,  it  can  be  resolved  into  the 
sum  of  hoinogeneous  parts 

Ui  +  Uo  +  Us+  ...  ', 

and   then,    again    except   as    to    an   additive    constant,  v    can    similarly  be 
expressed  as  a  sum  of  homogeneous  parts 

V1  +  V2  +  V3+  .... 
It  is  easy  to  prove  that 

by  means  of  which  the  value  of  v  can  be  obtained. 

The   case,   when    u   is   homogeneous   of    zero   dimensions,   presents   no 
difficulty ;    for  then  we  have 

u  —  h  +  aO, 

V  =  c  —  a  log  r, 
where  a,  b,  c  are  constants. 

Similarly  for  other  special  cases ;   and,  in  the  most  general  case,  only 
a  quadrature  is  necessary. 


10.] 


EXAMPLE    OF   RIEMANN  S   DEFINITION 


13 


The  tests  of  functional  dependence  of  one  complex  variable  on  another  are 
of  effective  importance  in  the  case  when  the  supposed  dependent  variable 
arises  in  the  form  u  +  iv,  where  it  and  v  are  real ;  the  tests  are,  of  course, 
superfluous  when  to  is  explicitlj^  given  as  a  function  of  z.  When  iv  does 
arise  in  the  form  u  +  iv  and  satisfies  the  conditions  of  functionality,  perhaps 
the  simplest  method  (other  than  by  inspection)  of  obtaining  the  explicit 
expression  in  terms  of  z  is  to  substitute  z  —  iy  for  x  in  u  +  iv ;  the  simplified 
result  must  be  a  function  of  z  alone. 

11.  Conversely,  when  w  is  explicitly  given  as  a  function  of  z  and  it 
is  divided  into  its  real  and  its  imaginary  parts,  these  parts  individually 
satisfy  the  foregoing  conditions  attaching  to  w,  and  v.  Thus  log  r,  where  r 
is  the  distance  of  a  point  z  from  a  point  a,  is  the  real  part  of  log {z  —  a); 
it  therefore  satisfies  the  equation 

d~it      d'-ic 

da;-      dy- 

Again,  </>,  the  angular  coordinate  of  z  relative  to  the  same  point  a,  is 
the  real  part  of  —i]og(z  —  a)  and  satisfies  the  same  equation :  the  more 
usual  form  of  0  being  tan~^  {{y  —  yQ)l{x  -  x^)],  where  a  =  x^-\-iy^.  Again,  if 
a  point  z  be  distant  r  from  a  and  r  from  6,  then  log(?Y?"'),  being  the  real 
part  of  log  \{z  -  a)\{z-  6)],  is  a  solution  of  the  same  equation. 

The  following  example,  the  result  of  which   will  be  useful  subsequently*    uses  the 
property  that  the  value  of  the  derivative  is  independent  of  the  differential  element. 


Consider  a  function 

where  c'  is  the  inverse  of 

Then 

I 
?<  =  los;  I 


?i  +  ?'2;  =  w  =  loo; ,. 

with  regard  to  a  circle,  centre  the  origin  0  and  radius  R. 

y 


z  —  c 
z  —  d 


so    the    curves,    u  =  constant. 


(fig.  4)  Oc  =  i\  xOc  =  a,  so  that  c 

then  if 

\z-c  I  _'i\ 
\  z  —  c'  \      R 


are    circles.       Let 
R^     ■ 


Fig.  4. 


the  values  of  X  for  points  in  the  interior  of  the  circle  of  radius  R  vary  from  zero,  when 
the  circle  ?{  =  constant  is  the  point  c,  to  unity,  when  the  circle  ■«  =  constant  is  the  circle  of 
radius  R.  Let  the  point  ^(  =  ^e"*)  be  the  centre  of  the  circle  determined  by  a  value  of  X, 
and  let  its  radius  be  p  (  =  ^i/iV).     Then  since 


cM  _r         cN 
7M~R        7N' 


we  have 


r-irp-6 
—  +p-6 


=  *=^ 


e^-p- 


R^ 


-U-p 


*  In  §  217,  in  connection  with  the  investigations  of  Schwarz,  by  whom  the  result  is  stated, 
Ges.  Werke,  t.  ii,  p.  183.  ' 


14  EXAMPLES  [11. 

Whence  Xig(/^^-r^)       ^^^M^-X^) 

Now  if  dn  be  an  element  of  the  normal  drawn  inwards  at  z  to  the  circle  A^zM,  we  have 
dz=dx-\-idy—  —  dn .  cos  ■<^  —idn  .  sin  i\r 
=  —e^'^dn, 

where  •^{  =  zKx')  is  the  argument  of  z  relative  to  the  centre  of  the  circle.     Hence,  since 

dw        I  1 

dz      z  —  c      z  —  c'^ 

du      .dv      dw      (   \  1    \    ^i^ 

we  have  ^~  +  *^~  =  -7-=    > )^    • 

dn       dn     dn      \z  —  g      z  —  c  J 

But  z  =  6e'''+pe'^\ 

so  that  z-c=  ^"^^  ( -ffe*^*' - Xre»0, 

and  2  -  c'  =  -  ^^^,  (Xre-^*  -  i?e"^) ; 

r  It'^  —  r^X" 

,        ,  du      .  dv      B?     7-2X2  ^i  {Z.  1  _  ?L  1  1 

and  therefore       -  +  ^-  =  -^-^e    \R^^^^i_j^ai     X^,#_x^,«/- 

Hence,  equating  the  real  parts,  it  follows  that 

du_ {m-r^-X^f 

dn~     XR{R^-r^){R^-2ErXcos{f-a)  +  X^r^'}' 

the  differential  element  dn  being  drawn  inwards  from  the  circumference  of  the  circle. 

The  application  of  this  method  is  evidently  effective  when  the  curves  «= constant, 
arising  from  a  functional  expression  of  w  in  terms  of  z,  are  a  family  of  non-intersecting 
algebraical  curves. 

Kv.  1.     Prove  that,  if  Zi  and  zo  denote  two  complex  variables, 

1^1  +  22  !^  i%l  +l^2l»        !%-%|^|2ll~i22|- 

Ex.  2.     Find  the  values  of  u  and  v  when  w  is  defined  as  a  function  of  z  in  the  following 

cases : — 

(i)     z  =  {w  +  if; 

(ii)     z  =  {l  +  GOSw)e^'^; 

,...,      1  -(1-2)4 

(m)     ^^ ^^  =e"'  w2  logi^,. 

l  +  {l-z)i 
In  each  case,  trace  the  curves  2i  =  a,  v  =  c,  regarded  as  loci  in  the  plane  of  x,  y. 

Ex.  3.     Shew  that  x~  -y^  —  2ixy  is  not  a  function  of  z;  and  that 
x'^  —  Zo:;y'^  +  i{Zx'^y-y^)  +  ax 
is  a  function  of  z  only  when  a  =  0. 

Ex.  4.     Shew  that  a  possible  value  of  u  is 

[x  -y){x'^  +  Axy  -\-y'^); 
and  determine  the  associated  value  of  w  in  terms  of  z. 

Determine  also  the  value  of  w  in  teyms  of  z  when  the  preceding  expression  is  the  value 
of  u  —  v. 

Ex.  5.     Find  the  value  of  v,  and  of  w  in  terms  of  z,  when 

sin  a; 

u  —  — :r— . 

cosh  y  —  cos  X 


11-]  DEFINITIONS  15 

Ex.  6.     Prove  that,  when  x  and  y  are  regarded  as  functions  of  n  and  v  (with  the 
foregoing  notation),  the  relations 

9^_3^        Zx  _     dy 

B%      32_^_  32y      d^y 

8^2  +  a^-O,      9^2  +  9^  =  0, 

are  satisfied. 

^A'.  7.     Shew  that,  if  J  and  B  are  any  two  fixed  points  in  a  plane,  if  P  is  any  variable 
point  {x,  y),  and  if  6  denotes  the  angle  APB,  then 

dx^      dy^ 

(Jonstruct  the  function  of  z,  =x  +  iy,  of  which  6  is  the  real  part,  and  also  the  function 
of  3  of  which  id  is  the  imaginary  part. 

Ex.  8.     Given  X,  a  function  of  x  and  y ;  shew  that  cf){X)  can  be  the  real  part  of 
a  function  of  z  if  the  quantity 

^dx^      dy^J       \\dxj        \dyj  J 
is  expressible  in  terms  of  X  alone. 

Verify  that  the  condition  is  satisfied  when  \  =  x+{x^+y^)i ;  and  obtain  the  function 
of  z  which  has  0  (X)  for  its  real  part. 

12.  As  the  tests  which  are  sufficient  and  necessaiy  to  ensure  that  a 
complex  quantity  is  a  function  of  z  have  been  given,  we  shall  assume  that 
all  complex  quantities  dealt  with  are  functions  of  the  complex  variable 
(§§  6,  7).  Their  characteristic  properties,  their  classification,  and  some  of 
the  simpler  applications  will  be  considered  in  the  succeeding  chapters. 

Some  initial  definitions  and  explanations  will  now  be  given. 

(i).  It  has  been  assumed  that  the  function  considered  has  a  differential 
coefficient,  that  is,  that  the  rate  of  variation  of  the  function  in  any  direction 
is  independent  of  that  direction  by  being  independent  of  the  mode  of  change 
of  the  variable.  We  have  already  decided  (§  8)  not  to  use  the  term  analytic 
for  such  a  function.  It  is  often  called  monogenic,  when  it  is  necessary  to 
assign  a  specific  name ;  but  for  the  most  part  we  shall  omit  the  name,  the 
property  being  tacitly  assumed*. 

We  can  at   once  prove   fi-om   the  definition   that,  when   the   derivative 

/      diu\        .        .     .     .      ,r,       n         ■  -n  dw      Idw 

Wi  (  =  -y-)  exists,  it  IS  itself  a  function,     lor  Wj  =-^  =  -  ^  are  equations 

*  This  is  in  fact  done  by  Eiemann,  who  calls  such  a  dependent  complex  simply  a  function. 
Weierstrass,  however,  has  proved  (see  §  85,  post)  that  the  idea  of  a  monogenic  function  of  a  complex 
variable  and  the  idea  of  dependence  expressible  by  arithmetical  operations  are  not  coextensive. 
The  definition  is  thus  necessary;  but  the  practice  indicated  in  the  text  will  be  adopted,  as  non- 
monogenic  functions  will  be  of  relatively  rare  occurrence. 


16  DEFINITIONS  [12. 

which,  when  satisfied,  ensure  the  existence  of  w^ ;  hence 

1  dwi  _  1  9  fdw\ 

i  dy       i  dy  \dx) 

_  3  /I  3w^ 

dx  \i  dy  J 

dx 
shewing,  as  in  §  8,  that  the  derivative  -^  is  independent  of  the  direction  in 
which  dz  vanishes.     Hence  w-^  is  a  function  of  z. 

Similarly  for  all  the  derivatives  in  succession. 

(ii).  Since  the  functional  dependence  of  a  complex  is  ensured  only  if  the 
value  of  the  derivative  of  that  complex  be  independent  of  the  manner  in 
which  the  point  z  -[-dz  approaches  to  coincidence  with  z,  a  question  naturally 
suggests  itself  as  to  the  effect  on  the  character  of  the  function  that  may  be 
caused  by  the  manner  in  which  the  variable  itself  has  come  to  the  value  of  z. 

If  a  function  has  only  one  value  for  each  given  value  of  the  variable, 
whatever  be  the  manner  in  which  the  variable  has  come  to  that  value,  the 
function  is  called  uniform*.  Hence  two  different  paths  from  a  point  a  to  a 
point  z  give  at  z  the  same  value  for  any  uniform  function;  and  a  closed 
curve,  beginning  at  any  point  and  completely  described  by  the  ^r-variable 
will  lead  to  the  initial  value  of  tu,  the  corresponding  ty-curve  being  closed,  if  z 
has  not  passed  through  any  point  which  makes  lu  infinite. 

The  simplest  class  of  uniform  functions  is  constituted  by  rational 
functions. 

(iii).  If  a  function  has  more  than  one  value  for  any  given  value  of  the 
variable,  or  if  its  value  can  be  changed  by  modifying  the  path  in  which 
the  variable  reaches  that  given  value,  the  function  is  called  multiform^;. 
Characteristics  of  curves,  which  are  graphs  of  multiform  functions  corre- 
sponding to  a  ^-curve,  will  hereafter  be  discussed. 

One  of  the  simplest  classes  of  multiform  functions  is  constituted  by 
algebraical  irrational  functions,  that  is,  functions  defined  by  an  irresoluble 
algebraic  equation /(w,  z)  =  0,  where/  is  a  polynomial  in  w  and  z. 

The  rational  functions  in  (ii)  occur  when /is  of  only  the  first  degree  in  w. 

(iv).  A  multiform  function  has  a  number  of  different  values  for  the  same 
value  of  z,  and  these  values  vary  with  z :  the  aggregate  of  the  variations  of 
any  one  of  the  values  is  called  a  branch  of  the  function.  Although  the 
function  is  multiform  for  unrestricted  variation  of  the  variable,  it  often 
happens  that  a  branch  is  uniform  when  the  variable  is  restricted  to 
particular  regions  in  the  plane. 

*  Also  monodromic,  or  monotropic;  with  Grerman  writers  the  title  is  eindeutig,  occasionally, 
einandrig. 

+  Also  polytropic  ;  with  German  writers  the  title  is  viehrdeutig. 


12.]  DEFINITIONS  17 

(v).  A  point  in  the  plane,  at  which  two  or  more  branches  of  a  multiform 
function  assume  the  same  value,  and  near  which  those  branches  are  inter- 
changed (§  94,  Note)  by  appropriate  modification  in  the  path  of  z,  is  called  a 
hranch-point*  of  the  function.  The  relations  of  the  branches  in  the  immediate 
vicinity  of  a  branch-point  will  be  discussed  hereafter. 

(vi).  A  function,  which  is  monogenic,  uniform  and  continuous  over  any 
part  of  the  ^r-plane,  is  called  holomorphic-f  over  that  part  of  the  plane.  When 
a  function  is  called  holomorphic  without  any  limitation,  the  usual  implication 
is  that  the  character  is  preserved  over  the  whole  of  the  plane  which  is  not  at 
infinity. 

The  simplest  example  of  a  holomorphic  function  is  a  polynomial  in  the 
variable. 

(vii).  A  root  (or  a  zero)  of  a  function  is  a  value  of  the  variable  for  which 
the  function  vanishes. 

The  simplest  case  of  occurrence  of  roots  is  in  a  rational  integral 
function,  various  theorems  relating  to  which  (e.g.,  the  number  of  roots 
included  within  a  given  contour)  will  be  found  in  treatises  on  the  theory 
of  equations. 

(viii).  The  infinities  of  a  function  are  the  points  at  which  the  value  of 
the  function  is  infinite.  Among  them,  the  simplest  are  the  poles^  of  the 
function,  a  pole  being  an  infinity  such  that  in  its  immediate  vicinity  the 
reciprocal  of  the  function  is  holomorphic. 

Infinities  other  than  poles  (and  also  the  poles)  are  called  the  singular 
points,  or  the  singularities,  of  the  function :  their  classification  must  be 
deferred  until  after  the  discussion  of  properties  of  functions. 

(ix).  A  function,  which  is  monogenic,  uniform  and,  except  at  poles, 
continuous,  is  called  a  meroniorphic  function^.  The  simplest  example  is  a 
rational  fraction. 

13.  The  following  functions  give  illustrations  of  some  of  the  preceding 
definitions. 

(a)     In  the  case  of  a  meromorphic  function 

*  Also  critical  point,  which,  however,  is  sometimes  used  to  include  all  special  points  of  a 
function;  with  German  writers  the  title  is  Verziopigungspunkt,  and  sometimes  Windungspunht. 
French  writers  use  point  de  ramification,  and  Italians  punto  di  giramento  and  punto  di 
diraniazione. 

t  Also  synectic. 

X  Also  polar  discontinuities ;  also  (§  32)  accidental  singularities. 

§  Sometimes  regular,  but  this  term  will  be  reserved  for  the  description  of  another  property  of 
functions. 


18"  EXAMPLES    ILLUSTRATING  [13. 

Avh©Fe-:F'  and  /"are  polynomials  in  z  without  a  common  factor,  the  roots  are 
the- roots  of  F  iz)  and  the  poles  are  the  roots  of /(^).  Moreover,  according, 
as  the-  degree  of  F  is  greater  or  is  less  than  that  of  /,  ^  =  oo  is  a  pole  or  a- 
zero  of  w;. 

(6)     If  w  be  a  polynomial  of  order  n,  then  each  simple  root  of  w  is  a 

br9.n;C^ -point  and  a  zero  of  w'",  where  m  is  a  positive  integer ;  z  =  qo  is 
a  pole  of  w  ;    and  z=qc    is  a  pole  but  not  a  branch-point  or  is  an  infinity 

(though  not  a  pole)  and  a  branch-point  of  w-  according  as  n  is  even  or  odd. 

• ,     (c)     In  the  case  of  the  function 

_    1 

,7  sn - 

z 

(the. notation  being  that  of  Jacobian  elliptic  functions),  the  zeros  are  given  by 

-  =  iK'  ■\-  '2'mK  +  2m' iK', 


iK'  -I-  ImK  +  'hyi'iK'  +  ^, 


for  all  positive  and  negative  integral  values  of  7)i  and  of  m  .     If  we  take 

'  ■  ■  1 

z 

where  t,  may  be  restricted  to  values  that  are  not  large,  then 

so  that,  in  the  neighbourhood  of  a  zero,  %v  behaves  like  a  holomorphic 
function.  There  is  evidently  a  doubly-infinite  system  of  zeros  ;  they  are 
distinct  from  one  another  except  at  the  origin,  where  an  infinite  number 
practically  coincide. 

The  infinities  of  w  are  given  by 

-      -  ^=1nK^%iiK\ 

z 

for  all  positive  and  negative  integral  values  of  n  and  of  n .     If  we  take 

z 

then  l=(-l)-sn^, 

,      w 

so  that,  in  the  immediate  vicinity  of  ^  =  0,  —  is  a   holomorphic  function. 

Hence  ^  =  0  is  a  pole  of  w.  There  is  thus  evidently  a  doubly-infinite  system 
of  poles ;  they  are  distinct  from  one  another  except  at  the  origin,  where  an 
infinite  number  practically  coincide.      But  the   origin  is  not  a  pole ;   the 


13.]  THE   DEFINITIONS  19 

function,  in  fact,  is  there  not  determinate,  for  it  has  an  infinite  number  of 
zeros  and  an  infinite  number  of  infinities,  and  the  variations  of  value  are  not 
necessarily  exhausted  by  zeros  and  infinities. 

For  the  function  — -  ,  the  origin  is  a  point  which  will  hereafter  be  called 

sn- 

z 

an  essential  singularity. 

Ex.     Obtain  essential  singularities  of  the  functions 

e^,      sinh-,      tanlis. 

2 


2—2 


CHAPTER   II. 

Integration  of  Uniform  Functions. 

14.  The  definition  of  an  integral,  that  is  adopted  when  the  variables 
are  complex,  is  the  natural  generalisation  of  that  definition  for  real  variables 
in  which  it  is  regarded  as  the  limit  of  the  sum  of  an  infinite  number  of 
infinitesimally  small  terms.     It  is  as  follows : — 

Let  a  and  z  be  any  two  points  in  the  plane ;  and  let  them  be  connected 
by  a  curve  of  specified  form,  which  is  to  be  the  path  of  variation  of  the 
independent  variable.  Let  f{z)  denote  any  function  of  ir;  if  any  infinity 
of  f(z)  lie  in  the  vicinity  of  the  curve,  the  line  of  the  curve  will  be  chosen 
so  as  not  to  pass  through  that  infinity.  On  the  curve,  let  any  number  of 
points  Zi,  Z.2,  ...,  Zn  in  succession  be  taken  between  a  and  z;  then,  if  the  sum 

(Z^  -  a)f{a)  +  (Z2  -  2i)f{Zi)  +...+{z-  Zn)f{Zn) 

have  a  limit,  when  n  is  indefinitely  increased  so  that  the  infinitely  numerous 
points  are  in  indefinitely  close  succession  along  the  whole  of  the  curve  from 
a  to  z,  that  limit  is  called  the  integral  oif{z)  between  a  and  z.  It  is  denoted, 
as  in  the  case  of  real  variables,  by 

''f{z)dz. 

It  is  known*  that  the  value  of  the  integral  of  a  function  of  a  real  variable 
between  limits  a  and  h  is  independent  of  the  manner  in  which,  under  the 
customary  definition,  the  interval  between  a  and  h  is  divided  up.  Assuming 
this  result,  we  infer  at  once  that,  the  same  property  holds  for  the  complex 
integral 

f{z)dz; 


for,  if  f{z)  =  a  +  iv,  where  u  and  v  are  real, 

/ {z)  dz  =  u dx  —  vdy  +  iudy  +  ivdx, 
and  each  of  the  integrals 

judx,     jvdy,     Judy,     Jvd.r, 

*  Harnack's  Introduction  to  the  Calculus,  (Cathcart's  translation),  §§  103,  142. 


14.]  DEFINITIONS   AS   TO   CONVERGENCE  21 

taken  between  limits  corresponding  to  the  extremities  of  the  curve,  is  inde- 
pendent of  the  way  in  which  the  range  is  divided  up. 

The  limit,  as  the  value  of  the  integral,  is  associated  with  a  particular 
curve:  in  order  that  the  integral  may  have  a  definite  value,  the  curve 
(called  the  path  of  integration)  must,  in  the  first  instance,  be  specified*. 
The  integral  of  any  function  whatever  may  not  be  assumed  to  depend  in 
general  only  upon  the  limits. 

We  have  to  deal  with  converging  series;  it  is  therefore  convenient  to  state  the 
definitions  of  the  terms  used.  For  proofs  of  the  statements,  developments,  and  appli- 
cations in  the  theory  of  convergence,  as  well  as  the  various  tests  of  convergence,  see 
Bromwich's  Theory  of  infinite  series,  Carslaw's  Fottri&r's  series  and  integrals,  Hobson's 
Functions  of  a  real  variable,  and  Pringsheim's  article  in  the  Encycloplidie  der  mathema- 
tischen   Wissensckaften,  t.  i,  pp.  49 — 146,  where  full  references  are  given. 

A  series,  represented  by 

«!,    a.j,   «3, ...  ad  inf., 

is  said  to  converge,  when  the  limit  of  •S'^,,  where 

'^n  =  «l  +  «2  + •••+««) 

as  n  increases  indefinitely,  is  a  unique  finite  quantity,  say  *S'.  When,  in  the  same  circum- 
stances, the  limit  of  S^  either  is  infinite  or,  if  finite,  is  not  unique  (that  is,  may  be  one  of 
several  quantities),  the  series  is  saidt  to  diverge. 

The  necessary  and  sufficient  condition  that  the  series 

«1,    «2J     «3,  ••• 

should  converge  is  that,  corresponding  to  every  finite  positive  quantity  e  taken  as  small 
as  we  please,  an  integer  m  can  be  found  such  that 

for  all  integers  n  such  that  n  ^  m,  and  for  every  positive  integer  r. 

When  the  series 
converges,  the  series 

converges ;  and  it  is  said  to  converge  absolutely.  When  the  series  of  moduli  |  aj  | ,  |  (Xg  | ,  |  as  | , . . . , 
does  not  converge,  though  the  series  a^,  a.^,  a-,, ...  converges,  the  convergence  of  the  latter 
is  said  to  be  conditional.  In  a  conditionally  converging  series,  the  order  of  the  terms 
must  be  kept :  derangement  of  the  ordei'  can  lead  to  different  limits ;  and  any  assigned 
sum,  as  a  limit,  can  be  obtained  by  appropriate  derangement.  In  an  absolutely  converging- 
series,  the  order  of  the  terms  can  be  deranged  without  affecting  the  limit  to  which  the 
series  converges  ;  the  convergence  is  sometimes  called  unconditional. 

These  definitions  apply  to  all  infinite  series,  whatever  be  the  source  of  their  terms. 
AVhen  the  terms  depend  upon  a  vai'iable  quantity  z,  and  the  convergence  of  th.e  series  is 
considered  as  z  varies,  we  have  further  classifications.     Denote  the  series  by 

/i(4   /2(^),   /3(4-ad  inf., 

*  This  specification  is  tacitly  supplied  when  the  variables  are  real:  the  variable  point  moves 
along  the  axis  of  x. 

t  Sometimes  the  series,  such  that  the  limit  of  <S',^  when  n  is  infinitely  large  is  one  of  a 
number  of  finite  quantities  (depending  upon  the  way  in  which  .S'„  is  formed),  are  called  oscillating. 


22  THEOREMS   ON  [14. 

and  suppose  that  it  converges  for  all  values  of  z  within  a  definite  region.  When  any 
small  quantity  S  has  been  chosen,  and  a  positive  integer  m  can  be  determined,  such  that 

2    }\{z)    <S 
v=n 

for  every  value  of  n  ^  m  and  for  all  values  of  s  in  the  region,  the  convergence  is  said  to  be 
uniform  (sometimes  continuous). 

Convergence  maybe  uniform  without  being  absolute;  it  can  be  absolute  without  being 
uniform. 

When  a  series  converges  for  all  values  of  z  such  that  |  s  |  <  '/',  but  -not  for  \z\>  r,  then 
the  circle,  centre  the  origin  of  the  variable  z  and  radius  equal  to  r,  is  called  the  circle  of 
convergence :   and  the  i-adius  is  sometimes  called  the  radius  of  convergence.      A  series 

such  as 

flfi,    «i2,    a.2Z^,  ...'Ad  inf., 

converges  absolutely  within  its  circle  of  convergence,  though  not  necessarily  on  its 
circumference.  It  does  not  necessarily  converge  unifonnly  within  its  circle  of  convergence ; 
but  if  /■'  is  a  positive  quantity,  less  than  the  radius  of  convergence  by  a  finite  quantity 
which  can  be  taken  small,  the  series  converges  uniformly  within  the  circle  of  radius  r' 
concentric  with  its  circle  of  convei-gence. 

Again,  when  a  uniformly  converging  series  is  integrated  term  by  term  over  a  finite 
range,  the  resulting  series  also  converges  uniformly.  But  a  uniformly  converging  series 
can  be  differentiated  term  by  term  only  if  the  series  of  derivatives  converges. 

15.     Some  inferences  can  be  made  from  the  definition  of  an  integral. 

(I.)  The  integral  along  any  path  from  a  to  z  passing  through  a  point  ^  is 
the  sum  of  the  integrals  froTii  a  to  ^  and  from  ^  to  z  along  the  same  path. 
Analytically,  this  is  expressed  by  the  equation 

{"^  f{z)dz=\^  f{z)dz+\y{z)dz, 

J  a  -la  J  ^ 

the  paths  on  the  right-hand  side  combining  to  form  the  path  on  the  left. 

(II.)  When  the  path  is  described  in  the  reverse  direction,  the  sign  of  the 
integral  is  changed :  that  is, 

jy(z)dz=-jj\z)dz, 

the  curve  of  variation  between  a  and  z  being  the  same. 

(III.)  The  integral  of  the  sum  of  a  finite  number  of  terms  is  equal  to 
the  sum  of  the  integrals  of  the  separate  terms,  the  path  of  integration  being 
the  sam,e  for  all. 

(IV.)  If  a  function  f(z)  be  finite  and  continuous  along  any  finite  line 
between  two  points  a  and  z,  the  integral   \    f{z)dz  is  finite. 

J  a 


15]  INTEGRATION  23 

Let  /  denote  the  integral,  so  that  we  have  /  as  the  limit  of 

n 

hence  |  /  j  =  limit  of     2  {Zr+,  -  z,)f{z,.) 

< %\z,^^-Z,.\\f{Zr)\. 

Because  f{z)  is  finite  and  continuous,  its  modulus  is  finite  and  therefore 
must  have  a  superior  limit,  say  M,  for  points  on  the  line.     Thus 

so  that  !  -^  i  <  limit  of  M%  \  z,.+i  —  z,.  | 

<MS, 

where  S  is  the  finite  length  of  the  path  of  integration.  Hence  the  -modulus 
of  the  integral  is  finite ;  the  integral  itself  is  therefore  finite. 

No  limitation  has  been  assigned  to  the  path,  except  finiteness  in  length ; 
the  proposition  is  still  true  when  the  curve  is  a  closed  curve  of  finite  lesngth. 

Hermite  and  Darboux  have  given  an  expression  for  the  integral  which 
leads  to  the  same  result.     We  have  as  above 

I=\^f{z)dz,  ■        ■ 

and  1-^!  <  I    l/(^)  I  \dz\ 

■J  a 

=  d\''\f{z)\\dz\, 

where  ^  is  a  real  positive  quantity  less  than  unity.  The  last  integral  involves 
only  real  variables ;  hence*  for  some  point  ^  lying  between  a  and  z,  we  have 

J  a  J  a 

SO  that  |/|  =  ^^|/(|)|. 

It  therefore  follows  that  there  is  some  argument  a  such  that,  if  X  =  6e"^, 

This  form  proves  the  finiteness  of  the  integral ;  and  the  result  is  the 
generalisation t  to  complex  variables  of  the  theorem  of  mean  value  just 
quoted  for  real  variables. 

*  By  what  is  usually  called  the  "  First  theorem  of  mean  value,"  in  the  integral  calculus;  for 
a  proof,  see  Garslaw's  Fourier^s  series  and  integrals,  §  39. 

t  Hermite,  Gours  a  la  faculte  des  sciences  de  Paris  (4™'--ed.,  1891),  p.  59,  where  the  reference 
to  Darboux  is  given. 


24  FUNDAMENTAL   THEOREM  [15. 

(V.)  When  a  function  is  expressed  as  a  uniformly  converging  series,  the 
integral  of  the  function  along  any  path  of  finite  length  is  the  sum  of  the 
integrals  of  the  terms  of  the  series  along  the  same  path,  provided  that  path 
lies  ivithin  the  circle  of  convergence  of  the  series: — a  result,  which  is  an 
extension  of  (III.)  above. 

Let  Uq  -\-  Ui  +  U2  +  ...  be  the  converging  series  ;  take 

f{z)  =  Uf,  +  l(i+  ...  +  Un  +  R, 

where  j  R  \  can  be  made  infinitesimally  small  with  indefinite  increase  of  n, 
because  the  series  converges  uniformly.  Then  by  (III.),  or  immediately  from 
the  definition  of  the  integral,  we  have 

(    f(z)  dz  =       u^dz  +  I    u^dz  +  ...  +  1    Undz  +  1    Rdz, 

the  path  of  integration  being  the  same  for  all  the  integrals.     Hence,  if 

@  =  I    f[z)dz  —  '1   I    u^dz, 

we  have  @  =  I    Rdz. 

.  •       ■  J  a 

Let  R'  be  the  greatest  value  of  \R\  for  points  in  the  path  of  integration 
from  a  to  z,  and  let  S  be  the  length  of  this  path,  so  that  S  is  finite ; 
then,  by  (IV.), 

I  e  1  <  SR. 

Now  S  is  finite ;  and,  as  n  is  increased  indefinitely,  the  quantity  R'  tends 
towards  zero  as  a  limit  for  all  points  within  the  circle  of  convergence  and 
therefore  for  all  points  on  the  path  of  integration  provided  that  the  path  lie 
within  the  circle  of  convergence.  When  this  proviso  is  satisfied,  1  ©  |  becomes 
infinitesimally  small  and  therefore  also  0  becomes  infinitesimally  small,  with 
indefinite  increase  of  n.  Hence,  under  the  conditions  stated  in  the  enuncia- 
tion, we  have 

I    f{z)dz—  ^   \    Umdz  =  0, 

J  a  III  =  0  -'  a 

which  proves  the  proposition. 

16.     The  following  lemma*  is  of  fundamental  importance. 

Let  any  region  of  the  plane,  on  which  the  5- variable  is  represented,  be 
bounded  by  one  or  more  simple f  curves  which  do  not  meet  one  another: 
each  curve  that  lies  entirely  in  the  finite  part  of  the  plane  will  be  considered 
to  be  a  closed  curve. 

*  It  is  proved  by  Eiemann,  Ges.  Werke,  p.  12,  aud  is  made  by  him  (as  also  by  Cauchy)  the 
basis  of  certain  theorems  relating  to  functions  of  complex  variables. 

t  For  the  immediate  purpose,  a  curve  is  called  simple,  if  it  have  no  multiple  points.  The 
aim,  in  constituting  the  boundary  from  such  curves,  is  to  prevent  the  superfluous  complexity  that 
arises  from  duplication  of  area  on  the  plane.  If,  in  any  particular  case,  multiple  points  existed, 
a  method  of  meeting  the  difficulty  would  be  to  take  each  simple  looii  as  a  boundary. 


16.] 


IN   INTEGRATION 


25 


Ifp  and  q  be  any  two  functiom  of  x  and  y,  luhich,  for  all  points  within  the 
region  or  along  its  boundary,  are  uniform,  finite  and  continuous,  then  the 
integral 

extended  over  the  whole  area  of  the  region,  is  equal  to  the  integral 

jipdx  +  qdy), 

taken  ni  a  positive  direction  round  the  ivhole  boundary  of  the  region. 

(As  the  proof  of  the  proposition  does  not  depend  on  any  special  form  of 
region,  we  shall  take  the  area  to  be  (fig.  5)  that  which  is  included  by  the 
curve  QiPiQsP.:  and  excluded  by  P:Q:P^Q,,  and  excluded  by  P^P,^.  The 
positive  directions  of  description  of  the  curves  are  indicated  by  the  arrows ; 
and  for  integration  in  the  area  the  positive  directions  are  those  of  increas- 
ing X  and  increasing  y.) 


First,  suppose  that  both  p  and  q  are  real.     Then,  integrating  with  regard 
to  X,  we  have* 

\\^^dxdy=j{qdyl 

where  the  brackets  imply  that  the  limits  are  to  be  introduced.  When  the 
limits  are  introduced  along  a  line  CQiQi'...  parallel  to  the  axis  of  x,  then, 
since  CQiQi'...  gives  the  direction  of  integration,  we  have 

[qdy]  =  -  q.dyi  +  qi'dy/ -  q4y.  +  qldy.'  - q.dy-,  +  q^dy^', 
where  the  various  differential  elements  are  the  projections  on  the  axis  of  y 
of  the  various  elements  of  the  boundary  at  points  along  C'QiQ/.... 

*  It  is  in  this  integration,  and  in  the  corresponding  integration  for  p,  that  the  properties  of 
the  function  q  are  assumed.  Any  deviation  from  uniformity,  finiteness  or  continuity  within  the 
region  of  integration  would  render  necessary  some  equation  different  from  the  one  given  in 
the  text. 


26  FUNDAMENTAL   LEMMA  [16. 

Now  when  integration  is  taken  in  the  positive  direction  round  the  whole 
boundary,  the  part  of  jqdy  arising  from  the  elements  of  the  boundary  at  the 
points  on  GQ-^Q-^ . . .  is  the  foregoing  sum.  For  at  Q^'  it  is  qs'dys  because  the 
positive  element  dys,  which  is  equal  to  CD,  is  in  the  positive  direction  of 
boundary  integration ;  at  Q^  it  is  —q^dy..  because  the  positive  element  dy.,, 
also  equal  to  CD,  is  in  the  negative  direction  of  boundary  integration ; 
at  Q.2  it  is  qidy.i,  for  similar  reasons;  at  Q^  it  is  —  q^dyo,  for  similar  reasons; 
and  so  on.     Hence 

corresponding  to  parallels  through  G  and  D  to  the  axis  of  x,  is  equal  to 
the  part  of  /  qdy  taken  along  the  boundary  in  the  positive  direction  for  all 
the  elements  of  the  boundary  that  lie  between  those  parallels.  Then  when 
we  integrate  for  all  the  elements  GD  by  forming  j\_qdy\  an  equivalent  is 
given  by  the  aggregate  of  all  the  parts  oi  Jqdy  taken  in  the  positive  direction 
round  the  whole  boundary  ;  and  therefore 


\j^dxdy=Jqdy, 


on  the  suppositions  stated  in  the  enunciation. 
Again,  integrating  with  regard  to  y,  we  have 

IJ  £da;dy=f[pda;] 

—  —  jj^dx^  +  pi'dxj'  —  jjodxo  +  p-zdooo  —  Psdxs  +  p^dx^, 

when  the  limits  are  introduced  along  a  line  BP^P^...  parallel  to  the  axis 
of  y :  the  various  differential  elements  are  the  projections  on  the  axis  of  a;  of 
the  various  elements  of  the  boundary  at  points  along  BP^P^.... 

It  is  proved,  in  the  same  way  as  before,  that  the  part  of  —  f  pdx  arising 
from  the  positively- described  elements  of  the  boundary  at  the  points  on 
BP^Pi  ...  is  the  foregoing  sum.  At  P-,'  the  part  of  fpdx  is  —  pjdxs,  because 
the  positive  element  dx./,  which  is  equal  to  AB,  is  in  the  negative  direction 
of  boundary  integration ;  at  P,,  it  is  p^dxs,  because  the  positive  element 
dx^,  also  equal  to  AB,  is  in  the  positive  direction  of  boundary  integration; 
and  so  on  for  the  other  terms.     Consequently 

-  [pdaf\, 

corresponding  to  parallels  through  A  and  B  to  the  axis  of  y,  is  equal  to 
the  part  of  J  pdx  taken  along  the  boundary  in  the  positive  direction  for  all 
the  elements  of  the  boundary  that  lie  between  those  parallels.  Hence 
integrating  for  all  the  elements  AB,  we  have  as  before 

dp 


I! 


dxdy  =  —  jpdx ; 


and  therefore  \\\d~d)  ^^""^^  ~  j{pdx  +  qdy). 


16.]  IN    INTEGRATION  27 

Secondly,  suppose  that  p  and  q  are  complex.  When  they  are  resolved 
into  real  and  imaginary  parts,  in  the  forms  p'  +  ip"  and  q  +  iq'  respectively, 
then  the  conditions  as  to  uniformity,  finiteness  and  continuity,  which  apply 
to  p  and  q,  apply  also  to  p,  q',  p",  q" .     Hence 

^^d  \\<^£-^)dxdy^^f{p''dx  +  <^'dy\  

and  therefore  ^^{^^^  -  ^^  dxdy  =  j{pdx  +  qdy) :  . 

which  proves  the  proposition. 

No  restriction  on  the  properties  of  the  functions  p  and  q  at  points 
that  lie  without  the  region  is  imposed  by  the  proposition.  They  may  have 
infinities  outside,  they  may  cease  to  be  continuous  at  outside  points,  or  they 
may  have  branch-points  outside ;  but  so  long  as  they  are  finite  and  continuous 
everywhere  inside,  and  in  passing  from  any  one  point  to  any  other  point 
always  acquire  at  that  other  the  same  value  whatever  be  the  path  of  passage 
in  the  region,  that  is,  so  long  as  they  are  uniform  in  the  region,  the  lemma 
is  valid. 

17.     The  following  theorem  due.  to  Cauchy  *  can  now  be  proved  : — 

If  a  function  f{z)  he  holomo^yhic  throughout  any  region  of  the  z-plane, 
then  the  integral  jf{z)dz,  taken  round  the  whole  boundary  of  that  region,  is  zero. 
We  apply  the  preceding  result  b}^  assuming 

owing  to  the  character  of /(^),  these  suppositions  are  consistent  with  the 
conditions  under  which  the  lemma  is  valid.  Since  _p  is  a  function  of  z,  we 
have,  at  every  point  of  the  region, 

dp  _1  dp 

dec      i  dy' 

and  therefore,  in  the  present  case, 

dq  _  .dp  _dp^ 

dx ~    dx      dy' 
There  is  no  discontinuity  or  infinity  of  p  or  q  within  the  region ;  hence 

|-|)«y=0, 

*  For  an  account  of  the  gradual  development  of  the  theory  and,  in  particular,  for  a 
statement  of  Cauchy's  contributions  to  the  theory  (with  references),  see  Casorati,  Teorica 
delle  funzioni  di  variahili  complesse,  pp.  64—90,  102—106.  The  general  theory  of  functions, 
as  developed  by  Briot  and  Bouquet  in  their  treatise  Theorie  dcs  fonctions  elliptiques,  is  based 
upon  Cauchy's  method. 


//' 


28  cauchy's  theorem  [17. 

the  integral  being  extended  over  the  region.     Hence  also 

jXpdx  +  qdy)  =  0, 
when  the  integral  is  taken  round  the  whole  boundary  of  the  region.     But 

pdx  +  qdy  --=  pdx  +  ipdy 
=  pdz 
=f{z)dz, 

and  therefore  //(^)  ^^  =  0- 

the  integral  being   taken  round   the  whole  boundary  of  the  region  Avithin 

which  f{z)  is  holomorphic. 

It   should    be  noted    that    the    theorem  requires  no   limitation    on    the 
character  of  f(z)  for  points  z  that  are  not  included  in  the  region. 

The  result  can  also  be  established  by  a  slightly  different  use  of  the 

original  theorem.     Writing 

f{z)  =  u  +  iv, 

where,  after  the  hypotheses  concerning  f(z),  the  real  functions  u  and  v 
are  uniform,  finite,  and  continuous  for  all  points  within  the  region  or  along 
the  boundary,  we  have 

Jf(z)  dz  =  f  {u  +  iv)  (dx  +  idy) 

=  j{udx—  vdy)  +  ij{vdx  +  udy). 

Owing  to  the  character  of  u  and  v,  we  have 


j  {udx  -  vdy)  =  j j  {-  ~  -  °^J  dxdy, 


taken  over  the  whole  region  ;   but 

du  _     ov 
dy        dx' 
and  therefore 

f(udx  —  vdy)  =  0, 
Similarly 

j(vda^  +  udy)^jj(^£-^~)dxdy, 

taken  over  the  whole  region  ;  but 

du      dv 
dx     dy ' 
and  therefore 

J{vdx  +  tidy)  =  0. 

Hence,  with  the  assumptions  made  as  to  f{z),  we  have 

Jf(z)dz==0. 

Some  important  propositions  can  be  derived  by  means  of  the  theorem,  as 
follows. 


18.] 


INTEGRATION    OF   HOLOMORPHIC    FUNCTIONS 


29 


18.      When  a  function  f{z)  is  holomoiyhic   over  any  continuous  region 

of  the  plane,  the  integral  \  f{z)dz  is  a  holomorphic  function  of  z,  provided  the 

points  z  and  a  as  ivell  as  the  whole  path  of  integration  lie  within  that  region. 

The  general  definition  (§  14)  of  an  integral  is  associated  with  a  specified 
path  of  integration.  In  order  to  prove  that  the  integral  is  a  holomorphic 
function  of  z,  it  will  be  necessary  to  prove  (i)  that  the  integral  acquires  the 
same  value  in  whatever  way  the  point  z  is  attained,  that  is,  that  the  value  is 
independent  of  the  path  of  integration,  (ii)  that  it  is  finite,  (iii)  that  it 
is  continuous,  anVi  (iv)  that  it  is  monogenic. 

Let  two  paths  a<yz  and  a^z  between  a  and  z  be  drawn  (fig.  6)  in  the 
continuous  region  of  the  plane  within  which  f\z)  is 
holomorphic.  The  line  a^^z^a  is  a  contour  over  the  area 
of  which  f{z)  is  holomorphic  ;  and  therefore  jf{z)  dz 
vanishes  when  the  integral  is  taken  along  o.yz^a. 
Dividing  the  integral  into  two  parts  and  implying  by 
Zy,  z^  that  the  point  z  has  been  reached  by  the  paths 
a.^z,  a^z  respectively,  we  have 


Fig.  6. 


and  therefore 


J  a 


dz 


f{z)dz  =  0, 
f{z)dz 


=  rf{z)dz. 

J  a 

Thus  the  value  of  the  integral  is  independent  of  the  way  in  which  z  has 
acquired  its  value  ;  and  therefore  I  f{z)  dz  is  uniform  in  the  region.  Denote 
it  by  F{z). 

Secondly,  f(z)  is  finite  for  all  points  in  the  region.  After  the  result 
of  §  17,  we  naturally  consider  only  such  paths  between  a  and  z  as  are  finite  in 
length,  the  distance  between  a  and  z  being  finite.  Hence,  (§  15,  IV.)  the 
integral  F{z)  is  finite  for  all  points  z  in  the  region. 

Thirdly,  let  z'  (=z  +  Sz)  be  a  point  infinitesimally  near  to  z ;  and  consider 
rz' 
I  f{z)  dz.     By  what  has  just  been  proved,  the  path  from  a  to  z'  can  be  taken 

J  a 

a^zz' ;   therefore 

'^f{z)dz=\'f{z)dz+\'^'f{z)dz 


or 


so  that 


■z+Ss  fz  rz+Sz 

f{z)dz-      f{z)dz=  f{z)dz, 

J  a  -J  z 

fz  +  5z 

F(z  +  8z)-F(z)=  f{z)dz. 


30  INTEGRATION   OF  [IS- 

Now  at  points  in  the  infinitesimal  line  from  z  to  /,  the  value  of  the 
continuous  function  y(0)  differs  only  by  an  infinitesimal  quantity  from  its 
value  at  z  ;  hence  the  right-hand  side  is 

{t\z)^e\lz, 

where  j  e  j  is  an  infinitesimal  quantity  vanishing  with  hz.     It  therefore  follows 

that 

F{z^Zz)-F{z) 

is  an  infinitesimal   quantity  with   a  modulus  of  the   same   order  of  small 
quantities  as  \^z\.     Hence  F{z)  is  continuous  for  points  ^  in  the  region. 

Lastly,  we  have 

F{z^lz)-F{z) 


and  therefore 


Iz 

F(z  +  Sz)-F(z) 


hz 

has  a  limit  when  hz  vanishes ;  and  this  limit,  f{z),  is  independent  of  the 
way  in  which  hz  vanishes.  Hence  F  {z)  has  a  differential  coefficient ;  the 
integral  is  monogenic  for  points  z  in  the  region. 

Thus  F  {z),  which  is  equal  to 


/: 


f{z)  dz, 


is  uniform,  finite,  continuous,  and  monogenic ;  it  is  therefore  a  holomorphic 
function  of  z. 

As  in  §  16  for  the  functions  p  and  q,  so  here  for  f(z),  no  restriction  is 
placed  on  properties  of  f(z)  at  points  that  do  not  lie  within  the  region ; 
so  that  elsewhere  it  may  have  infinities,  or  discontinuities,  or  branch -points. 
The  properties,  essential  to  secure  the  validity  of  the  proposition,  are 
(i)  that  no  infinities  or  discontinuities  lie  within  the  region,  and  (ii)  that  the 
same  value  of  f{z)  is  acquired  by  whatever  path  in  the  continuous  region 
the  variable  reaches  its  position  z. 

Corollary.  No  change  is  caused  in  the  value  of  the  integral  of  a 
holomorphic  function  between  two  points  tuhen  the  path  of  integration 
between  the  points  is  deformed  in  any  manner,  provided  only  that,  during  the 
deformation,  no  part  of  the  path  passes  outside  the  boundary  of  the  7'egion 
within  which  the  function  is  holomorphic. 

This  result  is  of  importance,  because  it  permits  the  adoption  of  special 
forms  of  the  path  of  integration  without  affecting  the  value  of  the  integral. 

19.  When  a  function  f  (z)  is  holomorphic  over  a  part  of  the  plane 
bounded  by  two  simple  curves  {one  lying  within  the  other),  equal  values  of 
Jf{z)dz  are  obtained  by  integrating  round  each  of  the  curves  in  a  direction, 
which — relative  to  the  whole  area  enclosed  by  each  of  them — is  positive. 


19.]  HOLOMORPHIC    FUNCTIONS      '  31 

The  ring-formed  portion  of  the  plane  (fig.  1,  p.  3)  which  lies  between 
the  two  curves  is  a  region  over  which  f(z)  is  holomorphic;  hence  the  integral 
jf(z)dz  taken  in  the  positive  sense  round  the  whole  of  the  boundary  of 
the  included  portion  is  zero.  The  integral  consists  of  two  parts  :  first,  that 
round  the  outer  boundary  the  positive  sense  of  Avhich  is  BEF;  and  second, 
that  round  the  inner  boundary  the  positive  sense  of  which  for  the  portion  of 
area  between  ABC  and  DEF  is  A  CB.  Denoting  the  value  of //(^)  dz  round 
DEF  by  {DEF),  and  similarly  for  the  other,  we  have 

{ACB)^{DEF)  =  Q. 
The  direction  of  an  integral  can  be  reversed  if  its  sign  be  changed,  so  that 
{AGB)  =  -  (ABC) ;  and  therefore 

(ABC)  =  (DEF). 
But  (ABC)  is  the  integral  jf(z)dz  taken  round  ABC,  that  is,  round  the 
curve  in  a  direction  which,  relative  to  the  area  enclosed  by  it,  is  positive. 

The  proposition  is  therefore  proved. 

The  remarks  made  in  the  preceding  case  as  to  the  freedom  from  limitations 
on  the  character  of  the  function  at  places  not  within  the  bounded  area  are 
valid  also  in  this  case. 

Corollary  I.  When  the  integral  of  a  function  is  taken  round  the  whole 
of  any  simple  curve  in  the  plane,  no  change  is  caused  in  its  value  hy  continuously 
deforming  the  curve  into  any  other  simple  curve  provided  the  function  is 
holomorphic  over  the  part  of  the  plane  in  luhich  the  deformation  is  effected. 

Corollary  II.  When  a  function  f(z)  is  holomorphic  over  a  continuous 
portion  of  a  plane  bounded  by  any  number  of  simple  non-intersecting  curves, 
all  but  one  of  which  are  external  to  one  another  and  the  remaining  one  of 
which  encloses  them  all,  the  value  of  the  integral  Jf(z)  dz  taken  positively  round 
the  single  external  curve  is  equal  to  the  sum  of  the  values  taken  round  each 
of  the  other  curves  in  a  direction  tuhich  is  positive  relative  to  the  area  enclosed 
hy  it. 

These  corollaries  are  of  importance  in  many  instances,  as  will  be  seen 
later.  The  simplest  instances  arise  in  finding  the  value  of  the  integrals  of 
meromorphic  functions  round  a  curve  which  encloses  one  or  more  of  the 
poles;  the  fundamental  theorem,  also  due  to  Cauchy,  for  these  integrals  is 
the  following. 

20.  Let  f{z)  denote  a  function  xvhich  is  holomorphic  over  any  region  in 
the  z-plane,  and  let  a  denote  any  point  within  that  region ;  then 

liri  J  z  —  a 
the  integral  being  taken  positively  round  the  tuhole  boundary  of  the  region. 

With  a  as  centre  and  a  very  small  radius  p,  describe  a  circle  C,  which 
will  be  assumed  to  lie  wholly  within  the  region ;  this  assumption  is  justifiable 


32  ■  INTEGRATION    OF  [20. 

because  the  point  a  lies  within  the  region.  Because /(^)  is  holomorphic  over 
the  assigned  region,  the  function  f(z)l(2:  —  a)  is  holomorphic  over  the  whole  of 
the  region  excluded  by  the  small  circle  C.  Hence,  by  Corollary  II.  of  §  19,  we 
have 


Bj  z  —  a  c-i  z  —  a 

the  notation  implying  that  the  integrations  are  taken  positively  round  the 
whole  boundary  B  and  round  the  circumference  of  C  respectively. 

For  points  on  the  circle  G,  let  2  —  a  =  pe^',  so  that  6  is  the  variable  for 
the  circumference  and  its  range  is  from  0  to  27r ;  then  we  have 

dz 


=  icid. 


z  —  a 


Along  the  circle  f(z)=f(a+  pe^^);  the  quantity  p  is  very  small  and  f(z)  is 
finite  and  continuous  over  the  whole  of  the  region,  so  that/(a  + pe^*)  differs 
from  f(a)  only  by  a  quantity  which  vanishes  with  p.  Let  this  difference 
be  e,  which  is  a  continuous  small  quantity ;  thus  j  e  j  is  a  small  quantity 
which,  for  every  point  on  the  circumference  of  C,  vanishes  with  p.     Then 

{fi^^dz  =  ir  \f(a)+e}de 
cJZ-a  Jo 


27^^/(a)  +  i  [""  edd. 
J  0 


If  E  denote  the  value  of  the  integral  on  the  right-hand  side,  and  tj  the 
greatest  value  of  the  modulus  of  e  along  the  circle,  we  have,  as  in  §  15, 

/•27r 

\E\<\      \e\dd 
J  0 

r27r 

<  7}dd 

J  0 

<  27r»7. 

Now  let  the  radius  of  the  circle  diminish  to  zero.  Then  r?  also  diminishes 
to  zero  and  therefore  \E\,  necessarily  positive,  becomes  less  than  any  finite 
quantity  however  small,  that  is,  E  is  itself  zero ;  and  thus  we  have 

lli^dz=2'rrif(a), 

Q]  Z  Ci 

which  proves  the  theorem. 

When  a  is  not  a  zero  oif{z),  this  result  is  the  simplest  case  of  the  integral 

f(z) 
of  a  meromorphic  function.      The  subject  of  integration  is  '^~^— ,  a  function 

which  is  monogenic  and  uniform  throughout  the  region  and  which,  every- 
where except  at  z  =  a,  is  finite  and  continuous ;  moreover,  z  =  a  is  a  pole. 


20.]  MEROMORPHIC   FUNCTIONS  -  33 

because  in  the  immediate   vicinity  of  a  the  reciprocal   of   the   subject  of 
integration,  viz.  {z  —  a)l f{z),  is  holomorphic. 

The  theorem  may  therefore  be  expressed  as  follows : — 

li  g  {z)  be  a  meromorphic  function,  which  in  the  vicinity  of  a  can  be 
expressed  in  the  form  f(z)/{z  —  a)  where  f(a)  is  not  zero,  and  which  at  all 
other  points  in  a  region  enclosing  a  is  holomorphic,  then 

2 — •  1 9  (^)  ^^  =  limit  of  (^  —  a)  ^  (z)  when  z=  a, 

the  integral  being  taken  round  a  curve  in  the  region  enclosing  the  point  a. 

The  pole  a  of  the  function  g  (z)  is  said  to  be  simple,  or  of  the  first  order, 
or  of  multiplicity  unity. 

Corollary.  The  more  general  case  of  a  meromorphic  function  with  a 
finite  number  of  poles  can  easily  be  deduced.  '  Let  these  be  ttj,  ...,  a^,  each 
assumed  to  be  simple ;  and  let 

G  (z)  =  {z-  tti) (z  -  a.2). ..{z  -  an). 
Let  f{z)  be  a  holomorphic  function  within  a  region  of  the  sr-plane  bounded 
by  a  simple  contour  enclosing  the  n  points  aj,  tta,  ...,  a„,  no  one  of  which  is  a 
zero  off{z).     Then  since 


G  {z)       r=l  G' (Clr)  ^  —  Cf'r' 

-  f(z)        -        1       f{z) 

we  have  7:9 —  =  2  7.7^ — .-  -^      ^  . 

Or  {Z)        r=l  tr   {ar)  Z  —  Ur 

We  therefore  have 

jG{z)  r=i  G' {ar)  h  -  ar 

each  integral  being  taken  round  the  boundary.    But  the  preceding  proposition 
gives 

7(^) 


z  —  a, 


dz  =  ^'irif{ar), 


because  f{z)  is  holomorphic  over  the  whole  region  included  in  the  contour; 
and  therefore 


jGiz)'^'-'^''\t^G'{ary 


the  integral  on  the  left-hand  side  being  taken  in  the  positive  direction*. 

The  result  just  obtained  expresses  the  integral  of  the  meromorphic 
function  round  a  contour  which  includes  a  finite  number  of  its  simple  poles. 
It   can   be   obtained   otherwise    from   Corollary   II.   of  §  19,    by  adopting 

*  We  shall  for  the  future  assume  that,  if  no  direction  for  a  complete  integral  be  specified,  the 
positive  direction  is  taken. 

F.  F.  3 


34  INTEGRATION   OF  [20. 

a  process  similar  to  that  adopted  above,  viz.,  by  making  each  of  the  curves  in 
that  Corollary  circles  round  the  points  a^,  ...,  a^,  vrith  radii  sufficiently  small 
to  secure  that  each  circle  is  outside  all  the  others. 

Ex.  1.     A  function  f{z)  is  holomorphic  over  an  area  bounded  by  a  simple  closed 
curve;  and  a,  h,  c  are  three  points  within  the  area.     Find  the  value  of  the  integral 


1  r      /(^) 


dz 


2Tri  J  {z  —  a)  {z  —  b)  {z  —  c) 
taken  round  the  curve ;  and  shew  what  it  becomes 

(i)    when  a  and  b  coincide, 
(ii)    when  a,  b,  c  coincide. 

Ex.  2.     Let  aS'(-)  denote  the  sum  of  any  set  of  selected  terms  of  the  series 

and  let  /(0  =  «o  +  aiC  +  «2CH..., 

where  /{()  is  a  holomorphic  function  of  f  within  the  range;  shew  that  the  sum  of  the 

same  set  of  terms  selected  from  /(f)  can  be  expressed  in  the  form 


27ri  J     z 


ha]dz 


21.  The  preceding  theorems  have  sufficed  to  evaluate  the  integral  of 
a  function  vv^ith  a  number  of  simple  poles.  We  now  proceed  to  obtain 
further  theorems,  which  can  be  used  among  other  purposes  to  evaluate 
the  integral  of  a  function  with  poles  of  order  higher  than  the  first. 

We  still  consider  a  function  f(z)  which  is  holomorphic  within  a.  given 
region.  Let  a  be  a  point  within  the  region  which  is  not  a  zero  of  f(z) ; 
we  have 

Let  a  +  8a  be  any  other  point  within  the  region,  so  that,  if  a  be  near  the 
boundary,  \Ba\  is  to  be  chosen  less  than  the  shortest  distance  from  a  to 
the  boundary ;   then 

f(a  +  So)  =  7s— •    — ~ — ^  dz, 

and  therefore 

/(a+ s«) -/(«)  =  2^./ (- ^-1^  +  ^::^J/(.)  <«. 

1    ff     Sa  J^_     J/Wd.,- 


27ri  }  \{z  —  ay     (z  —  of  {z  —  a  —  ha)) 
the  integral  being  in  every  case  taken  round  the  boundary. 

Since  /  (2^)  is  monogenic,  the  definition  of /'(a),  the  first  derivative  of 
/{a),  gives  f  (a)  as  the  limit  of 

f{a  +  ha)-f{a)^ 

Sa  '  . 


21.]; 


MEROMORPHIC    FUNCTIONS 


35 


when  Sa  ultimately  vanishes ;  hence  we  may  take 
f(a  +  8a)-f{a)    ' 


Ba 


=f'(a)  +  a, 


where  o-  is  a  quantity  which  vanishes  with  Ba  and  is  therefore  such  that  |  a 
also  vanishes  with  Sa.     Hence 


dividing  out  by  8a  and  transposing,  we  have 


f{z)dz; 


•^    ^^      27rij(^-a)'  ^iTi'{z-af{z-a-ha) 


dz. 


As  yet,  there  is  no  limitation  on  the  value  of  Sa ;  we  now  proceed  to  a 
limit  by  making  a-^ha  approach  to  coincidence  with  a,  viz.,  by  making  Sa 
ultimately  vanish.  Taking  moduli  of  each  -of  the  members  of  the  last 
equation,  we  have 


/'(»)-.-^:f/^ 


/(^) 


27ri }  (z  —  a)' 


dz 


-<T  + 


Ba 


/(^) 


2Tri  j  (z  —  ay  (z  —  a  -  Ba) 


dz 


<    0-1  + 


Let  the  greatest  modulus  of 


Ba\ 
'Itt 


f{^) 


dz 


{z  —  ay(z  —  a  —  Ba) 

for  points  z  along  the 


(z  —  ay{z  —  a  —  Ba) 

boundary  be  M,  which  is  a  finite  quantity  on  account  of  the  conditions 
applying  to  f(z)  and  of  the  fact  that  the  points  a  and  a  +  Ba  lie  within 
the  region  and  are  not  on  the  boundary.     Then,  by  §  15, 


dz 


<MS, 


}  (z  —  af{z  —  a  —  Ba) 
where  8  is  the  whole  length  of  the  boundary,  a  finite  quantity.     Hence 

1     f  /(^) 


fid) 


dz 


1         I  \^Cb\    Turci 

<  1  0-   +  L-J  MS. 
'      '        27r 


27ri  .'  {z  —  af 

When  we  proceed  to  the  limit  in  which  Ba  vanishes,  we  have  |  Sa  |  =  0 
and  I  o- 1  =  0,  ultimately  ;  hence  the  modulus  on  the  left-hand  side  ultimately 
vanishes,  and  therefore  the  quantity  to  which  that  modulus  belongs  is  itself 
zero,  that  is, 


so  that 


•^    ^  27rt .'  {z  —  af 

'    ^  '      2Tri  J  {z  —  af 


This  theorem  evidently  corresponds  in  complex  variables  to  the  well- 
known  theorem  of  differentiation  with  respect  to  a  constant  under  the 
integral  sign  when  all  the  quantities  concerned  are  real. 

3—2 


36  PROPERTIES   OF  [2L 

Proceeding  in  the  same  way,  we  can  prove  that 

ha  ZTTi }  {z  —  ay 

where  ^  is  a  small  quantity  which  vanishes  with  Sa.  Moreover  the  integral 
on  the  right-hand  side  is  finite,  for  the  subject  of  integration  is  everywhere 
finite  along  the  path  of  integration  which  itself  is  of  finite  length.  Hence, 
first,  a  small  change  in  the  independent  variable  leads  to  a  change  of  the 
same  order  of  small  quantities  in  the  value  of  the  function  /'  (a),  which 
shews  that  /'  (a)  is  a  continuous  function.     Secondly,  denoting 

f(a  +  Sa)-f(a) 

by  Sf  (a),  we  have  the  limiting  value  of  ~ — -  equal  to  the'  integral  on 

the  right-hand  side  when  Sa  vanishes,  that  is,  the  derivative  of  /'  (a)  has 
a  value  independent  of  the  form  of  Ba  and  therefore  /'  (a)  is  monogenic. 
Denoting  this  derivative  by  /"  (a),  we  have 

27ri  '  {z  —  of  ! 

Thirdly,  the  function  /'  (a)  is  uniform :    for  it  is  the  limit  of  the  value 

of  "^-^^ ~ — -^  ^     ;  and  both /(a)  and /(a-|-6'a)  are  uniform.     Lastly,  it 

is  finite;  for  (§  15)  it  is  the  value  of  the  integral  ^ — -.  |  y^^^)  ^^  which 

the  length  of  the  path  is  finite  and  the  subject  of  integration  is  finite  at 
every  point  of  the  path. 

Hence  /'  (a)  is  continuous,  monogenic,  uniform,  and  finite,  throughout 
the  whole  of  the  region  in  Avhich  f(z)  has  these  properties:  it  is  a  holo- 
morphic  function.     Hence  : — 

When  a  function  is  holomorphic  in  any  region  of  the  plane  bounded  by 
a  simple  curve,  its  derivative  is  also  holomorphic  within  that  region. 

And,  by  repeated  application  of  this  theorem : — 

When  a  function  is  holomorphic  in  any  region  of  the  plane  bounded  by 
a  simple  curve,  it  has  an  unlimited  number  of  successive  derivatives  each  of 
which  is  holomorphic  within  the  region. 

All  these  properties  have  been  shewn  to  depend  solely  upon  the  holo- 
morphic character  of  the  fundamental  function ;  but  the  inferences  relating 
to  the  derivatives  have  been  proved  only  for  points  within  the  region  and 
not  for  points  on  the  boundary.  If  the  foregoing  methods  be  used  to  prove 
them  for  points  on  the  boundary,  they  require  that  a  consecutive  point  shall 
be  taken  in  any  direction ;  in  the  absence  of  knowledge  concerning  the 
fundamental  function  for  points  outside  (even  though  just  outside),  no 
inferences  can  be  drawn  justifiably. 


21.]  HOLOMORPHIC    FUNCTIONS  87 

An  illustration  of  this  statement  is  furnished  by  the  hypergeometric 
series  which,  together  with  all  its  derivatives,  is  holomorphic  within  a  circle 
of  radius  unity  and  centre  the  origin.  The  series  converges  everywhere 
on  the  circumference,  provided  7  >  a  +  y8.  But  the  corresponding  condition 
for  convergence  on  the  circumference  ceases  to  be  satisfied  for  some  one  of 
the  derivatives  and  for  all  which  succeed  it :  as  such  functions  do  not  then 
converge,  the  circumference  of  the  circle  must  be  excluded  from  the  region 
within  which  the  derivatives  are  holomorphic. 

Ex.     Let  F{z)  and  G  (z)  denote  two  functions  of  z,  holomorphic  in  a  region  enclosing 
the  point  a,  which  is  a  zero  of  G  (z)  and  a  non-zero  of  F{z) ;  prove  that 

1     [  ^{^)     ._^'{(t)G'{a)-F{a)G"{a) 
27nj{Giz)}^'^'  {G'(a)Y 

when  a  is  a  simple  rOot  of  G{z)=0,  and  that 


J-.f: 
2m  J 


'^(')  ^,  _  ^^'  («)  <^"  (^)  -  2i^(^)  (^"'  (^) 
G{z)  S{G"ia)f 


when  a  is  a  double  root  of   G  (z)  =  0,  both  integrals  being  taken  round  a  small  contour 
which  encloses  a  but  no  other  zero  of  G{z). 

22.     Expressions  for  the   first   and    the   second   derivatives   have   been 
obtained. 

By  a  process  similar  to  that  which  gives  the  value  of /'(a),  the  derivative 
of  order  n  is  obtainable  in  the  form 


the  integral  being  taken  round  the  whole  boundary  of  the  region  or  round 
any  curves  which  arise  from  deformation  of  the  boundary,  provided  that  no 
point  of  the  curves  in  the  final  form  of  the  boundary  or  in  any  intermediate 
form  of  the  boundary  is  indefinitely  near  to  a. 

In  the  case  when  the  curve  of  integration  is  a  circle,  no  point  of  which 
circle  may  lie  outside  the  boundary  of  the  region,  we  have  a  modified  form 
for  /("'  (a). 

For  points  along  the  circumference  of  the  circle  with  centre  a  and  radius 
r,  let  z  —  a  =  re^*,  so  that,  as  before, 

z  —  a 
then  0  and  27r  being  taken  as  the  limits  of  0,  we  have 

^  t        r27r 

fw  (a)  =  ^        e-"^^/(a  +  re«'")  dd. 
•^       ^  2irr^  Jo 

Let  M  be  the  greatest  value  of  the  modulus  oi  f{z)  for  points  on  the 


38  PROPERTIES   OF  [22. 

circumference  (or,  as  it  may  be  convenient  to  consider,  for  points  on  or  within 
the  circumference) :  then 

I  /■*"'  (a)  I  <  ^r^        I  e-''^'  I  \f(^  +  ^«^')  1  ^^ 
\j      V  y  I      27rr'Mo 

27rr'* .'  0 
Now,  let  a  function  ^  (^)  be  defined  by  the  equation 

'^i.^)  =  — : — I' 


r 
evidently  it  can  be  expanded  in  a  series  of  ascending  powers  of  ^  —  a  which 
converges  within  the  circle.     The  series  is 

Hence 


■  n\- 


so  that,  if  the  value  of  the  nth  derivative  of  ^iz),  when  z  =  a,  be  denoted 
by  (^f'^'  (a),  we  have  |/<")  (a)  j  <  <^"^'  (a). 

These  results  can  be  extended  to  functions  of  more  than  one  variable : 
the  proof  is  similar  to  the  foregoing  proof.  When  there  are  two  variables, 
say  z  and  /,  the  results  may  be  stated  as  follows : — 

For  all  points  z  within  a  given  simple  curve  C  in  the  2^-plane  and  all 

points  /  within  a  given  simple  curve  C"  in  the  /-plane,  let  f{z,z')  be  a 

holomorphic  function;   then,  if  a  be  any  point  within  C  and  a'  any  point 

within  C,  . 

n\n'\n  f{z,  z')  _  d-+-'f{a,  aQ 

where  n  and  n'  are  any  integers  and  the  integral  is  taken  positively  round  the 
two  curves  G  and  C\ 

If  M  be  the  greatest  value  of  \f(z,  z')\  for  points  z  and  /  within  their 
respective  regions  when  the  curves  C  and  C"  are  circles  of  radii  r,  ?•'  and 

centres  a,  a',  then 

a"+"7(a,  a')  ,    ,  ,    M 

and  if  ^  (z,  z')  = 


M 

<■  I  ^ 

M 


(-^-^K-^)' 


22.]  HOLOMORPHIC   FUNCTIONS  39 

then 


B'^+^'/(a,  a')  I      3'^+'*' cf) (z,  z) 


da^dd'''      I  dz'^dz''"-'      '.  .  . 

when^  =  a  and /=  a'  in  the  derivative  of  ^(^, /).  .       '      • 

A  function  cfj,  related  in  this  manner  to  a  function  /  in  association  with 
which  it  is  constructed,  is  sometimes  called*  a  dominant  function. 

23.  All  the  integrals  of  meroinorphic  functions  that  have  been  considered 
have  been  taken  along  complete  curves :  it  is  necessary  to  refer  to  integrals 
along  curves  which  are  lines  only  from  one  point  to  another.  A  single 
illustration  will  suffice  at  present. 

Consider  the  integral       'LAJ_  ^^ ;  the  function  f{z)  is 

supposed  holomorphic  in  the  given  region :  z  and  Zg  are 
any  two  points  in  that  region.  Let  some  curves  joining 
z  to  Zq  be  drawn  as  in  the  figure  (fig.  7). 

f(z) 

Then is  holomorphic  over  the  whole  area  en-      '       -c-     „ 

z  —  a  ^  J^ig.  7. 

closed  by  Zo^zSz^:   and  therefore  we  have   j"^        dz  =  0,  the  integral  being 

taken  round  the  boundary  of  that  area.     Hence,  as  in  the  earlier  case,  we  have 

j  ,„  2  -  a  J  zo  ^  -a- 

The  point  a  lies  within  the  area  enclosed  by  Zq'^z^Zq,  and  the  function 

is  holomorphic,  except  in  the  immediate  vicinity  of  ^  =  a ;  hence 

"^  ~  ^  ,  M  --,,■.  1  ■ 

\l^^dz=Uif{a),  ■ 

the  integral  on  the  left-hand  side  being  taken  round  z^f^z^z^,.     Accordingly 

i  z,z-a  ]z,z-a 

f(z) 
We  denote  by  g  {z),  so  that  g  {z)  is  a  function  which  has  one  pole  a 

in  the  region  considered. 

The  preceding  results  are  connected  only  with  the  simplest  form  of 
meromorphic  functions ;  other  simple  results  can  be  derived  by  means  of  the 
other  theorems  proved  in  §§  17 — 21.  Those  which  have  been  obtained  are 
sufficient  however  to  shew  that :  The  integral  of  a  meromovphic  function 
Jg  (z)  dz,  from  one  point  to  another  of  the  region  of  the  function,  is  not  in 
general  a  uniform  function.  The  value  of  the  integral  is  not  altered  by 
any  deformation  of  the  path  which  does  not  meet  or  cross  a  pole  of  the 

*  Poincare  uses  the  term  majorante. 


40  GENERAL  PROPOSITIONS  [23. 

function ;  but  the  value  is  altered  when  the  path  of  integration  is  so 
deformed  as  to  pass  over  one  or  more  poles.  Therefore  it  is  necessary  to 
specify  the  path  of  integration  when  the  subject  of  integration  is  a  mero- 
morphic  function ;  only  partial  deformations  of  the  path  of  integration  are 
possible  without  modifying  the  value  of  the  integral. 

24.  The  following  additional  propositions*  are  deduced  from  limiting 
cases  of  integration  round  complete  curves.  In  the  first,  the  curve  becomes 
indefinitely  small ;  in  the  second,  it  becomes  infinitely  large.  And  in  neither, 
are  the  properties  of  the  functions  to  be  integrated  limited  as  in  the  preceding 
propositions,  so  that  the  results  are  of  wider  application. 

I.  If  f{z)  he  a  function  which,  whatever  he  its  character  at  a,  has  no 
infinities  and  no  hranch-points  in  the  immediate  vicinity  of  a,  the  value  of 
Jf(z)  dz  taken  round  a  small  circle  with  its  centre  at  a  tends  towards  zero 
when  the  circle  diminishes  in  magnitude  so  as  ultimately  to  be  merely  the 
point  a,  provided  that,  as  \z  —  a\  diminishes  indefinitely,  the  limit  of  {z  -  a)f{z) 
tend  uniformly  to  zero. 

Along  the  small  circle,  initially  taken  to  be  of  radius  r,  let 


so  that 

and  therefore 

Hence 


z  —  a=  re 

ai 

=  idd, 

z  —  a 

ff{z)dz 

r27r 

=  i       {z  — 
Jo 

a)f(z) 

d0. 

Jf{z)dz\ 

JO 

«)/(^) 

dd 

r2n 
.  0 

a)f{^) 

de 

/•27r 

<     Mde 

Jo 

where  M'  is  the  greatest  value  of  M,  the  modulus  of  (z  —  a)f(z),  for  points 
on  the  circumference.  Since  {z  —  a)f{z)  tends  uniformly  to  the  limit  zero 
as  \z-a\  diminishes  indefinitely,  \ff{z)dz\  is  ultimately  zero.  Hence  the 
integral  itself  jf{z)  dz  is  zero,  under  the  assigned  conditions. 

Note.  If  the  integral  be  extended  over  only  part  of  the  circumference  of 
the  circle,  it  is  easy  to  see  that,  under  the  conditions  of  the  proposition, 
the  value  of  jf{z)  dz  still  tends  towards  zero. 

*  The  form  of  the  first  two  propositions,  which  is  adopted  here,  is  due  to  Jordan,  Cours 
d' Analyse,  t,  ii,  §  256. 


24.]  IN   INTEGRATION  41 

Corollary.  If  {z  —  a)f{z)  tend  uniformly  to  a  limit  k  as  \z  —  a\ 
diminishes  indefinitely,  the  value  of  ^f{z)dz  taken  round  a  small  circle,  centre 
a,  tends  towards  27rik  in  the  limit. 

r      ^2, 
Thus  the  value  of   I j ,  taken  round  a  very  small  circle  centre  a,  where  a  is 

]  {a?  -  s2)2 

not  the  origin,  is  zero:  the  value  of  / ^ round  the  same  circle  is  ^  ( -V. 

Neither  the  theorem  nor  the  corollary  will  apply  to  a   function,  such  as  sn , 

1 
which   has   the  point   a   for  an   essential   singularity :   the  value   of  (2  -  a)  sn ,  as 

|z-a|  diminishes  indefinitely,  does  not  tend  (§  13)  to  a  uniform  limit.     As  a  matter  of 

fact,  the  function  sn has  an  infinite  number  of  poles  in  the  immediate  vicinity  of  a 

as  the  limit  z=a\s,  being  reached. 

II.  Whatever  he  the  character  of  a  function  f  {z)  for  infinitely  large  values 
of  z,  the  value  of  jf{z)  dz,  taken  round  a  circle  tvith  the  origin  for  centre,  tends 
towards  zero  as  the  circle  becomes  infinitely  large,  provided  that,  as  \  z\ 
increases  indefinitely,  the  limit  of  zf{z)  tend  uniformly  to  zero. 

Along  a  circle,  centre  the  origin  and  radius  R,  we  have  z  =  Re^^,  so  that 

^  =  ide, 

z 

and  therefore  !f{^)  dz  =  i\     zfiz)  dO. 

Jo 


Hence  |  ^f{z)  dz  \  = 


zf{z)de 
<  r\zf(z)\de 

Jo 


0 
■2ir 

<  I     Mde 


where  M'  is  the  greatest  value  of  M,  the  modulus  of  zfiz),  for  points  on 
the  circumference.  When  R  increases  indefinitely,  the  value  of  M'  is  zero 
on  the  hypothesis  in  the  proposition  ;  hence  |  jf{z)  dz  \  is  ultimately  zero. 
Therefore  the  value  of  jf{z)  dz  tends  towards  zero,  under  the  assigned  con- 
ditions. 

Note.  If  the  integral  be  extended  along  only  a  portion  of  the  circumfer- 
ence, the  value  of  //(^)  dz  still  tends  towards  zero. 

Corollary.  If  zfiz)  tend  uniformly  to  a  limit  k  as  \z\  increases 
indefinitely,  the  value  of  jf{z)  dz,  taken  round  a  very  large  circle,  centre  the 
origin,  tends  towards  27rik. 

Thus  the  value  of  j  (1  -  z^)~idz  round  an  infinitely  large  circle,  centre  the  origin,  is  zero 
if  n  >  2,  and  is  £77  it  n  =  2. 


42  GENERAL   PROPOSITIONS  [24. 

III.  If  all  the  infinities  and  the  hranch-points  of  a  function  lie  in  a  finite 
region  of  the  z-plane,  then  the  value  of  jf{z)  dz  round  any  simple  curve,  which 
includes  all  those  points,  is  zero,  provided  the  value  of  zf{z),  as  \z\  increases 
indefinitely,  tends  uniformly  to  zero. 

The  simple  curve  can  be  deformed  continuously  into  the  infinite  circle 
of  the  preceding  proposition,  without  passing  over  any  infinity  or  any 
branch-point ;  hence,  if  we  assume  that  the  function  exists  all  over  the  plane, 
the  value  of  J  f(z)dz  is,  by  Cor.  I.  of  §  19,  equal  to  the  value  of  the  integral 
round  the  infinite  circle,  that  is,  by  the  preceding  proposition,  to  zero. 

Another  method  of  stating  the  proof  of  the  theorem  is  to  consider 
the  corresponding  simple  curve  on  Neumann's  sphere  (§  4).  The  surface 
of  the  sphere  is  divided  into  two  portions  by  the  curve*:  in  one  portion  lie 
all  the  singularities  and  the  branch-points,  and  in  the  other  portion  there  is 
no  critical  point  whatever.  Hence  in  this  second  portion  the  function  is  holo- 
morphic  ;  since  the  area  is  bounded  by  the  curve  we  see  that,  on  passing  back 
to  the  plane,  the  excluded  area  is  one  over  which  the  function  is  holomorphic. 
Hence,  by  §  19,  the  integral  round  the  curve  is  equal  to  the  integral  round 
an  infinite  circle  having  its  centre  at  the  origin  and  is  therefore  zero,  as 
before. 

Corollary.  If,  under  the  same  circumstances,  the  value  of  zf(z),  as 
I  z  I  increases  indefinitely,  tend  uniformly  to  k,  then  the  value  of  Jf(z)  dz  round 
the  simple  curve  is  lirik. 

Thus  the  value  of  / ^  along  any  simple  curve,  which  encloses  the  two  points 

J  {a^-z^)2  ^ 

a  and  —  a,  is  27r ;  the  value  of 

dz 


{(1-22)(1-F02)}4 


round  any  simple  curve  enclosing  the  four  points  1,  —1,  y,  —  y,  is  zero,  h  being  a  non- 


,  _i 


vanishing  constant;   and  the  value  of  \ {\  —  z'^'^)~ ^ dz,  taken  round  a  circle,  centre  the 
origin  and  radius  greater  than  unity,  is  zero  when  n  is  an  integer  greater  than  1. 

dz 


But  the  value  of 


i{(^-' 


■ei)(2-e2)(s-e3)]2 

round  any  circle,  which  has  the  origin  for  centre  and  includes  the  three  distinct  points 
gj,  62?  ^3,  is  not  zero.  The  subject  of  integration  has  3=oo  for  a  branch-point,  so  that  the 
condition  in  the  proposition  is  not  satisfied  ;  and  the  reason  that  the  result  is  no  longer 
valid  is  that  the  deformation  into  an  infinite  circle,  as  described  in  Cor.  I.  of  §  19, 
is  not  possible  because  the  infinite  circle  would  meet  the  branch-point  at  infinity. 

*  The  fact  that  a  single  path  of  integration  is  the  boundary  of  two  portions  of  the  surface 
of  the  sphere,  within  which  the  function  may  have  different  characteristic  properties,  will  be 
used  hereafter  {§  104)  to  obtain  a  relation  between  the  two  integrals  that  arise  according  as  the 
path  is  deformed  within  one  portion  or  within  the  other. 


25.]  EXAMPLES  48 

25.  The  further  consideration  of  integrals  of  functions,  that  do  not  possess 
the  character  of  uniformity  over  the  whole  area  included  by  the  curve  of  in- 
tegration, will  be  deferred  until  Chap.  IX.  Some  examples  of  the  theorems 
proved  in  the  present  chapter  will  now  be  given. 

Ex.  1.  It  is  sufficient  merely  to  mention  the  indefinite  integrals  (that  is,  integrals  from 
an  arbitrary  point  to  a  point  z)  of  rational  integral  functions  of  the  variable.  After  the 
preceding  explanations  it  is  evident  that  they  follow  the  same  laws  as  integrals  of  similar 
functions  of  real  variables. 

Ex.  2.     Consider  the  integral   /  ,  taken  round  a  simple  curve. 

When  n  is  0,  the  value  of  the  integral  is  zero  if  the  curve  do  not  include  the  point  a, 
and  it  is  2ni  if  the  curve  include  the  point  a. 

When  n  is,  &  positive  integer,  the  value  of  the  integral  is  zero  if  the  curve  do  not 
include  the  point  a  (by  §  17);  and  the  value  of  the  integral  is  still  zero  if  the  curve  do 
include  the  point  a  (by  §  22,  for  the  function  /(s)  of  the  text  is  1  and  all  its  derivatives 
are  zero).  Hence  the  value  of  the  integral  round  any  curve,  which  does  not  pass  through 
a,  is  zero. 

We  can  now  at  once  deduce,  by  §  20,  the  result  that,  if  a  holomorphic  function  be 
constant  along  any  simple  closed  curve  ivithin  its  region.,  it  is  constant  over  the  whole 
area  within  the  curve.  For  let  t  be  any  point  within  the  curve,  z  any  point  on  it,  and  C 
the  con.stant  value  of  the  function  for  all  the  points  z ;  then 


ZTTl   J    Z—l 


o-s^..  ;f^'&, 


■^W^a^'/i 


the  integral  being  taken  round  the  curve,  so  that 

dz 
't 

since  the  point  t  lies  within  the  curve. 

Ex.  3.     The  integral  - — .   I  /'  (2)  log  - — -  dz  is  taken  round  a  circle,  centre  the  origin 
ZtvZ  J  z  —  1 

and  radius  greater  than  unity ;  and  the  function  f{z)  is  holomorphic  everywhere  within 

the  circle.     Prove  that  the  value  of  the  integral  is 

Ex.  4.     Consider  the  integral  \e~^'^dz. 

In  any  finite  part  of  the  plane,  the  function  e"^  is  holomorphic;  therefore  (§  17)  the 
integral  round  the  boundary  of  a  rectangle 
(fig.  8),  bounded  by  the  lines  x=±a,  y=0, 
y  =  h,  is  zero :  and  this  boundary  can  be 
extended,  provided  the  deformation  remain 
in  the  region  where  the  function  is  holo- 
morphic. Now  as  a  tends  towards  infinity, 
the  modulus  of  e"^^,  being  e-^'^^v'^,  tends 
towards   zero   when    y  remains    finite ;    and  p.     g 

therefore    the    preceding    rectangle    can    be 

extended  towards  infinity  in  the  direction  of  the  axis  of  x,  the  side  h  of  the  rectangle 
remaining  unaltered. 


44  ExAMPLEtl  IN  [25. 

Along  A' A,  we  have  z=x:  so  that  the  value  of  the  integral  along  the  part  A' A  of  the 

boundary  is  I       e   ^"  dx. 
J  -a 

Along  AB,  we  have  z  =  a-\-iy,  so  that  the  value  of  the  integral  along  the  part  AB 
is  i  I    e-(«  +  *2')^<i?/. 

Along  BB',  we  have  z=x  +  ib,  so  that  the  value  of  the  integral  along  the  part  BB' 

is    /       e-(='  +  ^)"dx. 
J  a 

Along  B'A\  we  have  z=  —a  +  iy,  so  that  the  value  of  the  integral  along  the  part 
B'A'  is  i  I     e-(-«  +  '>)=(iy. 

The  second  of  these  portions  of  the  integral  is  e~"^ .  ■i .  I    e'-i'-'^'^y'-dy,  which  is  easily  seen 

jo 
to  be  zero  when  the  (real)  quantity  a  is  infinite. 

Similarly  the  fourth  of  these  portions  is  zero. 

Hence  as  the  complete  integral  is  zero,  we  have,  on  passing  to  the  limit, 

/•  00  /-co 

whence  ^''^  e~'^^-^'-'"=dx=  j      e-''^  dx  =  7r^, 

or  I       e"^'^  (cos  2bx  —  ^  sin  2bx)  dx=  iT^e~^'^ ; 

and  therefore,  on  equating  real  parts,  we  obtain  the  well-known  result 

e-^^  cos  'ihxdx  =  -n\e-^''. 


L 


This  is  only  one  of  numerous  examples*  in  which  the  theorems  in  the  text  can  be 
applied  to  obtain  the  values  of  definite  integrals  with  real  limits  and  real  variables. 

Ex.  5.  By  taking  the  integral  ^e~^''dz  along  the  perimeter  of  a  sector  of  a  circle 
between  the  radii  of  a  circle  given  ^  =  0,  6  =  \ir,  and  the  intercepted  part  of  the  circum- 
ference of  radius  r  which  is  ultimately  increased  without  limit,  establish  the  value 
(-|-7r)2  for  each  of  Fresnel's  integrals 


/: 


cos  vP'  du,       \      sin  ifi  dii. 
J  0 


Ex.  6.     Prove  that,  when  a^  +  b'^  <  1,  the  value  of  the  integral 


/, 


2i"  a  cos  x  +  13  sin  x  +  y  ^  ^ 


0     (X  cos  ^+6  sin  a- +1 

for  real  values  of  x  within  the  range,  is 

27r  f    _         aa  +  b^         ] 

(l"_  a2 _  52)i  V     (1  -  a2  -  ¥)i  +  V' 

*  See  Briot  and  Bouquet,  Theorie  des  fonctions  elliptiques,\  (2nd  ed.),  pp.  141  et  sqq.,  from 
which  examples  4  and  8  are  taken. 


25.]  CONTOUR   INTEGRATION  45 

Ex.  7.     Evaluate  the  following  integrals  by  the  process  of  contour  integration : — 

,.,        f  ■"  cos  ax  ^         ^  .  . 

(1;       I        TT, — ,,  ,  o — 7x  «-^5  where  a  is  real  ; 

....         /"  °°  cos  a^  —  COS  6x    -  ,....         /"  °°    eax_Qhx 

(u)     j^    ^^ c?^;  (m)     ^_^-^-^dx, 

where  a  and  h  are  real  and  lie  between  0  and  1 ; 

r— — -  c?a7,    where    0  <  a  <  1. 
_ao  1-f  e* 

r  5.11—1 
^jj;.  8.     Consider  the  integral  /  ^ —  dz.,  where  n  is  a  real  positive  quantity  less  than 

unity. 

The  only  infinities  of  the  subject  of  integration  are  the  origin  and  the  point  - 1 ; 
the  branch-points  are  the  origin  and  ^  =  00 .  Everywhere  else  in  the  plane  the  function- 
behaves  like  a  holomorphic  function ;  and,  therefore,  when  we  take  any  simple  closed 
curve  enclosing  neither  the  origin  nor  the  point  —1,  the  integral  of  the  function  round 
that  curve  is  zero. 

Choose  the  curve,  so  that  it  lies  on  the  positive  side  of  the  axis  of  x  and  that  it  is 
made  up  of  : — 

(i)    a  semicircle  C3  (fig.  9),  centre  the  origin  and  radius  R  which  is  made  to  increase 
indefinitely  : 

(ii)    two  semicircles,  Cj  and  C2,  with  their  centres  at  0  and  —1  respectively,  and  with 
radii  r  and  r',  which  ultimately  are  made  infinitesimally  small : 

(iii)  the  diameter  of  C3  along  the  axis  of  x  excepting  those  ultimately  infinitesimal 
portions  which  are  the  diameters  of  c^  and  of  c^. 

The  subject  of  integration  is  uniform  within  the  area  thus  enclosed  although  it 
is  not  uniform  over  the  whole  plane.  We  shall  take  that  value  of  2"-i  which  has  its 
argument  equal  to  {n-\)6,  where  6  is  the  argument  of  z. 


Fig.  9. 

The  integral  round  the  boundary  is  made  up  of  four  parts. 

(a)     The  integral  round  C3.     The  value  of  z.  j-^,  as  1 2 1  increases  indefinitely,  tends 

uniformly  to  the  limit  zero  ;  hence,  as  the  radius  of  the  semicircle  is  increased  indefinitely, 
the  integral  round  G^  vanishes  (§  24,  il.,  Note). 

^n  —  l 

(6)     The  integral  round  Ci.     The  value  of  z.  ^--— ,  as  |  2  |  diminishes  indefinitely, 

tends  uniformly  to  the  limit  zero;  hence  as  the  radius  of  the  semicircle  is  diminished 
indefinitely,  the  integral  round  q  vanishes  (§  24,  i..  Note):  ; 


46  EXAMPLES   IN  [25. 

(c)  The  integral  round  03.  The  value  of  (1  +  z)  —-  ,  as  j  1+  z  |  diminishes  indefinitely 
for  points  in  the  area,  tends  uniformly  to  the  Hmit  (-l)»-i,  i.e.,  to  the  limit  e^'^~'^>''^. 
Hence  this  part  of  the  integral  is 

In-\)m    i _^±_ 

being  taken  in  the  direction  indicated  by  the  arrow  round  c^,  the  infinitesimal  semicircle. 
Evidently  — —  =id6  and  the  limits  are  tt  to  0,  so  that  this  part  of  the  whole  integral  is 


=  nre 


ymi 


id)     The  integral  along  the  axis  of  x.     The  parts  at  - 1  and  at  0  which  form  the 
diameters  of  the  small  semicircles  are  to  be  omitted  ;  so  that  the  value  is 


This  is  what  Cauchy  calls  the  principal  value  of  the  integral 


dx. 


Since  the  whole  integral  is  zero,  we  have 


f 


.»-i 


^•7re'*'^'+  I       f— -  dx  =  0. 

ol+^ 


-1  /"O 


and  Q  =  \^  fzi^^'^ 

principal  values  being  taken  in  each  case.    Then,  taking  account  of  the  arguments,  we  have 

/•"(-^A'i-i  ,-,,„,  f'^x"'-'^dx 

jo      1--^  Jo      1-^ 


_g(»-l)7ri 


Since  i'n-e^'''+ P  +  P'  =  0,  we  have 

so  that 

P—Q  cos  mr  —  TT  sin  mr,     Q  sm  nir  =  tt  cos  mr. 

Hence  |     ^f--dx=P=7rcosecn7r, 

J  0  ^  +* 

/' ""  x^  ~  ^ 
dx  =  Q=  IT  cot  nir. 
0  1-^ 


'-    Ex.  9.     In  the  same  way  it  may  be  proved  that 


/: 


dx=  -^  —  2  0)'"^    'e 


.    IT 
t 

where  n  is  an  integer,  a  is  positive  and  w  is  e  ^w. 


25.]  CONTOUR   INTEGRATION  47 

Ex.  10.  By  considering  the  integral  Je-^2»-i(^2  round  the  contour  of  the  sector  of  a 
circle  of  radius  r,  bounded  by  the  radii  (9  =  0,  ^  =  a,  where  a  is  less  than  Jtt  and  n  is  positive, 
it  may  be  proved  that 

/•OO 

i      K-le~''''°'"cos(/3  +  rsina)}c^r  =  r(%)cos(i3  +  ?ia), 
on  proceeding  to  the  limit  when  r  is  made  infinite.  (Briot  and  Bouquet.) 

JE^x.  11.  By  considering  the  integral  j{z^  -  l)^  z''^^-^-'^  dz,  taken  round  a  semicircle, 
prove  that 

r  sin»  0  e^^  dd  =  ^   .    ,,    -^f  n(m) 

provided  the  real  part  of  m  is  greater  than  -  1. 
Similarly  deduce  the  value  of 

/    sin™  ^  cos™  (9  e"*^  c?<9, 

where  the  real  parts  of  m  and  n  are  each  greater  than  - 1,  from  a  consideration  of  the 
integral 

taken  round  a  semicircle. 

(Many  of  the  results  stated  in  de  Haan,  Nouoelles  tables  dHntegrales  definies,  can  be 
obtained  in  a  similar  manner.) 

r   dz 
Ex.  12.     Consider  the  integral  I  -^ — - ,  where  n  is  an  integer.    The  subject  of  integration 

is  meroraorphic ;  it  has  for  its  poles  (each  of  which  is  simple)  the  n  points  w''  for  r=0, 
1,  ...,  TO  —  1,  where  w  is  a  primitive  ?ith  root  of  unity ;  and  it  has  no  other  infinities  and  no 

branch-points.     Moreover  the  value  of  -^ — - ,  as  |  ^  |  increases  indefinitely,  tends  uniformly 

to  the  limit  zero  ;  hence  (§  24,  iii.)  the  value  of  the  integral,  taken  round  a  circle  centre 
the  origin  and  radius  >  1,  is  zero. 

This  result  can  be  derived  by  means  of  Corollary  II.  in  §  19.  Surround  each  of  the 
poles  with  an  infinitesimal  circle  having  the  pole  for  centre ;  then  the  integral  round 
the  circle  of  radius  >  1  is  equal  to  the  sum  of  the  values  of  the  integral  round  the 
infinitesimal  circles.     The  value  round  the  circle  having  aT  for  its  centre  is,  by  §  20, 


Ittx  (limit  of -^^^ — -  ,  when  0=0)'"  j 


27r^■ 
=  —  < 

n 

Hence  the  integral  round  the  large  circle 

2W-1 


2    0)™-'" 
n    r=0 


=0. 


-^-— -^  c?2,  taken  round  a  semicircle,  prove  that 
/■*    cosao;  ,       TT    .  /""    sin  a.r  tt 


provided  a  is  positive. 


48  EXAMPLES   IN  [25. 

Ex.  14.     Taking  as  the  definition  of  Bernoulli's  numbers  that  they  are  the  coefficients 
in  the  expansion 

^        _£_!_£_     y     I  _-\  \m-l      "»'      r,,2m-\ 

e^-1      x^2~„%i^      ^        {2m)  I 

prove  (by  contour  integration)  that 

_2(2m)  !   °°      1 
™~(27r)2'«  ,ill^' 

In  the  same  way,  obtain  expressions  for  the  coefiicients,  in  the  expansion  in  powers  of  x, 
of  the  quantity 

gxy 


e^-1 

(Hermite.) 

Ex.   15.     In  all  the  preceding  examples,  the  poles   that  have   occurred  have  been 

simple  :  but  the  results  proved  in   §  21  enable  us  to  obtain  the  integrals  of  functions 

which    have   multiple   poles  within   an   area.      As   an   instance,    consider  the   integral 

dz 

— ^  round  any  curve  which  includes  the  point  i  but  not  the  point   —  ^,  these 


/: 


(1+22)" 

points  being  the  two  poles  of  the  subject  of  integration,  each  of  multiplicity  n  +  1. 


We  have  seen  that  /(»)  (a)  =  -^.     ,    ■'  \i^,  dz 


where  /  (s)  is  holomorphic  throughout  the  region  bounded  by  the  curve  round  which  the 
integral  is  taken. 

In  the  present  case  a  is  i,  and  f(z)  =  -, r-— -r  ;  so  that 


nl  (2  +  i)2«  +  i' 


and  therefore 
Hence  we  have 


/ 


dz  2ni  .,,,..        2n\    it 

(H-22^"  +  i      n\-'      ^'     n\7i,\2^^' 


In  the  case  of  the  integral  of  a  function  round  a  simple  curve  which  contains  several 
of  its  poles,  we  first  (§  20)  resolve  the  integral  into  the  sum  of  the  integrals  round  simple 
curves  each  containing  only  one  of  the  points,  and  then  determine  each  of  the  latter 
integrals  as  above. 

Another  method,  that  is  sometimes  possible,  makes  use  of  the  expression  of  the  uniform 
function  in  partial  fractions.  After  Ex.  2,  we  need  retain  only  those  fractions  which  are  of 
the  form  A/{z  —  a) :  the  integral  of  such  a  fraction  is  2TriA,  and  the  value  of  the  whole 
integral  is  therefore  27ri2A.  It  is  thus  sufficient  to  obtain  the  coefficients  of  the  inverse 
first  powers  which  arise  when  the  function  is  expressed  in  partial  fractions  corresponding 

to  each  pole.     Such  a  coefficient  A,  being  the  coefficient  of  in  the  expansion  of  the 

function,  is  called  by  Cauchy  the  residue  of  the  function  relative  to  the  point. 


For  example, 

1  „  f    1 


(23+1)2         ■•>    [^+1         2  +  ^0^2  +  0)2/^9    1(2+1)'        (2  +  to)2  ^  (2  +  0)2)-'/ ' 


h 


25.]  CONTOUR   INTEGRATION  49 

so  that  the  residues  relative  to  the  points  —1,  -co,  —a>^  are  |,  fw,  fw^  respectively. 
Hence  if  we  take  a  semicircle,  of  radius  >1  and  centre  the  origin  with  its  diameter 
along  the  axis  of  y,  so  as  to  lie  on  the  positive  side  of  the  axis  of  y,  the  area  between  the 
semi-circumference  and  the  diameter  includes  the  two  points  -  a>  and  —  m^ ;  and  therefore 
the  value  of 

dz 

taken  along  the  semi-circumference  and  the  diameter,  is 

that  is,  the  value  is  -  ^  tti. 

Ex.   16.      Let  ?f  denote    j         I      -^  , '   J  dzdz',  f  being  a  rational  mtegral  function 
J  (C)  j  (C)  22  -  1  *  ^ 

2vlm„2™s'''  of  the  complex  variables  z,  z',  the  integrations  being  taken  in  the  positive  sense 

round  the  closed  contours  C,  C,  of  which  C  is  a  circle  of  unit  radius  with  its  centre 

at  the  origin.     Shew  that  «  =  0  if  C"  lies  wholly  inside  C,  or  if  C  and  C  lie  wholly  outside 

one  another,  and  that  ii=  —4Tr^'SAnim  (w=0,  1,  2,  ...)  if  C  completely  surrounds   C. 

Discuss  also  the  value  of  u  if  C  is  a  circle  passing  through  the  points   ±i  but  not 

coinciding  with  C,  ?ind  f{z,  z')=f{-z,  —z'). 

(Math.  Trip.,  Part  II.,  1898.) 

Note.  For  further  applications  of  Cauchy's  theory  of  residues,  together  with  many 
references  to  Cauchy's  own  results,  Lindelof 's  monograph  Ze  calcul  des  residus  (Gauthier- 
ViUars,  1905)  may  be  consulted. 


F.  F, 


CHAPTER  III. 


Expansion  of  Functions  in  Series  of  Powers. 


26;  We  are  now  in  a  position  to  obtain  the  two  fundamental  theorems 
relating  to  the  expansion  of  functions  in  series  of  powers  of  the  variable : 
they  are  due  to  Cauchy  and  Laurent  respectively. 

Cauchy's  theorem  is  as  follows*  : — 

When  a  function  is  holomorphic  over  the  area  of  a  circle  of  centre  a,  it  can 
he  expanded  as  a  series  of  positive  integral  powers  of  z  —  a,  converging  for  all 
points  within  the  circle. 

Let  z  be  any  point  within  the  circle ;  describe  a  concentric  circle  of 
radius  r  such  that 

\z  —  a\  =  p  <r<R, 

where  R  is  the  radius  of  the  given  circle.  If  t 
denote  a  current  point  on  the  circumference  of  the 
new  circle,  we  have 


■^  ^  '      27rijt  —  z 


j_r/(o 

27riJ  t  —  a 


dt 


1  - 


a 


t  —  a 


Fig.   10. 


the  integral  extending  along  the  whole  circumference  of  radius  r.     Now 

,\n+i 


=  1  + 


1  - 


t  —  a 


a      fz  —  a\ 


t  —  a 


t  —  a 


\t-a/ 
+  . 


t  —  a 
so  that,  by  §  15  (III.),  we  have 

27n  J  t  —  a  2'7Ti  J  (t  —  a)- 


+ 


z  —  a 
t  —  a 


(z-ari    /(o 


i  j(t- 


277*     J  (t  —  a) 


dt 


+ 


Ljmf^-j'Y^'dt. 

lirij  t  —  z  \t  —  aj 


*  Exercices  d' Analyse  et  de  Physique  Mathematique,  t.  ii,  pp.  50  et  seq.;  the  memoir  was  first 
made  public  at  Turin  in  1832. 


26.]  cauchy's  theorem  on  the  expansion  of  a  function  51 

Now f(t)  is  holomorphic  over  the  whole  area  of  the  circle;  hence,  if  t  be 
not  actually  on  the  boundary  of  the  region  (§§21,  22),  a  condition  secured  by 
the  hypothesis  r  <  R,  we  have 


and  therefore 


f(^)=f(a)  +  i.-a)f(a)  + +  ^fZl^/ <.  (a)  +  ^-i^^)""  f/CO        dt 


w!  27rt      j  t  -  2  (t  -  a)'^+^' 

Let  the  last  term  be  denoted  by  L.     Since  \2—  a\=  p  and  \t  —  a\  =  r; 

it  is  at  once  evident  that  \t-z\^7'  —  p.     Let  M  be  the  greatest  value  of 

\f(t)  I  for  points  along  the  circle  of  radius  r ;  then  M  must  be  finite,  owing  to 

the  initial  hypothesis  relating  to/(^).     Taking 

t  —  a  =^  re^*, 
so  that  dt  =  i  (t  —  a)  dO, 


p'^+^ 
we  have  1^1  =  ^ — 

'  ZTT 


fit)    dd 


0  t-zit-aj' 


<o   ^TT — A  \f{t)\de 


n+i 

M 


-f-n  ^,,  _  p^ 


0 


-r^(i-? 


Now  r  was  chosen  to  be  greater  than  p ;   as  h  becomes  infinitely  large, 

i-\       becomes  infinitesimally  small.     Also  ilf  (1  —  ^j     is  finite.     Hence  as 

n  increases  indefinitely,  the  limit  of  j  Z  | ,  necessarily  not  negative,  is  in- 
finitesimally small  and  therefore,  in  the  same  case,  L  tends  towards  zero. 

It  thus  appears,  exactly  as  in  §  15  (V.),  that,  when  n  is  made  to  increase 
without  limit,  the  difference  between  the  quantity y"(^)  and  the  first  n+\ 
terms  of  the  series  is  ultimately  zero  ;  hence  the  series  is  a  converging  series 
having/ (2;)  as  the  limit  of  the  sum,  so  that 

f{z)=f{a)-v{z-a)f'{a)  +  ^-^^f"ia)+ +  ^-i^/<'^)  (a)+ , 


which  proves  the  proposition  under  the  assigned  conditions.      It  is  the  form 
of  Taylor's  expansion  for  complex  variables. 

Note.  A  series,  such  as  that  on  the  right-hand  side  and  not  necessarily 
arising  through  the  expansion  of  a  given  function  /  (2^),  is  frequently  denoted 
by  P  (^^  —  a),  where  P  is  a  general  symbol  for  a  converging  series  of  positive 
integral  powers  of  ^  —  a  :  it  is  also  sometimes*  denoted  by  P  (^  |  a).  Con- 
formably with  this  notation,  a  series  of  negative  integral  powers  of  ^  —  a 

*  Weierstrass,  Ges.  Werke,  t.  ii,  p.  77. 

4—2 


52  cauchy's  theokem  on  the  expansion  of  a  function  [26. 

would  be  denoted  by  P  ( j :  a  series  of  negative  integral  powers  of  z 

either  by  -P  (- )  or  by  P{z\'Xi ),  the  latter  implying  a  series  proceeding  in 

positive   integral    powers   of  a  quantity  which  vanishes  when  z  is  infinite, 
that  is,  in  positive  integral  powers  of  z~^. 

If,  however,  the  circle  can  be  made  of  infinitely  great  radius  so  that  the 
function  f[z)  is  holomorphic  over  the  finite  part  of  the  plane,  the  equivalent 
series  is  denoted  by  G  {z  —  a),  and  it  converges  over  the  whole  plane*. 
Conformably  with  this  notation,  a  series  of  negative  integral  powers  of  z  —  a 

which  converges  over  the  whole  plane  is  denoted  by  G  i j . 

£Jx.     If  the  expansion,  taken  in  the  form  ao  +  «i2  +  «22^+  •••  j  be  vahd  over  the  whole  of 
the  finite  part  of  the  plane,  then  the  limit  of 


1^ 


as  m  increases  indefinitely,  is  zero.     More  generally,  if  the  circle  of  convergence  of  the 
series  be  of  radius  r,  then  the  limit  of  the  preceding  quantity  is  1/r.  (Cauchy.) 

27.  The  following  remarks  on  the  proof  and  on  inferences  from  it  should 
be  noted. 

(i)  In  order  that  (t  —  z)~^  may  be  expanded  in  the  required  form,  the 
point  z  must  be  taken  actually  within  the  area  of  the  circle  of  radius  R; 
and  therefore  the  convergence  of  the  series  F  {z  —  a)  is  not  established  for 
points  on  the  circumference. 

(ii)  The  coefficients  of  the  powers  of  ^  —  a  in  the  series  are  the 
values  of  the  function  and  its  derivatives  at  the  centre  of  the  circle ;  and  the 
character  of  the  derivatives  is  sufficiently  ensured  (§  21)  by  the  holomorphic 
character  of  the  function  for  all  points  within  the  region.  It  therefore 
follows  that,  if  a  function  be  holomorphic  within  a  region  bounded  by  a 
circle  of  centre  a,  its  expansion  in  a  series  of  ascending  powers  of  z  —  a, 
which  converges  for  all  points  within  the  circle,  depends  only  upon  the  values 
of  the  function  and  its  derivatives  at  the  centre. 

Conversely,  a  converging  power-series  in  z  —  a,  having  assigned 
coefficients  f(a),  f  (a),  •  •  • ,  defines  a  uniform  function  within  the  radius 
of  convergence  of  the  series. 

But  instead  of  having  the  values  of  the  function  and  of  all  its  derivatives 
at  the  centre  of  the  circle,  it  will  suffice  to  have  the  values  of  the  holomorphic 
function  itself  over  any  region  at  a  or  along  any  line  through  a,  the  region 
or  the  line  being  not  merely  a  point.  The  values  of  the  derivatives  at  a  can 
be  found  in  either  case  ;  for  / '  (6)  is  the  limit  of  [fyb  +  hh)  -f{b)}/Sb,  so  that 
the  value  of  the  first  derivative  can  be  found  for  any  point  in  the  region  or 
on  the  line,  as  the  case  may  be ;  and  so  for  all  the  derivatives  in  succession. 
*  It  then  is  often  called  an  integral  function. 


27.]  DARBOUX'S   EXPRESSION  53 

(iii)  The  form  of  Maclaurin's  series  for  complex  variables  is  at  once 
derivable  by  supposing  the  centre  of  the  circle  at  the  origin.  We  then 
infer  that,  if  a  function  he  holomorphic  over  a  circle,  centre  the  origin,  it  can  he 
represented  in  the  form  of  a  series  of  ascending,  positive,  integral  powers  of  the 
variable  given  hy 

/(^)=/(o)  +  ^/'(o)  +  |:,/"(o)  +  ..., 

where  the  coefficients  of  the  various  powers  of  z  are  the  values  of  the  derivatives 
of  f{z)  at  the  origin;  and  the  series  converges  for  all  points  loithin  the  circle. 

Thus,  the  function  e^  is  holomorphic  over  the  finite  part  of  the  plane ; 
therefore  its  expansion  is  of  the  form  G{z).  The  function  log  (1  +z)  has  a 
singularity  at  —  1  :  hence  within  a  circle,  centre  the  origin  and  radius  unity, 
it  can  be  expanded  in  the  form  of  an  ascending  series  of  positive  integral 
powers  of  z,  it  being  convenient  to  choose  that  one  of  the  values  of  the 
function  which  is  zero  at  the  origin.  Again,  tan~^  z-  has  singularities  at  the 
four  points  ^  =  —  1,  which  lie  on  the  same  circumference ;  choosing  the  value 
at  the  origin  which  is  zero  there,  we  have  a  similar  expansion  in  a  series, 
converging  for  points  within  the  circle. 

Similarly  for  the  function  (1  +  ^)",  which  has  —  1  for  a  singularity  unless 
n\B  Q.  positive  integer. 

(iv)  Darboux's  method*  of  derivation  of  the  expansion  of  f{z)  in 
positive  powers  oi  z  —  a  depends  upon  the  expression,  obtained  in  §  15  (IV.), 
for  the  value  of  an  integral.     When  applied  to  the  general  term 


27riJ\t- 


=  L  say,  it  gives  L  =  \r  (^^)      fi^X 

where  ^  is  some  point  on  the  circumference  of  the  circle  of  radius  r,  and  X  is 

a  complex  quantity  of  modulus  not  greater  than  unity.    The  modulus  of  y 

is  less  than  a  quantity  which  is  less  than  unity ;  the  terms  of  the  series  of 
moduli  are  therefore  less  than  the  terms  of  a  converging  geometric  progress- 
ion, so  that  they  form  a  converging  series ;  the  limit  of  \L\,  and  therefore 
of  L,  can,  with  indefinite  increase  of  n,  be  made  zero  and  Taylor's  expansion 
can  be  derived  as  before. 

Ex.  1.    Prove  that  the  arithmetic  mean  of  all  values  ofz~'^  2  a^z",  for  points  lying  along 

v  =  0 

a  circle  j  2  |  =?'  entirely  contained  in  the  region  of  continuity,  is  a„.    (Rouche,  Gutzmer.) 

Prove  also  that  the  arithmetic  mean  of  the  squares  of  the  moduli  of  all  values  of 
00 
2  avZ",  for  points  lying  along  a  circle  \z\—r  entirely  contained  in  the  region  of  continuity, 

v=0 

is  equal  to  the  sum  of  the  squares  of  the  moduli  of  the  terms  of  the  series  for  a  point  on 
the  circle.  (Gutzmer.) 

*  Liouville,  3*"»«  S^r.,  t.  ii,  (1876),  pp.  291—312. 


54  Laurent's  expansion  of  [27. 

Ex.  2.     Prove  that  the  function  2  a«0»", 

w  =  0 

is  finite  and  continuous,  as  well  as  all  its  derivatives,  within  and  on  the  boundary  of  the 
circle  |  ^  |  =  1,  provided  |  a  |  <  1.  (Fredholm.) 

Ex.  3.     The  radii  of  convergence  of  the  series 

f{z)=aQ+aiZ  +  a2Z^  +  ...,      g  {z')  =  bQ  +  biz' ■\-h<i^^  + ..., 
are  p  and  p  ;  prove  that  pp  is  the  radius  of  convergence  of  the  series 

h  {z")  =  ao^o  +  «i&is"  +  a^o,^''^  +  . . . . 
Denoting  the  singularities  of  /(z)  by  Sj,  Sg,...,  and  those  of  g  {^)  by  s{,  s^,...,  prove 
that  the  singularities  of  h  (s")  are  given  by  s^s„',  for  all  values  of  m  and  n.     (Hadamard.) 

Ex.  4.     (See  also  Ex.  2,  §  20.)     It  is  possible  to  express  the  sum  of  selected  terms 
in  the  form  of  a  definite  integral.     Thus,  writing 

for  m  =  \,  2,...,  consider  the  finite  series 

^=Co  +  Ci(z-a)  +  ...+c„(s-a)'^ 


^J^i{^---...rr-^Yu< 


t  —  a  \t  —  aj 


z  —  a 
a 


z  -  a 
t  —  a 


■dt 


=j_  [M  _jii 

•'  1  — 

2in  J  t-z  \       \t-aj      J 

■'  ^  '  2tti  J  t-z  \t-aj 

Ex.  5      Establish  the  following  results  in  a  similar  manner  :- 

(i)     Cp(2-a)P  +  Cp  +  i(z-a)P  +  i  +  ...+Cg(3-a)9 


^Mm-(:-^r'>''' 


(ii)     CQ  +  C2{z-af  +  Ci{z-aY  +  .,. 

ziri  J  {t-z){t  +  z-'ia) 


fm^___a,^ 


(iii)     Ci  +  C3{z-aY  +  Cs{z-ay+:.. 


(t-z)  {t+z-2a) 

28.     Laurent's  theorem  is  as  follows*  : — 

A  function,  which  is  holomorphic  in  a  part  of  the  plane  bounded  hy  two 
concentric  circles  with  centre  a  and  finite  radii,  can  be  expanded  in  the  form 
of  a  double  series  of  integral  poiuers,  positive  and  negative,  of  z—  a;  and  the 
series  converges  in  the  part  of  the  plane  between  the  circles. 

*  Comptes  Rendus,  t.  xvii,  (1843),  p.  939. 


28.3; 


A   FUNCTION   IN   SERIES 


5§' 


Let  z  be  any  point  within  the  region  bounded  l)y  the  two  circles  of  radii/ 
U  and  R'  \    describe   two   concentric   circles   of 
radii  r  and  r' ,  such  that 

R>r>  \z—  a\>r'  >  R. 
Denoting  by  t  and  by  6-  current  points  on  the 
circumference  of  the  outer  and  of  the  inner 
circles  respectively,  ^nd  considering  the  space 
which  lies  between  them  and  includes  the  point 
z,  we  have,  by  §  20, 


fi^)=^mjt-^-m<is...(,. 


Fig-  11. 


27rijt  —  z  2'7ri's  —  z 

a  negative  sign  being  prefixed  to  the  second  integral  because  the  direction 
indicated  in  the  figure  is  the  negative  direction  for  the  description  of  the 
inner  circle  regarded  as  a  portion  of  the  boundary. 

Now  we  have  ,         :  i. 

^z  -  ay*+i 


t 


t-z 


a     ^      z 
=  1  + 


t  —  a 


a      I  z  —  a 
+ 


t  —  a 


+ 


+ 


fz  -  gV 
\t-a) 


+ 


t—a 
this  expansion  being  adopted  with  a  view  to  an  infinite  converging  series, 

is  less  than  unity  for  all  points  t;   and  hence,  by  §  15, 


because 


t  —  a 


t-z 


dt 


fit) 


^^l^at.iz-a^\J^d,^......,iz-arj^^^ 

^mi^r^dt.         :■  ■  'i: 

H  —  z\t  —  a)  •  :.      -  \ 


dt 


t  —  z\t  —  a^ 

Now  each  of  the  integrals,  which  are  the  respective  coefficients  of  powers  of 
^— a,  is  finite,  because  the  subject  of  integration  is  everywhere  finite. along 
the  circle  of  finite  radius,  by  §  15  (IV.).     Let  the  value  of 

be  271^'^,. :  the  quantity  Ur  is  not  necessarily  equal  to  f^^^(a)  -=-  r  !,  because  no 
knowledge  of  the  function  or  of  its  derivatives  is  given  for  a  point  within 
the  innermost  circle  of  radius  R'.     Thus 


— .  \f^dt  =  Uo  +  (z  -a)ui  +  (z  —  ayu^+  ,.,..,  +(^z -ay^\ 
'/(t)  fz  -  aY+'' 


27r' 


27ri  Jt  —  z  \t  —  a 
The  modulus  of  the  last  term  is  less  than 


dt. 


l-P  ^^^ 


66  Laurent's  expansion  of  [28. 

where  p  is  |^  — a|  and  M  is  the  greatest  value  of  \f{t)\  for  points  along 
the  circle.  Because  p  <  r,  this  quantity  diminishes  to  zero  with  indefinite 
increase  of  n ;   and  therefore  the  modulus  of  the  expression 

1     [fit) 


27ri 


1^-^  dt  —  Uo  —  {z  —  a)ui  — —  (z-  dY  Ur 


becomes  indefinitely  small  with  unlimited  increase  of  n.  The  quantity  itself 
therefore  vanishes  in  the  same  circumstances  ;   and  hence 

^r-~.  \{^dt  =  Uo  +  (z-a)ih  + +{z  -a)'^u,n+ , 

ZTTl  }t—  Z 

so  that  the  first  of  the  integrals  is  equal,  to  a  series  of  positive  powers.  This 
series  converges  within  the  outer  circle,  for  the  modulus  of  the  {m  +  1)*'^  term 
is  less  than 

which  is  the  (m  +  1)*^^  term  of  a  converging  series. 

As  in  §  27,  the  equivalence  of  the  integral  and  the  series  can  be  affirmed 
only  for  points  which  lie  within  the  outermost  circle  of  radius  R. 

Again,  we  have 

z  —  a     ^     s  —  a                   /s  —  aV^ 
=1  + -  + +    + 

s  —  z  z  —  a  \z  —  a/ 

1  — 

z  —  a 

this  expansion  being  adopted  with  a  view  to  an  infinite  converging  series, 


(' 

-a^ 

11+1 

u 

—  a 

I 

1  - 

s  - 

-  a 

because 


s  —  a 
z  —  a 


is  less  than  unity  for  all  points  s.     Hence 


-2s/0;*=.-^2s//<^)'^^+ +(;i=2s/(^ -">"/(») 


+  J-.  ffi^r*' /w  ds 

2iri  j\z  —  aJ       z  —  s 


The  modulus  of  the  last  term  is  less  than 

M'    /rY+' 


p 

where  M'  is  the  greatest  value  of  \f{s)  \  for  points  along  the  circle  of  radius 
/.  With  unlimited  increase  of  n,  the  modulus  of  this  last  term  is  ultimately 
zero  ;  and  thus,  by  an  argument  similar  to  the  one  which  was  applied  to  the 
former  integral,  we  have 

liriU-z  z-a      {z-af^ {z-a)^^ ' 

where  Vm  denotes  the  integral  J  (s  —  a)™~^/(s)  ds  taken  round  the  circle. 


28.]  A   FUNCTION   IN   SERIES  57 

As  in  the  former  case,  the  series  is  one  which  converges,  its  converg- 
ence being  without  the  inner  circle;  the  equivalence  of  the  integral  and 
the  series  is  valid  only  for  points  z  that  lie  without  the  innermost  circle  of 
radius  K . 


The  coefficients  of  the  various  negative  powers  ol  z—a  are  of  the  form 

\s-a) 


^i  f^T^  ^ 


(s  -  a)"* 

a  form  that  suggests  values  of  the  derivatives  of  f(s)  at  the  point  given  by 

-— —  =  0,  that  is,  at  infinity.     But  the  outermost  circle  is  of  finite  radius ; 

and  no  knowledge  of  the  function  at  infinity,  lying  without  the  circle,  is 
given,  so  that  the  coefficients  of  the  negative  powers  may  not  be  assumed 
to  be  the  values  of  the  derivatives  at  infinity,  just  as,  in  the  former  case,  the 
coefficients  m,.  could  not  be  assumed  to  be  the  values  of  the  derivatives  at  the 
common  centre  of  the  circles. 

Combining  the  expressions  obtained  for  the  two  integrals,  we  have 

f(z)  =  Uo  +  (z-a)  ?ii  +  {z  —  of  Uz+  ... 

-\-{z-a)-^v^  +  {z-a)-^v^-V  .... 

Both  parts  of  the  double  series  converge  for  all  points  in  the  region  between 
the  two  circles,  though  not  necessarily  for  points  on  the  boundary  of  the 
region.  The  whole  series  therefore  converges  for  all  those  points :  and  we 
infer  the  theorem  as  enunciated. 

Conformably  with  the  notation  (§  26,  Note)  adopted  to  represent  Taylor's 
expansion,  a  function  f{z)  of  the  character  required  by  Laurent's  Theorem 
can  be  represented  in  the  form 


P,{z-a)  +  FJ-^) 


the  series  Pa  converging  within  the  outer  circle  and  the  series  Pg  converging 
without  the  inner  circle ;  their  sum  converges  for  the  ring-space  between  the 
circles. 

29.     The  coefficient  Uo  in  the  foregoing  expansion  is 

27^^•  ]*-«"**' 
the  integral  being  taken  round  the  circle  of  radius  r.     We  have 

t—  a 


58  ;  iAURENT'S   THEOREM  .  [29. 

for  points  on  the  circle  ;  arid  therefore 

so  that  \ua\<  \  ^—Mt<  M', 

J  Lit 

M'  being  the  greatest  value  of  Mt,  the  modulus  of  f(t),  for  points  along  the 
circle.  If  M  be  the  greatest  value  of  [/(-S")  |  for  any  point  in  the  whole 
region  in  which  f(z)  is  defined,  so  that  M'  <  M,  then  we  have 

I  Uo  I  <  M, 
that  is,  the  modulus  of  the  term  independent  of  ^^  —  a  in  the  expansion  of 
f(z)  by  Laurent's  Theorem  is  less  than  the  greatest  value  of  \f(z)  \  at  points 
in  the  region  in  which  it  is  defined. 

Again,  (z  —  a)~'^f{z)  is  a  double  series  in  positive  and  negative  powers  of 
z  —  a.  the  term  independent  of  z  —  a  being  /^,„  ;  hence,  by  what  has  just  been 
proved,  j  m,,^  |  is  less  than  p~™M,  where  p  is  \z—  a\.  But  the  coefficient  Um 
does  not  involve  z,  and  for  any  point  z  we  can  therefore  choose  a  limit.  The 
lowest  limit  will  evidently  be  given  by  taking  z  on  the  outer  circle  of  radius 
R,  so  that  \t(>in\<  MR~^.  Similarly  for  each  coefficient  v^^ ;  and  therefore  we 
have  the  result : — 

If  fiz)  he  expanded  as  hy  Laurent's  Theorem  in  the  form 

00  00 

m=l  m  =  l 

then  .  \um\<MR-'^,     \v„i\<MR''^, 

luhere  M  is  the  greatest  value  of  \f{z)\  at  points  within  the  region  in  which 
f{z)  is  defined,  and  R  and  R'  are  the  r-adii  of  the  outer  and  the  inner  circles 
respectively.  ■ 

Corollary.  If  M(r)  denote  the  greatest  value  of  \f{z)  |  for  values  of  z 
on  the  circumference  of  the  circle  \z  —  a\  =  r,  then 

I  u^  \  <  r-J^M  (r),    -I  Vm  \  <  r'"'^M{r)  : 
which  may  be  lower  limits  than  the  preceding.     As  above,  we  have. 

"de 


-\tf<^ 


taken  round  the  circle  \z  —  a\  =  r;  so  that 

'de , ue 


Uo  \< 


\^\mH\~M{r)^M{r\ 


Similarly,  as  u^  is  the  term  independent  of  2^  —  a  in  the  Laurent  expansion 

of  {z  —  a)~''^f{z),  we  have 

\um\<  greatest  value  of  |  (^  —  a)~^f(z)  \  along  \z—  a\=r 
^r-^M{r); 
and  so  for  v„,- 


SO.J  EXPANSION   IN   NEGATIVE   POWERS  59 

30.  The  following  proposition  is  practically  a  corollary  from  Laurent's 
Theorem  : — 

Wheyi  a  function  is  holomorphic  over  all  the  plane  which  lies  outside  a 
circle  of  centre  a,  it  can  be  expanded  in  the  form  of  a  series  of  negative  integral 
powers  of  z  —  a,  the  series  converging  everywhere  in  that  part  of  the  plane. 

It  can  be  deduced  as  the  limiting  case  of  Laurent's  Theorem  when  the 
radius  of  the  outer  circle  is  made  infinite.  We  then  take  r  infinitely  large, 
and  substitute  for  t  by  the  relation 

t  —  a  =  re^\ 
so  that  the  first  integral  in  the  expression  (i),  p.  55,  for/(^)  is 

t  —  a 
Since  the  function  is  holomorphic  over  the  whole  of  the  plane  which  lies 
outside  the  assigned  circle,  f{t)  cannot  be  infinite  at  the  circle  of  radius  r 
when  that  radius  increases  indefinitely.  If  f{t)  tend  towards  a  (finite) 
limit  k,  which  must  be  uniform  owing  to  the  hypothesis  as  to  the  functional 
character  oif{z),  then,  since  the  limit  of  {t  —  z)l(t  —  a)  is  unity,  the  preceding 
integral  is  equal  to  k. 

The  second  integral  in  the  same  expression  (i),  p.  55,  for  f{z)  is 
unaltered  by  the  conditions  of  the  present  proposition ;   hence  we  have 

f{z)  =  k-\-{z—a)-^Vi-\-{z-a)-^-V2-\-...., 
the   series   converging  without  the  circle,   though  it  does  not  necessarily 
converge  on  the  circumference. 

The  series  can  be  represented  in  the  form 

1 


conformably  with  the  notation  of  §  26. 

Of  the  three  theorems  in  expansion  which  have  been  obtained,  Cauchy's 
is  the  most  definite,  because  the  coefficients  of  the  powers  are  explicitly 
obtained  as  values  of  the  function  and  of  its  derivatives  at  an  assigned 
point.  In  Laurent's  theorem,  the  coefficients  are  not  evaluated  into  simple 
expressions.  In  the  corollary  from  Laurent's  theorem  the  coefficients  are, 
as  is  easily  proved,  the  values  of  the  function  and  of  its  derivatives  for  infinite 
values  of  the  variable.  The  essentially  important  feature  of  all  the  theorems 
is  the  expansibility  of  the  function  in  converging  series  under  assigned 
conditions. 

31.  It  was  proved  (§21)  that,  when  a  function  is  holomorphic  in  any 
region  of  the  plane  bounded  by  a  simple  curve,  it  has  an  unlimited  number 
of  successive  derivatives  each  of  which  is  holomorphic  in  the  region.     Hence, 


60  EXPANSION  OF  FUNCTIONS  [31. 

by  the  preceding  propositions,  each  such  derivative  can  be  expanded  in 
converging  series  of  integral  powers,  the  series  themselves  being  deducible 
by  differentiation  from  the  series  which  represents  the  function  in  the  region. 
In  particular,  when  the  region  is  a  finite  circle  of  centre  a,  within  which 
f{z)  and  consequently  all  the  derivatives  of/(^)  are  expansible  in  converging 
series  of  positive  integral  powers  of  z  —  a,  the  coefficients  of  the  various 
powers  of  ^  —  a  are — save  as  to  numerical  factors — the  values  of  the 
derivatives  at  the  centre  of  the  circle.  Hence  it  appears  that,  when  a  function 
is  holomorphic  over  the  area  of  a  given  circle,  the  values  of  the  function  and  all 
its  derivatives  at  any  point  z  tvithin  the  circle  depend  only  upon  the  variable 
of  the  point  and  upon  the  values  of  the  function  and,  its  derivatives  at  the 
centre. 

32.  Some  of  the  classes  of  points  in  a  plane  that  usually  arise  in 
connection  with  uniform   functions  may  now  be  considered. 

(i)  A  point  a  in  the  plane  may  be  such  that  a  function  of  the  variable 
has  a  determinate  finite  value  there,  always  independent  of  the  path  by 
which  the  variable  reaches  a;  the  point  a  is  called  an  ordinary  point*  of 
the  function.  The  function,  supposed  continuous  in  the  vicinity  of  a,  is 
continuous  at  a :  and  it  is  said  to  behave  regularly  in  the  vicinity  of  an 
ordinary  point. 

Let  such  an  ordinary  point  a  be  at  a  distance  d,  not  infinitesimal,  from 
the  nearest  of  the  singular  points  (if  any)  of  the  function ;  and  let  a  circle  of 
centre  a  and  radius  just  less  than  d  be  drawn.  The  part  of  the  ^-plane  lying 
within  this  circle  is  called  f  the  domain  of  a ;  and  the  function,  holomorphic 
within  this  circle,  is  said  to  behave  regularly  (or  to  be  regular)  in  the  domain 
of  a.  From  the  preceding  section,  we  infer  that  a  function  and  its  derivatives 
can  be  expanded  in  a  converging  series  of  positive  integral  powers  of  ^  —  a 
for  all  points  z  in  the  domain  of  a,  an  ordinary  point  of  the  function :  and 
the  coefficients  in  the  series  are  the  values  of  the  function  and  of  its  derivatives 
at  a. 

The  property  possessed  by  the  series — that  it  contains  only  positive 
integral  powers  of  ^^  —  a — at  once  gives  a  test  which  is  both  necessary  and 
sufficient  to  determine  whether  a  point  is  an  ordinary  point.  If  the  point  a. 
he  ordinary,  the  limit  of  (z  ~  a)f(z)  necessarily  is  zero  when  z  becomes  equal 
to  a.  This  necessary  condition  is  also  sufficient  to  ensure  that  the  point  is 
an  ordinary  point  of  the  function  /(^),  supposed  to  be  uniform;  for,  since 
f(z)  is  holomorphic,  the  function  (z  —  a)f(z)  is  also  holomorphic  and  can  be 
expanded  in  a  series 

Uq  +  z<i  (z  —  a)  +  Uo  {z  —  of  -\- ..., 

*  Sometimes  a  regular  point. 

t  The  German  title  is  Umgebung,  the  French  is  domaine. 


32.]  CLASSES   OF   POINTS   DEFINED  61 

converging  in  the  domain  of  a.     The  quantity  u^  is  zero,  being  the  value 
of  {2  —  a)f{z)  at  a  and  this  vanishes  by  hypothesis ;   hence 

{z  -  a)f  {z)  =  {z  -  a)  [u^  +  Uo^{z  -  a)  +  . . .], 
shewing  that  f{z)  is  expressible  as  a  series  of  positive  integral  powers  of 
z-a  converging  within  the  domain  of  a,  or,  in  other  words,  that/(^)  certainly 
has  a  for  an  ordinary  point  in  consequence  of  the  condition  being  satisfied. 

(ii)  A  point  a  in  the  plane  may  be  such  that  a  function  f{z)  of  the 
variable  has  a  determinate  infinite  value  there,  always  independent  of  the 
path  by  which  the  variable  reaches  a,  the  function  behaving  regularly  for 

points  in  the  vicinity  of  a ;  then  ^.7^  has  a  determinate  zero  value  there,  so 

that  a  is  an  ordinary  point  of  -:F(~\-     '^^®  point  a  is  called  a  pole  (§  12) 

or  an  accidental  singularity*  of  the  function. 

A  test,  necessary  and  sufficient  to  settle  whether  a  point  is  a  pole  of 
a  function,  will  subsequently  (§  42)  be  given. 

(iii)  A  point  a  in  the  plane  may  be  such  that/(^)  has  not  a  determinate 
value  there,  either  finite  or  infinite,  though  the  function  is  definite  in  value 
at  all  points  in  the  immediate  vicinity  of  a  other  than  a  itself. 

Such  a  point  is  called  f  an  essential  singularity  of  the  function.  No 
hypothesis  is  postulated  as  to  the  character  of  the  function  for  points 
at  infinitesimal  distances  from  the  essential  singularity,  while  the  relation 
of  the  singularity  to  the  function  naturally  depends  upon  this  character  at 
points  near  it.  There  may  thus  be  various  kinds  of  essential  singularities 
all  included  under  the  foregoing  definition,  even  for  uniform  functions ; 
one  classification  is  effected  through  the  consideration  of  the  character  of 
the  function  at  points  in  their  immediate  vicinity.     (See  §  88.) 

One  sufficient  test  of  discrimination  between  an  accidental  singularity 
and  an  essential  singularity  is  furnished  by  the  determinateness  of  the  value 
at  the  point.  If  the  reciprocal  of  the  function  have  the  point  for  an  ordinary 
point,  the  point  is  an  accidental  singularity — it  is,  indeed,  a  zero  for  the 
reciprocal.  But  when  the  point  is  an  essential  singularity,  the  value  of  the 
reciprocal  of  the  function  is  not  determinate  there ;  and  then  the  reciprocal, 
as  well  as  the  function,  has  the  point  for  an  essential  singularity. 

In  these  statements  and  explanations,  it  is  assumed  that  the  essential 
singularity  is  an  isolated  point.  It  will  hereafter  be  seen  that  uniform 
functions  can  be  constructed  for  M^hich  this  is  not  the  case ;  thus  there  are 
uniform  functions  which  have  lines  of  essential  singularity.  For  the  present, 
we  shall  deal  only  with  essential  singularities  that  are  isolated  points. 

*  Weierstrass,   Ges.  TVerke,  t.  ii,  p.  78,  to  whom  the  name  is  due,  calls  it  aussenvesentliche 
singuldre  Stelle ;  the  term  non-essential  is  suggested  by  Mr  Cathcart,  Harnack,  p.  148. 
t  Weierstrass  calls  it  wesentUclie  singuldre  Stelle. 


62  EXAMPLES  [32. 

Ex.  1.     Consider  the  function  cos  -  in  the  vicinity  of  the  origin. 

z 

The  value  at  z  =  0  clearly  is  indeterminate;  but  it  tends  to  limits  that  depend  upon 
the  mode  by  which  z  approaches  the  origin. 

Thus  suppose  that  z  approaches  the  origin  along  the  axis  of  imaginary  quantities ;  and 
let  z  =  ai,  where  a  is  real  and  can  be  made  as  small  as  we  please.     Then 

1  '-         -'- 

cos  -  =  ^6"-  +^e    °-; 

if  a  be  positive  then  the  first  term,  and  if  a  be  negative  then  the  second  term,  can  be 
made  larger  than  any  assigned  finite  quantity  by  sufficiently  diminishing  a :  that  is, 

by  these  methods  of  approach  of  z  to  its  origin,  the  function  cos  -  ultimately  acquires  an 
infinite  value. 

Next  suppose  that  z  approaches  the  origin  along  the  axis  of  real  quantities,  and 
assume  it  to  have  positive  values,  (the  same  reasoning  applies  if  it  has  negative  values) ; 
in  particular,  consider  real  values  of  z,  such  that  0^2^/3,  where  /3  is  a  quantity  that  may 
be  assigned  as  small  as  we  please.     "When  ^  is  assigned,  take  any  positive  integer  m,  such 

that 

2 

SO  that  m  will  be  any  integer  lying  between  some  one  integer  (that  will  be  large,  in 
dependence  upon  the  value  of  /3)  and  infinity.     Let 

whei'e  ^  is  a  positive  quantity  such  that  0  ^  f  ^  tt  ;  then 

11 

and  so  0  <  2  <  iS.     For  such  values  we  have 

cos  -  =  ( —  1)"*"^  sin  {", 

and  therefore  with  the  range  of  ^  from  0  to  tt,  the  function  ranges  continuously  in 
numerical  value  between  0  and  1.  In  particular,  when  f=0,  the  function  has  a  zero 
value;  (also  when  C  =  '"'i  ^'^^  ^^i^  in  effect  gives  the  next  greater  value  of  m);  and  this 
holds  for  each  of  the  integers  m  so  assumed.  Hence  it  follows  that  within  the  range 
0^2^/3  for  real  values  of  z,  no  matter  how  small  the  real  quantity  j8  may  be  assigned, 

the  function  cos  -  has  an  unlimited  number  of  zeros ;  also  that,  within  the  same  range, 

the  function  cos  —  k  (where  k  is  a  real  quantity  not  greater  than  unity)  has  an  unlimited 
z  ? 

number  of  zeros. 

jEx.  2.     Consider  the  function  cos  -  in  the  vicinity  of  the  origin,  when  the  variable  z  is 

made  to  approach  the  origin  along  the  spiral  6  =  fir,  where  z  —  re^\  and  /;i  is  a  parametric 
quantity ;  and  shew  that,  in  the  immediate  vicinity  of  the  origin  along  this  path, 

sinh  u  ^    cos  -    <  cosh  2u. 

I         ■^  I 

Discuss  the  possibility  of  so  choosing  the  approach  of  z  to  the  origin  as,  for  values  of  z 
such  that  \z\  <  y  where  -y  is  a  quantity  that  may  be  made  as  small  as  we  please,  to 

make  cos  -  acquire  a  value  A  +  iB. 


32.]  EXAMPLES  63 

Ex.  3.     Shew  that   the  function  cosec  -  has  an  unlimited  number  of  poles  in  the 

immediate  vicinity  of  its  essential  singularity  2=0. 

1 
Ex.  4.     Consider  the  variations  in  value  of  the  function  e«  for  values  of  ^,  such  that 
I  2  I  is  not  greater  than  some  assigned  small  quantity  k. 

1 
Tn  particular,  consider  the  possibility  of  e^^  either  acquiring,  or  tending  to,  any  assigned 

1 
value  A.     The  values  of  z  for  which  e^  =  A  are  given  by 

-  =  2>?;7n'+log^4, 

z 

where  k  is  any  integer,  positive  or  negative.      Let  A  =  ae'^^,  where  a  and  a  ar^  real ; 
so  that 

-  =  {2kn  +  a)i+\oga. 

If  z=x  +  iy  as  usual,  then 

x  —  iy      ,^,  \  ■    1 

^2-j-p  =-- (2/?:7r  +  a)  I  +  log  a ; 

1^  , 

and  therefore  all  the  points,  for  which  e^  acquires  the  value  J.,  lie  upon  the  circle 

x^-\-y^=.- . 

^      log  a 

Accordingly,  we  consider  an  arc  of  this  circle  which  lies  within  the  circle 

1 
Not  every  point  on  the  arc  leads  to  the  value  A   of  e';  for  taking  any  point  (|,  77) 
on  it,  let 

J  log  a  =  2m7r  +  6, 

where  m  is  an  integer,  and  0  ^  6  <  2Tr ;  thus 

75 — %,= -i{2mTr  +  6)+loga, 
?■'  +  '? 

1 
so  that  the  value  of  e-  is  e^osa-i(2mn+9)^   =ae~^\  which  is  only  the  same  as  ae"*  for 

I    -\ 
particular  points.     It  is  however  clear  that  \e^\  is  the  same  for  all  points  on  the  circular 

arc. 

1 
The  values  of  z  for  which  ez  =  A  are  given  by 

1 


~'(2/?;7r  +  a)z  +  loga' 
where  k  is  an  integer.     It  is  manifest  that  a  value  of  k  (say  ki)  can  be  chosen  for  which 

i  2  I   <  K,  ■ 

1 

this  inequality  holding  for  all  values  of  k  greater  than  k^ :  so  that  the  function  e«  acquires 
the  value  A  at  an  unlimited  number  of  points  in  the  region  \z\  <  k.  Further,  by 
sufl&ciently  increasing  k,  we  can  make  |  2  |  smaller  than  any  assigned  quantity  however 

1 
small;  and  therefore  A  is  one  of  the  (unlimited  number  of)  values  of  e^  as  z  ultimately 
becomes  zero.  .  . 


64-  UNIFORM   FUNCTIONS   AT   AND   NEAR  [32. 

It  may  be  remarked  at  once  that  there  must  be  at  least  one  infinite 
value  among  the  values  which  a  uniform  function  can  assume  at  an  essential 
singularity.  For  if/(^)  cannot  be  infinite  at  a,  then  the  limit  of  {z  —  a)f{z) 
would  be  zero  when  z  —  a\  no  matter  what  the  non-infinite  values  oi  f{z) 
may  be,  and  no  matter  by  what  path  z  acquires  the  value  a ;  that  is,  the 
limit  would  be  a  determinate  zero.  The  function  (z  —  a)f{z)  is  regular  in 
the  vicinity  of  a :  hence  by  the  foregoing  test  for  an  ordinary  point,  the  point 
a  would  be  ordinary  and  the  value  of  the  uniform  function  f{z)  would  be 
determinate,  contrary  to  hypothesis.  Hence  the  function  must  have  at  least 
one  infinite  value  at  an  essential  singularity. 

Further,  a  uniform  function  must  be  capable  of  assuming  any  value  C 
at  an  essential  singularity.     For  an  essential  singularity  of /(^)  is  also  an 

essential  singularity  of  f(z)  —  G  and  therefore  also  of  tt-t-t — p.     The  last 

function  must  have  at  least  one  infinite  value  among  the  values  that  it 
can  assume  at  the  point;  and,  for  this  infinite  value,  we  have  f{z)  =  C 
at  the  point,  so  that  f{z)  assumes  the  assigned  value  G  at  the  essential 
singularity. 

Note.  This  result,  that  a  uniform  function  can  acquire  any  assigned 
value  at  an  isolated  essential  singularity,  is  so  contrary  to  the  general  idea  of 
the  one-valuedness  of  the  function,  that  the  function  is  often  regarded  as  not 
existing  at  the  point :  and  the  point  then  is  regarded  as  not  belonging  to  the 
region  of  significance  of  the  function.  The  difference  between  the  two  views 
is  largely  a  matter  of  definition,  and  depends  upon  the  difference  between 
two  modes  of  considering  the  variable  z.  If  no  account  is  allowed  to  be 
taken  of  the  mode  by  which  z  approaches  its  value  at  an  essential  singularity 
a,  the  function  does  not  tend  uniformly  to  any  one  value  there.  If  such 
account  is  allowed,  then  it  can  happen  (as  in  Ex.  4,  above)  that  z  may 
approach  the  va,lue  a  along  a  particular  path  through  a  limiting  series  of 
values,  in  such  a  way  that  the  function  can  acquire  any  assigned  value  in  the 
limit  when  z  coincides  with  a  after  the  specified  mode  of  approach. 

33.  There  is  one  important  property  possessed  by  every  uniform  funct- 
ion in  the  immediate  vicinity  of  any  of  its  isolated  essential  singularities ; 
it  was  first  stated  by  Weierstrass *,  as  follows: — In  the  immediate  vicinity  of 
an  isolated  essential  singularity  of  a  uniform  function,  there  are  positions  at 
which  the  function  differs  from  an  assigned  value  by  a  quantity  not  greater 
than  a  non-vanishing  magnitude  that  can  he  made  as  small  as  we  please. 

*  Weierstrass,  Ges.  Werke,  t.  ii,  pp.  122 — 124;  Durege,  Elemente  der  Theorie  der  Funktionen, 
p.  119;  Holder,  Math.  Ann.,  t.  xx,  (1882),  pp.  138—143;  Picard,  "  Memoire  sur  les  fonctions 
entieres,"  Annales  de  I'Ecole  Norm.  Sup.,  2""=  Ser.,  t.  ix,  (1880),  pp.  145 — 166,  which,  in  this 
regard,  should  be  consulted  in  connection  with  the  developments  in  Chapter  V.  See  also  §  62. 
Picard's  proof  is  followed  in  the  text. 


33.]  AN   ESSENTIAL   SINGULARITY  65 

Let  a  be  the  singularity,  G  an  assigned  value,  and  e  a  non-vanishing 
magnitude  which  can  be  chosen  arbitrarily  small  at  our  own  disposal ;  and  in 
the  vicinity  of  a,  represented  by 

\z  -a\<  p, 

consider  the  function  -t7~\ — p-     •'^^^  values  of  z  in  the  range 

0<\z  —  a\<  p, 
this  function  may  have  poles,  or  it  may  not. 

If  it  has  poles,  then  at  each  of  them  /(^)  -  0  =  0  :  that  is,  the  function 
f{z)  actually  attains  the  value  C,  so  that  the  difference  between  f{z)  and  G 
for  such  positions  is  not  merely  less  than  e,  it  actually  is  zero. 
If  it  has  no  poles,  then  the  function 

1 

is  regular  everywhere  through  the  domain 

0  <  I  ^  —  a  I  <  p, 
because  no  point  in  that  domain  is  either  a  pole  or  an  essential  singularity. 
Accordingly,  by  Laurent's  theorem,  it  can  be  expanded  in  that  domain  in  a 
converging  series  of  positive  and  negative  powers,  in  the  form 

=  Uq-\-  {z-a)u-i^  + -\-{z  —  aYun  + 


fi^)-G 


z  -  a      (z  —  af  {z  —  ay^ 


Choose  a  quantity  p'  such  that  0  <  p  <p.  The  series  of  positive  powers 
converges  everywhere  within  and  on  a  circle,  centre  a  and  radius  p' :  let  S{z^ 
denote  its  value  at  z.  The  series  of  negative  powers  converges  everywhere 
in  the  plane  outside  the  point  a ;  and  therefore  the  series 

z  —  a      {z  —  af 
converges  everywhere  outside  the  point  a  :  let  T(z)  denote  its  value,  so  that 

1    =«w  +  ^<^> 


Accordingly,  as   \S{z)\  is  finite  and  \T{z)\  not  zero — it  may  be  a  rapidly 
increasing  quantity  as  |  ^  —  a  |  decreases — choose  |  ^  —  «- 1  so  that,  while  not 

being  zero,  it  gives  the  modulus  of  the  right-hand  side  as  greater  than  - . 

As  z  —  a  occurs  in  a  denominator,  this  can  be  done.     Then,  for  such  a  value 

of  z, 

1 


and  therefore 

which  proves  the  theorem. 

F.  F. 


1 
>- 

e 


\f(^)-C\<e, 


66  CONTINUATIONS   OF   A   FUNCTION  [33. 

It  may  happen  that  the  function  attains  the  value  G  only  at  the  essential  singularity, 
where  G  is  one  of  its  unlimited  number  of  values.     Thus  to  find  the  zeros  of  the  function 

cosec  -  in  the  vicinity  of  the  origin,  we  must  have  sin  -  infinite  at  them ;  this  can  only 

occur  when  z  becomes  zero  along  the  axis  of  imaginaries,  and  cannot  occur  for  any  value 
of  z  such  that  |  z  |  >  0.  Such  a  value  is  called  an  exceptional  value ;  the  discussion  of 
exceptional  values  is  effected  by  Picard  in  his  memoir  quoted. 

Ex.     Discuss  the  character  of  the  functions  cos  (l/s),  tan  (l/z)  for  values  of  |  2  j  which 

1 

are  very  small ;  and  the  character  of  the  functions  tan  s,  e^^,  ^""■e^,  e      ^,  z  log  z,  for  values 

of  1 2  I  which  are  very  large. 

34.  Let  f{z)  denote  the  function  represented  by  a  series  of  powers 
Pi  {z  —  a),  the  circle  of  convergence  of  which  is  the  domain  of  the  ordinary 
point  a,  and  the  coefficients  in  which  are  the  values  of  the  derivatives  of 
f  {z)  at  a.  The  region  over  which  the  function  f{z)  is  holomorphic  may 
extend  beyond  the  domain  of  a.  although  the  circumference  bounding  that 
domain  is  the  greatest  of  centre  a  that  can  be  drawn  within  the  region. 
The  region  evidently  cannot  extend  beyond  the  domain  of  a  in  all  directions. 

Take  an  ordinary  point  h  in  the  domain  of  a.  The  value  at  h  of  the 
function  f{z)  is  given  by  the  series  Pj  (b  —  a),  and  the  values  at  b  of  all  its 
derivatives  are  given  by  the  derived  series.  All  these  series  converge  within 
the  domain  of  a  and  they  are  therefore  finite  at  h ;  and  their  expressions 
involve  the  values  at  a  of  the  function  and  its  derivatives. 

Let  the  domain  of  b  be  formed.  The  domain  of  b  may  be  included  in 
that  of  a,  and  then  its  bounding  circle  will  touch  the  bounding  circle  of  the 
domain  of  a  internally.  If  the  domain  of  b  be  not  entirely  included  in  that 
of  a,  part  of  it  will  lie  outside  the  domain  of  a;  but  it  cannot  include  the 
whole  of  the  domain  of  a  unless  its  bounding  circumference  touch  that  of 
the  domain  of  a  externally,  for  otherwise  it  would  extend  beyond  a  in  all 
directions,  a  result  inconsistent  with  the  construction  of  the  domain  of  a. 
Hence  there  must  be  points  excluded  from  the  domain  of  a  which  are  also 
excluded  from  the  domain  of  b. 

For  all  points  z  in  the  domain  of  b,  the  function  can  be  represented  by 
a  series,  say  P^{z—b),  the  coefficients  of  which  are  the  values  at  b  of  the 
function  and  its  derivatives.  Since  these  values  are  partially  dependent 
upon  the  corresponding  values  at  a,  the  series  representing  the  function  may 
be  denoted  by  P^  {z  ~  b,  a). 

At  a  point  z  in  the  domain  of  b  lying  also  in  the  domain  of  a,  the  two 
series  Pi  (z  —  a)  and  Po  (z  —  b,  a)  must  furnish  the  same  value  for  the 
function  f{z) ;  and  therefore  no  new  value  is  derived  from  the  new  series  Pg 
which  cannot  be  derived  from  the  old  series  Pi.  For  all  such  points  the  new 
series  is  of  no  advantage ;  and  hence,  if  the  domain  of  b  be  included  in  that 
of  a,  the  construction  of  the  series  P^{z-b,a)  is  superfluous.  Thus,  in 
choosing  the  ordinary  point  b  in  the  domain  of  a  we  choose  a  point,  if 
possible,  that  will  not  have  its  domain  included  in  that  of  a. 


34]  OVER   ITS   REGION   OF   CONTINUITY  6Y 

At  a  point  z  in  the  domain  of  h,  which  does  not  lie  in  the  domain  of  a, 
the  series  Pg  iz  —  b,  a)-  gives  a  value  for  /(z)  which  cannot  be  given  by 
Pi  (z-a).  The  new  series  P^  then  gives  an  additional  representation  of  the 
function ;  it  is  called*  a  continuation  of  the  series  which  represents  the  function 
in  the  domain  of  a.  The  derivatives  of  Pg  give  the  values  of  the  derivatives 
of  f{z)  for  points  in  the  domain  of  h. 

It  thus  appears  that,  if  the  whole  of  the  domain  of  h  be  not  included  in 
that  of  a,  the  function  can,  by  the  series  which  is  valid  over  the  whole 
of  the  new  domain,  be  continued  into  that  part  of  the  new  domain  excluded 
.from  the  domain  of  a. 

Now  take  a  point  c  within  the  region  occupied  by  the  combined  domains 
of  a  and  h ;  and  construct  the  domain  of  c.  In  the  new  domain,  the 
function  can  be  represented  by  a  new  series,  say  Po  {z  —  c),  or,  since  the 
coefficients  (being  the  values  at  c  of  the  function  and  of  its  derivatives) 
involve  the  values  at  a  and  possibly  also  the  values  at  h  of  the  function 
and  of  its  derivatives,  the  series  representing  the  function  may  be  denoted 
by  P3  {z  —  c,  a,  h).  Unless  the  domain  of  c  include  points,  which  are  not 
included  in  the  combined  domains  of  a  and  h,  the  series  P3  does  not  give 
a  value  of  the  function  which  cannot  be  given  by  Pj  or  P^:  we  therefore 
choose  c,  if  possible,  so  that  its  domain  will  include  points  not  included  in 
the  earlier  domains.  At  such  points  z  in  the  domain  of  c  as  are  excluded 
from  the  combined  domains  of  a  and  h,  the  series  P3  {z  —  c,  a,  b)  gives  a  value 
for  f(z)  which  cannot  be  derived  from  P^  or  Pg ;  and  thus  the  new  series 
is  a  continuation  of  the  earlier  series. 

Proceeding  in  this  manner  by  taking  successive  points  and  constructing 
their  domains,  we  can  reach  all  parts  of  the  plane  connected  with  one 
another  where  the  function  preserves  its  holomorphic  character;  their 
combined  aggregate  is  called  *|-  the  region  of  continuity  of  the  function. 
With  each  domain,  constructed  so  as  to  include  some  portion  of  the  region  of 
continuity  not  included  in  the  earlier  domains,  a  series  is  associated,  which  is 
a  continuation  of  the  earlier  series  and,  as  such,  gives  a  value  of  the  function 
not  deducible  from  those  earlier  series ;  and  all  the  associated  series  are 
ultimately  deduced  from  the  first. 

Each  of  the  continuations  is  called  an  Element  of  the  function.  The 
aggregate  of  all  the  distinct  elements  is  called  a  monogenic  analytic  function : 
it  is  evidently  the  complete  analytical  expression  of  the  function  in  its  region 
of  continuity. 

Let  z  be  any  point  in  the  region  of  continuity,  not  necessarily  in  the 
circle  of  convergence  of  the  initial  element  of  the  function ;  a  value  of  the 

*  Biermann,    Theorie   der   analytischen    Functionen,    p.    170,   which   may   be   consulted   in 
connection  with  the  whole  of  §  34;   the  German  word  is  Fortsetzung. 
t  Weierstrass,  Ges.  Werke,  t.  ii,  p.  77. 

5—2 


68  REGION   OF   CONTINUITY   OF  [34. 

function  at  z  can  be  obtained  through  the  continuations  of  that  initial 
element.  In  the  formation  of  each  new  domain  (and  therefore  of  each  new 
element)  a  certain  amount  of  arbitrary  choice  is  possible ;  and  there  may, 
moreover,  be  different  sets  of  domains  which,  taken  together  in  a  set,  each 
lead  to  z  from  the  initial  point.  When  the  analytic  function  is  uniform,  as 
before  defined  (§  12),  the  same  value  at  z  for  the  function  is  obtained, 
whatever  be  the  set  of  domains.  If  there  be  two  sets  of  elements,  different^ 
obtained,  which  give  at  z  different  values  for  the  function,  then  the  ana- 
lytic function  is  multiform,  as  before  defined  (§  12) ;  but  not  every  change 
in  a  set  of  elements  leads  to  a  change  in  the  valu.e  at  2:  of  a  multiform 
function,  and  the  analytic  function  is  uniform  within  such  a  region  of  the 
plane  as  admits  only  equivalent  changes  of  elements. 

The  whole  process  is  reversible  when  the  function  is  uniform.  We  can 
pass  back  from  any  point  to  any  earlier  point  by  the  use,  if  necessary,  of 
intermediate  points.  Thus,  if  the  point  a  in  the  foregoing  explanation 
be  not  included  in  the  domain  of  h  (there  supposed  to  contribute  a  continu- 
ation of  the  first  series),  an  intermediate  point  on  a  line,  drawn  in  the 
region  of  continuity  so  as  to  join  a  and  h,  would  be  taken ;  and  so  on, 
until  a  domain  is  formed  which  does  include  a.  The  continuation,  associated 
with  this  domain,  must  give  at  a  the  proper  value  for  the  function  and  its 
derivatives,  and  therefore  for  the  domain  of  a  the  original  series  F^{z—a) 
will  be  obtained,  that  is,  Pi  iz  —  a)  can  be  deduced  from  P^  {z  —  b,  a)  the 
series  in  the  domain  of  b.  This  result  is  general,  so  that  any  one  of  the 
continuations  of  a  uniform  function,  represented  by  a  power-series,  can  be 
deduced  from  any  other ;  and  therefore  the  expression  of  such  a  function  in 
its  region  of  continuity  is  potentially  given  by  one  element,  for  all  the 
distinct  elements  can  be  deduced  from  any  one  element, 

35.  It  has  been  assumed  that  the  property,  characteristic  of  some  of  the 
uniform  functions  adduced  as  examples,  of  possessing  either  accidental  or 
essential  singularities,  is  characteristic  of  all  such  functions;  it  will  be  proved 
(§  40)  to  hold  for  every  uniform  function  which  is  not  a  mere  constant. 

The  singularities  limit  the  region  of  continuity ;  for  each  of  the  separate 
domains  is,  from  its  construction,  limited  by  the  nearest  singularity,  and  the 
combined  aggregate  of  the  domains  constitutes  the  region  of  continuity  when 
they  form  a  continuous  space*.  Hence  the  complete  boundary  of  the  region 
of  continuity  is  the  aggregate  of  the  singularities  of  the  function f. 

*  Cases  occur  in  -whicli  the  region  of  continuity  of  a  function  is  composed  of  isolated  spaces, 
each  continuous  in  itself,  but  not  continuous  into  one  another.  The  consideration  of  such  cases 
will  be  dealt  with  briefly  hereafter,  and  they  are  assumed  excluded  for  the  present :  meanwhile, 
it  is  sufficient  to  note  that  each  continuous  space  could  be  deduced  from  an  element  belonging  to 
some  domain  of  that  space  and  that  a  new  element  would  be  needed  for  a  new  space. 

t  See  Weierstrass,  Ges.  Werke,  t.  ii,  pp.  77 — 79;  Mittag-Leflier,  "  Sur  la  representation  analy- 
tique  des  fonclions  monogenes  uniformes  d'une  variable  independante, "  Acta  Math.,  t.  iv,  (1884), 
pp.  1  et  seq.,  especially  pp.  1 — 8. 


35.]  AN   ANALYTIC   FUNCTION  69 

It  may  happen  that  a  function  has  no  singularity  except  at  infinity ;  the 
region  of  continuity  then  extends  over  the  whole  finite  part  of  the  plane  but 
it  does  not  include  the  point  at  infinity. 

It  follows  from  the  foregoing  explanations  that,  in  order  to  know  a 
uniform  analytic  function,  it  is  necessary  to  know  some  element  of  the 
function,  which  has  been  shewn  to  be  potentially  sufficient  for  the  derivation 
of  the  full  expression  of  the  function  and  for  the  construction  of  its  region  of 
continuity.  But  the  process  of  continuation  is  mainly  descriptive  of  the 
analytic  function,  and  in  actual  practice  it  can  prove  too  elaborate  to  be 
effected*. 

To  avoid  the  continuation  process,  Mittag-Leffler  has  devised f  another 
method  of  representing  a  uniform  function.  Let  a  be  an  ordinary  point  of 
the  function,  and  let  a  line,  terminated  at  a,  rotate  round  it.  In  the  vicinity 
of  a,  let  the  element  of  the  function  be  denoted  hy  P{z  —  a);  and  imagine 
the  continuation  of  this  element  to  be  effected  along  the  vector  as  far  as 
possible.  It  may  happen  that  the  continuation  can  be  effected  to  infinity 
along  the  vector;  if  not,  there  is  some  point  a'  on  the  vector  beyond  which 
the  continuation  is  impossible.  In  the  latter  case,  the  part  of  the  vector  j 
from  a  to  infinity  is  excluded  from  the  range  of  variation  of  the  variable. 
Let  this  be  done  for  every  position  of  the  vector ;  then  the  part  of  the  plane, 
which  remains  after  these  various  ranges  have  been  excluded,  gives  a  star- 
shaped  figure,  which  is  a  region  of  continuity  of  the  uniform  function  of 
which  P{z  —  a)  is  the  initial  element.  The  function  manifestly  can  be 
continued  over  the  whole  of  this  star,  by  means  of  appropriate  elements ;  but 
there  is  no  indication  as  to  the  necessary  number  of  elements.  Instead  of 
using  the  elements  to  express  the  function,  Mittag-Leffler  constructs  a  single 
expression,  which  is  the  valid  representation  of  the  function  over  the  whole 
star;  the  expression  is  an  infinite  series  of  polynomials,  and  not  merely  a 
power-series. 

Thus  let  there  be  a  power-series 

h  +  h^{z-a)  +  ^^b<i{z-  af  +  --h{z-af+ ..., 

which  converges  uniformly  in  a  region  round  the  point  a ;  the  radius  of  convergence  of  the 

2 
series  is  r,  where  Ijr  is  the  upper  limit  of  the  quantities  (6„/n  \)n.     Let  the  star-shaped 
figure  be  constructed;  the  following  is  the  simplest  form  of  expression  as  obtained  by 
Mittag-Leflfler  to  represent,  over  the  whole  star,  the  function  of  which  the  foregoing  series 
is  an  element.     Let  the  quantity 

*  Some  examples  have  been  constructed  by  Prof.  M.  J.  M.  Hill,  Proc.  Lond.  Math.  Soc, 
vol.  XXXV,  (1903),  pp.  388—416. 

t  Exact  references  are  given  at  the  beginning  of  Chapter  VII. 
t  In  effect,  this  is  a  section,  in  the  sense  used  in  §  103. 


70 


CONTINUATION   OF   ANALYTIC   FUNCTION 


[35. 


which  is  a  polynomial,  be  denoted  by  g^  (z) ;  and  take 

On{z)=gn{z)-gn~l{z\    for  0=  1,  2,  . .. . 

Mittag-Leffler's  expression  is 

M  =  0 

and  it  converges  everywhere  within  the  star. 

Again,  an  element  representing  a  function  is  effective  only  within  its  own 
circle  of  convergence,  while  it  may  be  known  that  the  function  is  holomorphic 
over  some  closed  domain  which  touches  the  circle  of  convergence  externally. 
The  process  of  continuation  would  make  it  possible  to  obtain  the  analytical 
representation  over  the  whole  domain  by  means  of  appropriate  elements : 
but  again  there  is  no  indication  as  to  the  necessary  number  of  elements. 
Painleve*  has  shewn  how  to  construct  a  single  expression,  which  is  the  valid 
representation  of  the  function  over  the  whole  domain ;  this  expression  also  is 
an  infinite  series  of  polynomials,  and  not  merely  a  power-series. 

For  the  establishment  of  these  results,  we  refer  to  the  memoirs  quoted. 

36.  The  method  of  continuation  of  a  function,  by  means  of  successive 
elements,  is  quite  general ;  there  is  one  particular  continuation,  which  is 
important  in  investigations  on  conformal  representation.  It  is  contained  in 
the  following  proposition,  due  to  Schwarzi*: — 

//  an  analytic  function  w  of  z  he  defined  only  for  a  region  S'  in  the 
positive  half  of  the  z-plane,  and  if  continuous  real  values  of  w  correspond  to 
continuous  real  values  of  z,  then  w  can  be  continued  across  the  axis  of  real 
quantities. 

Consider  a  region  S",  symmetrical  with  8'  relative  to  the  axis  of  real 
quantities  (fig.  12).     Then  a  function  is 
defined  for  the  region  S"  by  associating 
a  value  Wq,  the  conjugate  of  w,  with  Zq, 
the  conjugate  of  z. 

Let  the  two  regions  be  combined 
along  the  portion  of  the  axis  of  oc  which 
is  their  common  boundary;  they  then 
form  a  single  region  S'  +  S". 

Consider  the  integrals 

1   f     w  1   r     ivo 

27nJs^z-^  2TnJs"Z,-c, 

taken  round  the   boundaries   of  8'  and  of  S"   respectively.      Since   tu   is 
continuous  over  the   whole  area  of  S'  as  well  as  along  its  boundary,  and 

*  Gomptes  Rendus,  t.  cxxvi,  (1898),  pp.  320,  321;  see  also  the  references  to  Painlev^  at 
the  beginning  of  Chapter  VII. 

t  Grelle,  t.  Ixx,  (1869),  pp.  106, 107,  and  Ges.  Math.  Abli.,  t.  ii,  pp.  66—68.  See  also  Darboux, 
Theorie  generale  des  surfaces,  t.  i,  §  130. 


dz„ 


36.1  DUE   TO   SCHWARZ  71 

likewise  w^  relative  to  8",  it  follows  that,  if  the  point  ^  be  in  8',  the  value  of 
the  first  integral  is  w  (^)  and  that  of  the  second  is  zero ;  while,  if  ^  be  in  8", 
the  value  of  the  first  integral  is  zero  and  that  of  the  second  is  Wq  (0-  Hence 
the  sum  of  the  two  integrals  represents  a  unique  function  of  a  point  in  either 
8'  or  ^S"'.     But  the  value  of  the  first  integral  is 

1     /'  ^    wdz        1     [^w  (x)  doc 

—  '  (C) +         '        ^  ^ 


"Itti  j  B  z  —  ^      27ri  J  ^     x  —  ^     ' 

the  first  being  taken  along  the  curve  BGA  and  the  second  along  the  axis 
AxB  ;  and  the  value  of  the  second  integral  is 


1    M  Wq  (x)  dx        1    f  ^  Wo  dzo 


the  first  being  taken  along  the  axis  BxA  and  the  second  along  the  curve 
ADB.     But 

Wo  (^)  =  U)  {x), 

because  conjugate  values  w  and  lu^  correspond  to  conjugate  values  of  the 
argument  by  definition  of  Wq,  and  because  lu  (and  therefore  also  iv^  is  real 
and  continuous  when  the  argument  is  real  and  continuous.  Hence  when  the 
sum  of  the  four  integrals  is  taken,  the  two  integrals  corresponding  to  the 
two  descriptions  of  the  axis  of  x  cancel ;  we  have  as  the  sum 

1    r  ^    wdz         1    [  ^  Wo  dzo 
'(C) z  +  7^-.    W- 


27r* j  B  z  —  ^     27ri J  ^  Zq—  ^' 

and  this  sum  represents  a  unique  function  of  a  point  in  8'  +  8".  These  two 
integrals,  taken  together,  are 

1     r  w'dz 

taken  round  the  whole  contour  of  aS'  +  8",  where  iv'  is  equal  to  w  (^)  in  the 
positive  half  of  the  plane  and  to  Wq  (^)  in  the  negative  half 

For  all  points  ^  in  the  whole  region  8'  +  8",  this  mtegral  represents  a 
single  uniform,  finite,  continuous  function  of  ^;  its  value  is  w{^)  in  the 
positive  half  of  the  plane  and  is  Wo(0  i^  *^®  negative  half;  and  therefore 
Wo  (0  is  the  continuation,  into  the  negative  half  of  the  plane,  of  the  function 
which  is  defined  hj  w{^)  for  the  positive  half 

For  a  point  c  on  the  axis  of  x,  we  have 

w(z)-w{c)  =  A{z-c)  +  B(z-cy+G(z-cy+...; 

and  all  the  coefiicients  A,  B,  C,  ...  are  real.  If,  in  addition,  w  be  such 
a  function  of  z  that  the  inverse  functional  relation  makes  z  a  uniform 
analytic  fiinction  of  w,  obviously  A  must  not  vanish.  Thus  the  functional 
relation  may  be  expressed  in  the  form 

w  (z)  -  lu  (c)  =  {z-c)P(z-  c), 
where  P(z  —  c)  does  not  vanish  when  z  =  c. 


CHAPTER   IV. 

General  properties  of  Uniform  Functions,  particularly  of  those 
WITHOUT  Essential  Singularities. 

37.  In  the  derivation  of  the  general  properties. of  functions,  which  will  be 
deduced  in  the  present  and  the  next  three  chapters  from  the  results  already 
obtained,  it  is  to  be  supposed,  in  the  absence  of  any  express  statement  to 
other  effect,  that  the  functions  are  uniform,  monogenic  and,  except  at  either 
accidental  or  essential  singularities,  continuous*. 

Theorem  I.  A  function,  which  is  constant  throughout  any  region  of  the 
plane  however  small,  or  which  is  constant  along  any  line  however  short,  is 
constant  throughout  its  region  of  continuity. 

For  the  first  part  of  the  theorem,  we  take  any  point  a  in  the  region  of  the 
plane  where  the  function  is  constant ;  and  we  draw  a  circle  of  centre  a  and 
of  any  radius,  taking  care  that  the  circle  remains  within  the  region  of 
continuity  of  the  function.     At  any  point  z  within  this  circle,  we  have 

f{z)=f{a)  +  {z-a)f'{a)  +  ^^^f"{a)-^..., 

a  converging  series  the  coefficients  of  which  are  the  values  of  the  function 
and  its  derivatives  at  a.     Let  a  point  a  +  ha  he  taken  in  the  region ;  then 

f  (a)  =  Limit  ol  '^-^ K ---^^  , 

which  is  zero  because  f{a  +  8a)  is  the  same  constant  as  f{a):  so  that  the 
first  derivative  is  zero  at  a.  Similarly,  all  the  derivatives  can  be  shewn  to 
be  zero  at  a ;  hence  the  above  series  after  its  first  term  is  evanescent, 
and  we  have 

that  is,  the  function  preserves  its  constant  value  throughout  its  region  of 
continuity. 

The  second  result  follows  in  the  same  way,  when  once  the  derivatives  are 
proved  zero.     Since  the  function  is  monogenic,   the   value  of  the  first  and 

*  It  will  be  assumed,  as  in  §  35  (note,  p.  68),  that  the  region  of  continuity  consists  of  a  single 
space.  Functions,  which  exist  in  regions  of  continuity  consisting  of  a  number  of  separated 
spaces,  will  be  discussed  in  Chap.  VII. 


37.]  ZEROS   OF   A   UNIFORM   FUNCTION  73 

of  each    of  the   successive    derivatives  will  be   obtained,    if  we   make  the 
differential  element  of  the  independent  variable  vanish  along  the  line. 

Now,  if  a  be  a  point  on  the  line  and  a  +  3a  a  consecutive  point,  we  have 
f{a  +  Sa)  =f{a) ;  hence/'  (a)  is  zero.  Similarly  the  first  derivative  at  any 
other  point  on  the  line  is  zero.  Therefore  we  have  /'  (a  +  Sa)  =/'  (a),  for 
each  has  just  been  proved  to  be  zero :  hence  /"  (a)  is  zero.  Similarly  the 
value  of  the  second  derivative  at  any  other  point  on  the  line  is  zero.  So  on 
for  all  the  derivatives :  the  value  of  each  of  them  at  a  is  zero. 

Using  the  same  expansion  as  before  and  inserting  again  the  zero  values 
of  all  the  derivatives  at  a,  we  find  that 

so  that  under  the  assigned  condition  the  function  preserves  its  constant  value 
throughout  its  region  of  continuity. 

It  should  be  noted  that,  if  in  the  first  case  the  area  and  in  the  second  the 
line  reduce  to  a  point,  then  consecutive  points  cannot  be  taken;  the  values 
at  a  of  the  derivatives  cannot  be  proved  to  be  zero  and  the  theorem  cannot 
then  be  inferred. 

Corollary  I.  If  two  functions  have  the  same  value  over  any  area  of 
their  common  region  of  continuity  however  small  or  along  any  line  in  that 
region  however  short,  then  they  have  the  same  values  at  all  points  in  their 
common  region  of  continuity. 

This  is  at  once  evident :  for  their  difference  is  zero  over  that  area  or  along 
that  line  and  therefore,  by  the  preceding  theorem,  their  difference  has  a 
constant  zero  value,  that  is,  the  functions  have  the  same  values,  everywhere 
in  their  common  region  of  continuity. 

But  two  functions  can  have  the  same  values  at  a  succession  of  isolated 
points,  without  having  the  same  values  everywhere  in  their  common  region 
of  continuity ;  in  such  a  case  the  theorem  does  not  apply,  the  reason  being 
that  the  fundamental  condition  of  equality  over  a  continuous  area  or  along 
a  continuous  line  is  not  satisfied. 

Corollary  II.  A  function  cannot  he  zero  over  any  area  of  its  i^egion 
of  continuity  however  small,  or  along  any  line  in  that  region  however  short, 
without  being  zero  everywhere  in  its  region  of  continuity. 

It  is  deduced  in  the  same  manner  as  the  preceding  corollary. 

If,  then,  there  be  a  function  which  is  evidently  not  zero  everywhere,  we 
conclude  that  its  zeros  are  isolated  points  though  such  points  may  he  multiple 
zeros. 

Further,  in  any  finite  area  of  the  region  of  continuity  of  a  function  that  is 
subject  to  variation,  there  can  he  at  most  only  a  finite  number  of  its  zeros,  when 


74  ■  ZEROS  OF  A  '[3T. 

710  point  of  the  boundary  of  the  area  is  an  essential  singularity.  For  if  there 
were  an  infinite  number  of  such  points  in  any  such  region,  there  must  be  a 
cluster  in  at  least  one  area  or  a  succession  along  at  least  one  line,  infinite  in 
number.  Either  they  must  then  constitute  a  continuous  area  or  a  continuous 
line  where  the  function  is  everywhere  zero :  which  would  require  that  the 
function  should  be  zero  everywhere  in  its  region  of  continuity,  a  condition 
excluded  by  the  hypothesis.  Or  they  must  be  so  close  to  some  point,  say  c, 
that  the  function  has  an  unlimited  number  of  zeros  within  a  region 

\z  —  c\<e, 

where  e  can  be  made  as  small  as  we  please  :  and  so  for  non-zero  values  of  the 
function.  After  the  general  properties  which  have  been  established,  and 
the  proposition  of  |  33,  it  is  clear  that  c  is  an  essential  singularity  of  the 
function,  contrary  to  the  hypothesis  as  to  the  region  of  continuity  of  the 
function. 

It  immediately  follows  that  the  points  within  a  region  of  continuity, 
at  which  a  function  assumes  any  the  same  value,  are  isolated  points ;  and 
that  only  a  finite  number  of  such  points  occur  in  any  finite  area. 

This  result  may  be  established  in  another  way. 

'Let  f{z)  be  a  uniform  monogenic  function;  we  proceed  to  shew  that, 
when /(a)  is  not  zero,  we  can  choose  a  region  round  a  in  which /"(^)  nowhere 
vanishes.     We  have 

/  {z)  =  tto  +  ai{z  —  a)+  ttg  {z  —  af  +  ..., 

where  ag  is  not  zero,  the  series  for/(^)  converging  absolutely  and  uniformly 

for  values  of  z  such  that 

\z  —  a\^r  <  R. 

Within  or  on  the  circle  r,  let  M  be  the  greatest  value  of 

la^  +  a-ziz-a)  +  ...\, 

so  that  M  is,  of  course,  finite.     Let 

\ao\  =  Ms, 

so  that  s  is  finite ;  and  take  values  of  z  such  that 

\2  —  a\^  cr  <  s. 
Then 

\f(z)  \^\ao\  —  \z-a\\ai  +  a2{z  —  a)  +  ...\ 

^ao-o-M 

^{s-a-)M, 

so  that,  at  no  place  within  this  region  can/(^)  vanish. 

Now  let  c  be  a  zero  off{z)  of  order  n,  so  that 

f(z)  =  {z-crgiz), 


37.]  UNIFORM   FUNCTION  75 

where  g  (c)  is  not  zero  and  g  {z)  is  uniform  and  monogenic.  By  what  has  just 
been  proved,  we  can  choose  a  region  round  c  such  that  g  {z)  has  no  zero  within 
it.  Then  obviously /(2^)  has  no  zero  within  that  region  except  at  the  place  c ; 
in  other  words,  the  zero  oif{z)  is  an  isolated  point. 

38.  Theorem  II.  The  multiplicity  m  of  any  zero  a  of  a  function  is 
finite  provided  the  zero  he  an  ordinary  point  of  the  function,  supposed  not  to  he 
zero  throughout  its  region  of  continuity ;  and  the  function  can  he  expressed  in 
the  form 

(z  -  ay  (f)  (z), 

where  ^  (z)  is  holomorphic  in  the  vicinity  of  a,  and  a  is  not  a  zero  of  (j)  {z). 

Let  f{2)  denote   the  function ;    since  a  is  a   zero,  we   have  /(a)  =  0. 

Suppose  that  /'  (a),  f"  (a),  vanish  :    in  the  succession  of  the  derivatives 

of  /,  one  of  finite  order  must  be  reached  which  does  not  have  a  zero  value. 
Otherwise,  if  all  vanish,  then  the  function  and  all  its  derivatives  would 
vanish  at  a;  the  expansion  of /(^)  in  powers  oi z  —  a  would  lead  to  zero  as 
the  value  of /(^),  that  is,  the  function  would  everywhere  be  zero  in  the 
region  of  continuity,  if  all  the  derivatives  vanish  at  a. 

Let,  then,  the  mth  derivative  be  the  first  in  the  natural  succession  which 
does  not  vanish  at  a,  so  that  m  is  finite.     Using  Cauchy's  expansion,  we  have 

(t-  —  ri'\'^  ( s  —  r7^™+^ 

f{z)  =  i^_^/ w  (a)  +  V         IX,  7'"^+^'  (a)  +  . . . 
^  V  /  m !,  (m+ 1) !  -^  ^ 

=  {z-  a)'"'  (f)  (z), 

where  ^  {z)  is  a  function  that  does  not  vanish  with  a  and,  being  the  quotient 

of  a  converging  series  by  a  monomial  factor,  is  holomorphic  in  the  immediate 

vicinity  of  a. 

Corollary  I.  If  infinity  he  a  zero  of  a  function  of  multiplicity  m  and 
at  the  same  time  he  an  ordinary  point  of  the  function,  then  the  function  can  he 
expressed  in  the  form 


^© 


where  <J3  (-)  is  a  function  that  is  continuous  and  different  from  zero  for  infinitely 
large  values  of  z. 

The  result  can  be  derived  from  the  expansion  in  §  30  in  the  same  way  as 
the  foregoing  theorem  from  Cauchy's  expansion. 

Corollary  II.  The  numher  of  zeros  of  a  function,  account  heing  taken  of 
their  multiplicity,  which  occur  within  a  finite  area  of  the  region  of  continuity 
of  the  function,  is  finite,  when  no  point  of  the  houndary  of  the  area  is  an 
essential  singularity. 

By  Corollary  II.  of  §  37,  the  number  of  distinct  zeros  in  the  limited  area 
is  finite,  and,  by  the  foregoing  theorem,  the  multiplicity  of  each  is  finite ; 


76  ZEROS   OF    A  [38. 

hence,  when  account  is  taken  of  their  respective  multiplicities,  the  total 
number  of  zeros  is  still  finite. 

The  result  is,  of  course,  a  known  result  for  a  polynomial  in  the  variables ; 
but  the  functions  in  the  enunciation  are  not  restricted  to  be  of  the  type  of 
polynomials. 

Note.  It  is  important  to  notice,  both  for  Theorem  II.  and  for  its  Corol- 
lary I.,  that  the  zero  is  an  ordinary  point  of  the  function  under  consideration ; 
the  implication  therefore  is  that  the  zero  is  a  definite  zero  and  that  in  the 
immediate  vicinity  of  the  point  the  function  can  be  represented  in  the  form 

P (^  -  a)  or  P  f -  I ,  the  function  P(a—  a)  or  P  (  —  j  being  always  a  definite 

zero. 

Instances  do  occur  for  which  this  condition  is  not  satisfied.  The  point 
may  not  be  an  ordinary  point,  and  the  zero  value  may  be  an  indeterminate 
zero ;  or  zero  may  be  only  one  of  a  set  of  distinct  values  though  everywhere 
in  the  vicinity  the  function  is  regular.     Thus  the  analysis  of  §  13  shews  that 

^  =  a  is  a  point  where  the  function  sn has  any  number  of  zero  values  and 

any  number  of  infinite  values,  and  there  is  no  indication  that  there  are  not 
also  other  values  at  the  point.  In  such  a  case  the  preceding  proposition  does 
not  apply ;  there  may  be  no  limit  to  the  order  of  multiplicity  of  the  zero,  and 
we  certainly  cannot  infer  that  any  finite  integer  m  can  be  obtained  such  that  . 

{z  -  a)-'"  <}}  (z) 

is  finite  at  the  point.  Such  a  point  is  (§§  32,  33)  an  essential  singularity  of 
the  function. 

39.  Theorem  III.  A  multiple  zero  of  a  function  is  a  zero  of  its 
derivative;  and  the  Tnultiplicity  for  the  derivative  is  less  or  is  greater  by 
unity  according  as  the  zero  is  not  or  is  at  infinity. 

If  a  be  a  point  in  the  finite  part  of  the  plane  which  is  a  zero  oi  f{z) 
of  multiplicity  n,  we  have 

f{z)  =  {z-arcl^{z\ 

and  therefore  /'  {z)  =  {z-  af-^  [ncj)  (z)  +  (z  -  a)  0'  (z)]. 

The  coefficient  of  (z  —  ay^~'^  is  holomorphic  in  the  immediate  vicinity  of  a  and 
does  not  vanish  for  a ;  hence  a  is  a  zero  for  f  (z)  of  decreased  multiplicity 
n  —  1. 

If  z  =  CO  be  a  zero  of  f(z)  of  multiplicity  r,  then 

f(z)  =  Z-rcf>(^y 


39.]  UNIFORM    FUNCTION  77 

where  ^  (-J  is  holomorphic  for  very  large  values  of  z  and  does  not  vanish  at 
infinity.     Therefore 

The  coefficient  of  z'''-^  is  holomorphic  for  very  large  values  of  z,  and  does 
not  vanish  at  infinity;  hence  ^=  go  is  a  zero  oi  f  {z)  of  increased  multiplicity 
r  +  1. 

Corollary  I.  If  a  function  be  finite  at  infinity,  then  ^  =  oo  is  a  zero  of  the 
first  derivative  of  multiplicity  at  least  two. 

Corollary  II.     If  a  be  a  finite  zero  of  f{z)  of  multiplicity  n,  we  have 

f(z)^     n      ^  <^'{z) 
f{z)      z-a      c}){z)' 

Now  a  is  not  a  zero  of  cb  (z);  and  therefore    .      !  is  finite,  continuous,  uniform 

and  monogenic  in  the  immediate  vicinity  of  a.  Hence,  taking  the  integral 
of  both  members  of  the  equation  round  a  circle  of  centre  a  and  of  radius 
so  small  as  to  include  no  infinity  and  no  zero,  other  than  a,  of  f  (z) — and 
therefore  no  zero  of  cf)  (z) — we  have,  by  former  propositions, 

^nij/iz)"^'-''- 

40.  Theorem  IV.  A  function  must  have  an  infinite  value  for  sovne  finite 
or  infinite  value  of  the  variable. 

If  ilf  be  a  finite  maximum  value  of  the  modulus  for  points  in  the  plane, 
then  (§  22)  we  have 

where  r  is  the  radius  of  an  arbitrary  circle  of  centre  a,  provided  the  whole  of 
the  circle  is  in  the  region  of  continuity  of  the  function.  But  as  the  function 
is  uniform,  monogenic,  finite  and  continuous  everywhere,  this  radius  can  be 
increased  indefinitely ;  when  this  increase  takes  place,  the  limit  of 

|/(-)(a)l 

is  zero,  and  therefore/"**  (a)  vanishes.  This  is  true  for  all  the  indices  1,  2,.., 
of  the  derivatives. 

Now  the  function  can  be  represented  at  any  point  z  in  the  vicinity  of  a 
by  the  series 

f{a)  +  {z-  a)f'  (a)  -f  ^^V"  («)+-, 


78  INFINITIES   OF   A  [40. 

which  degenerates,  under  the  present  hypothesis,  to /(a),  so  that  the  function 
is  everywhere  constant.  Hence,  if  a  function  has  not  an  infinity  somewhere 
in  the  plane,  it  must  be  a  constant. 

The  given  function  is  not  a  constant;  and  therefore  there  is  no  finite 
limit  to  the  maximum  value  of  its  modulus,  that  is,  the  function  acquires 
an  infinite  value  somewhere  in  the  plane. 

Corollary  I.  A  function  must  have  a  zero  value  for  some  finite  or 
infinite  value  of  the  variable. 

For  the  reciprocal  of  a  uniform  monogenic  analytic  function  is  itself  a 
uniform  monogenic  analytic  function ;  and  the  foregoing  proposition  shews 
that  this  reciprocal  must  have  an  infinite  value  for  some  value  of  the 
variable,  which  therefore  is  a  zero  of  the  original  function. 

Corollary  II.     A  function  must  assume  any  assigned  value  at  least  once. 

Corollary  III.  Every  function  which  is  not  a  mere  constant  must  have 
at  least  one  singula^^ity,  either  accidental  or  essential.  For  it  must  have 
an  infinite  value :  if  this  be  a  determinate  infinity,  the  point  is  an  accidental 
singularity  (§  32) :  if  it  be  an  infinity  among  a  set  of  values  at  the  point,  the 
point  is  an  essential  singularity  (§§  32,  33). 

41.  Among  the  infinities  of  a  function,  the  simplest  class  is  that  con- 
stituted by  its  poles  or  accidental  singularities,  already  defined  (§  32)  by  the 
property  that,  in  the  immediate  vicinity  of  such  a  point,  the  reciprocal  of 
the  function  is  regular,  the  point  being  an  ordinary  (zero)  point  for  that 
reciprocal. 

It  follows  from  this  property  that,  because  (§  37)  an  ordinary  zero  of  a 
uniform  function  is  an  isolated  point,  every  pole  of  a  uniform  function  is  also 
an  isolated  point :  that  is  to  say,  in  some  non-infinitesimal  region  round  a 
pole  a,  no  other  pole  of  the  function  can  occur. 

Theorem  V.     A  function,  which  has  a  point  c  for  an  accidental  singularity, 

can  he  expressed  in  the  form 

{z  -  c)-'^  (/)  {z), 

where  n  is  a  finite  positive  integer  and  ^  (z)  is  a  continuous  function  in  the 
vicinity  of  c. 

Since  c  is  an  accidental  singularity  of  the  function /(ir),  the  function  -;-— 

is  regular  in  the  vicinity  of  c  and  is  zero  there  (§  32).  Hence,  by  §  38,  there 
is  a  finite  limit  to  the  multiplicity  of  the  zero,  say  n  (which  is  a  positive 
integer),  and  we  have 


41.]  UNIFOEM   FUNCTION  79 

where  %  (z)  is  uniform,  monogenic  and  continuous  in  the  vicinity  of  c  and  is 
not  zero  there.  The  reciprocal  of  x  (2),  say  cf)  (z),  is  also  uniform,  monogenic 
and  continuous  in  the  vicinity  of  c,  which  is  an  ordinary  point  for  (f)  (z) ; 
hence  we  have 

f(z)  =  {z-c)-<}>(z), 
which  proves  the  theorem. 

The  finite  positive  integer  n  measures  the  multiplicity  of  the  accidental 
singularity  at  c,  which  is  sometimes  said  to  be  of  multiplicity  n  or  of 
order  n. 

Another  analytical  expression  for  f{z)  can  be  derived  from  that  which 
has  just  been  obtained.  Since  c  is  an  ordinary  point  for  ^  (z)  and  not  a  zero, 
this  function  can  be  expanded  in  a  series  of  ascending,  positive,  integral 
powers  of  ^  —  c,  converging  in  the  vicinity  of  c,  in  the  form 

cl>{z)  =  P(z-c) 

=  Uq  +  Ui{z  -  C)+  ...  +  Un-1  (Z  -  C)"'-^  +  Un{z-cY  ^-  ... 
=  Wo  +  U^  (^  -  C)  +  ...  +  Un-i  {Z  -  Cy-'  +  (Z-  Cy  Q(Z-  C), 

where  Q  (z  —  c),  a  series  of  positive,  integral,  powers  of  ^  —  c  converging  in  the 
vicinity  of  c,  is  a  monogenic  analytic  function  of  z.     Hence  we  have 

the  indicated  expression  for/ (2^),  valid  in  the  immediate  vicinity  of  c,  where 
Q,{z  -  c)  is  uniform,  finite,  continuous  and  monogenic. 

CoEOLLARY.  A  function,  which  has  z  =  00  for  an  accidental  singularity  of 
multiplicity  n,  can  he  expressed  in  the  form 


<)> 


ivliere  ^{-\is  a  continuous  function  for  very  large  values  of  \z\,  and  is  not 
zero  when  ^  =  00  .     It  can  also  he  expressed  in  the  form 

aoz""  +  a,z''-'  +  ...+  an-i ^  +  Q  (") .' 

where  Q  [— )  is  uniform,  finite,  continuous  and  monogenic  for  very  large  values 

of  \z\. 

The  derivation  of  the  form  of  the  function  in  the  vicinity  of  an  accidental 
singularity  has  been  made  to  depend  upon  the  form  of  the  reciprocal  of  the 
function. 

As  the  accidental  singularities  of  a  function  are  isolated  points,  there  is 
only  a  finite  number  of  them  in  any  limited  portion  of  the  plane. 


80  INFINITIES   OF   A  [42. 

42.  We  can  deduce  a  criterion  which  determines  whether  a  given 
singularity  of  a  uniform  function  f{z)  is  accidental  or  essential. 

When  the  point  is  in  the  finite  part  of  the  plane,  say  at  c,  and  a  finite 
positive  integer  n  can  be  found  such  that 

{z-crf{z) 

is  not  infinite  at  c,  then  c  is  an  accidental  singularity. 

When  the  point  is  at  infinity  and  a  finite  positive  integer  n  can  be  found 

such  that 

z-f{z) 

is  not  infinite  when  z=  oo  ,  then  z=<X)  is  an  accidental  singularity. 

If  the  condition  be  not  satisfied  in  the  respective  cases,  the  singularity 
at  the  point  is  essential.  But  it  must  not  be  assumed  that  the  failure  of  the 
limitation  to  finiteness  in  the  multiplicity  of  the  accidental  singularity  is 
the  only  source  or  the  complete  cause  of  essential  singularity. 

Since  the  association  of  a  single  factor  with  the  function  is  effective  in 
preventing  an  infinite  value  at  the  point  when  the  condition  is  satisfied, 
it  is  justifiable  to  regard  the  discontinuity  of  the  function  at  the  point 
as  not  essential,  and  to  call  the  singularity  either  non-essential  or  accidental 
(§  32). 

43.  Theorem  VI.  The  poles  of  a  function,  that  lie  in  the  finite  part 
of  the  plane,  are  all  the  poles  (of  increased  multiplicity)  of  the  derivatives  of 
the  function  that  lie  in  the  finite  part  of  the  plane. 

Let  c  be  a  pole  of  the  function  f(z)  of  multiplicity  p  :  then,  for  any  point 

z  in  the  vicinity  of  c, 

f(z)  =  {z-crP<f>{z), 

where  ^  (z)  is  holomorphic  in  the  vicinity  of  c,  and  does  not  vanish  for  z  =  c. 
We  have 

/'  (z)  =  (z-  c)--P  <^'  (z)  -p(z-  c)-P-'  <j)  (z) 

=  (2-c)-P-mZ-c)cj,'{z)-pcf>{z)} 

where  v  (z)  is  holomorphic  in  the  vicinity  of  c,  and  does  not  vanish  for  z  =  c. 

Hence  c  is  a  pole  of/'  (z)  of  multiplicity  p  +  1.  Similarly  it  can  be  shewn 
to  be  a  pole  of  /*''*  (z)  of  multiplicity  p  +  r. 

This  proves  that  all  the  poles  of /(^•)  in  the  finite  part  of  the  plane  are 
poles  of  its  derivatives.  It  remains  to  prove  that  a  derivative  cannot  have 
a  pole  which  the  original  function  does  not  also  possess. 

Let  a  be  a  pole  of  /'  (z)  of  multiplicity  m  :  then,  in  the  vicinity  of  a, 
f'iz)  can  be  expressed  in  the  form 


43.]  UNIFORM  FUNCTION  81 

where  yjr  (z)  is  holomorphic  in  the  vicinity  of  a  and  does  not  vanish  for  z  =  a. 
Thus 

and  therefore  /'  (z)  =  ^(^  +  _i>l_  +  . . . , 

so  that,  integrating,  we  have 

•^  ^  ^  (m  -  1)  (^  -  a)»^-i      (;?i  _  2)  (^  -  a)"*-2      •"• 

When  there  is  no  term  in  log  (z  —  a)  in  this  expression,  f(z)  is  uniform  : 
that  is,  a  is  a  pole  o{f(z).  When  there  is  a  term  in  log  (z  -  a),  then  f(z)  is 
not  uniform. 

An  exception  occurs  in  the  case  when  m  is  unity:  for  then 

/'W=^^t'(«)  +  ^V"(«)+.... 

the  integral  of  which  leads  to 

f{z)  =  ylr{a)\og(z-a)  +  ..., 

so  that  f(z)  is  no  longer  uniform,  contrary  to  hypothesis.  Hence  a  derivative 
cannot  have  a  simple  pole  in  the  finite  part  of  the  plane  ;  and  so  this  exception 
is  excluded. 

The  theorem  is  thus  proved. 

Corollary  I.     The  r^^  derivative  of  a  function  cannot  have  a  pole  in  the 
finite  part  of  the  plane  of  multiplicity  less  than  r  +  1. 

Corollary  II.     If  c  be  a  pole  of  f{z)  of  any  order  of  multiplicity  [i,  and 
if  f^''"'  {z)  he  expressed  in  the  form 

an  «1 


there  are  no  terms  in  this  expression  with  the  indices  —  1,  —  2,  . , . ,  —  r. 
Corollary  III.     If  c  be  a  pole  of  f{z)  of  multiplicity  p,  we  have 
f'{z)^  -p    ^  <\>'{z) 
f{z)       z-c       (ji(z)' 

where  ^  (z)  is  a  holomorphic  function  that  does  not  vanish  for  ^  =  c,  so  that 
J/  /  \ 

^  is  a  holomorphic  function  in  the  vicinity  of  c.     Taking  the  integral  of 
(l>{z) 

f  (z) 

■^-^-^  round  a  circle,  with  c  for  centre,  with  radius  so  small  as  to  exclude  all 

other  poles  or  zeros  of  the  function  f(z),  we  have 

A^[l(^dz  =  -p. 

Corollary  IV.     If  a  simple  closed  curve  include  a  number  N  of  zeros 
of  a  uniform  function  f{z)  and  a  number  P  of  its  poles,  in  both  of  which 

P    F.  6 


82:  INFINITIES   OF   A  [48. 

numbers  account  is  taken  of  possible  multiplicity,  and  if  the  curve  contain 
no  essential  singularity  of  the  function,  then 


liri  J  f  (z) 


_        27rijf{z) 

the  integral  being  taken  round  the  curve. 

.       f  (z)     .     . 
The  only  infinities  of  the  function  "^  ^ ,  .   within  the  curve  are  the  zeros 

and  the  poles  of  /(^^).  Round  each  of  these  draw  a  circle  of  radius  so  small 
as  to  include  it  but  no  other  infinity;  then,  by  Cor.  II.  §  19,  the  integral 
found  the  closed  curve  is  the  sum  of  the  values  when  taken  round  these 
circles.     By  the  Corollary  II.  §  39  and  by  the  preceding  Corollary  III.,  the 

sum  of  these  values  is 

=  Xn  —  1p 

It  is  easy  to  infer  the  known  theorem  that  the  number  of  roots  of  a 
polynomial  of  order  n  is  n,  as  well  as  the  further  result  that  2tt  {N  —  P) 
is  the  variation  of  the  argument  oi  f{z),  when  z  describes  the  closed  curve 
in  a  positive  sense. 

Ex.  1.  A  function  f{z)  is  uniform  over  an  area  bounded  by  a  contour;  it  has  no 
essential  singularity  within  that  area ;  and  it  has  no  zero  and  no  pole  on  the  contour. 
Prove  that  the  change  in  the  argument  oi  f{z\  as  z  makes  a  complete  description  of  the 
contour,  is  27r  (n  ~p),  where  n  is  the  number  of  zeros  and  p  is  the  number  of  poles  within 
the  area.  (Cauchy.) 

Ex.  2.  Prove  that,  if  F{2)  be  holomorphic  over  an  area  of  simple  contour,  which  con- 
tains roots  «!,  a2, ...  of  multiplicity  m^,  m^,...  and  poles  Cj,  C2, ...  of  multiplicity  p\,  P2-,--. 
respectively  of  a  function  f{z)  which  has  no  other  singularities  within  the  contour,  then  ' 

-\.  fF{z/-^dz=  2  mrF{a,)-  2  ^,i^(c,), 

^^*  J  J\^)  J-=r  r=l 

the  integral  being  taken  t'ound  the  contour. 

In  particular,  if  the  contour  contains  a  single  simple  root  a  and  no  -  singularity,  then 
that  root  is  given  by 

«=s— •  \^  -rri^z, 
27^^j    -f\z) 

the  integral  being  taken  as  before.  (Laurent.) 

Ex.  3.  Discuss  the  integral  in  the  preceding  example  when  F{z)  =  \ogz,  and  the  origin 
is  excluded  by  a  small  circle  of  radius  p,  less  than  the  smallest  of  the  quantities  |  a^  \  and 
\c^\.  (Goursat.) 

44.     Theorem  VII.     If  infinity  he  a  pole  of  f{z),  it  is  also  a  pole  of 
f'{z)  only  when  it  is  a  multiple  pole  of  f{z). 

Let  the  multiplicity  of,  the  pole  for  f{z)  be  n ;  then  for  very  large  values 
of  z  we  have 

f(,)  =  ,n^Q^ 


44.]  UNIFORM   FUNCTION  83 

where  (f>  is  holomorphic  for  very  large  values  of  z  and  does  not  vanish  at 
infinity;   hence 

/'W=...-.|„^(i)-i<^'(l)i 

The  coefiicient  of  z'^-^  is  holomorphic  for  very  large  values  of  z  and  does  not 
vanish  at  infinity ;  hence  infinity  is  a  pole  of/'  (z)  of  multiplicity  n  —  1. 

If  n  be  unity,  so  that  infinity  is  a  simple  pole  of /(^),  then  it  is  not  a 
pole  of  f'(z);   the  derivative  is  then  finite  at  infinity. 

45.     Theorem  VIII.     A  function,  which  has  no  singularity  in  a  finite 
part  of  the  plane,  and  has  z=  qc  for  a  pole,  is  a  polynomial  in  z. 

Let  n,  necessarily  a  finite  integer,  be  the  order  of  multiplicity  of  the  pole 
at  infinity :  then  the  function  f{z)  can  be  expressed  in  the  form 


tto^'^  +  tti^'^-i  + +  an-^z  +  Q  (-)  , 


where  Q\~]  is  a  holomorphic  function  for  very  large  values  of  z,  and  is  finite 


(or  zero)  when  z  is  infinite. 

Now  the  first  n  terms  of  the  series  constitute  a  function  which  has  no 
singularities  in  the  finite  part  of  the  plane:  and  y(2^)  has  no  singularities 

in  that  part  of  the  plane.     Hence  Q\-]  has  no  singularities  in  the  finite  part 

of  the  plane :  it  is  finite  for  infinite  values  of  z.     It  thus  can  never  have  an 
infinite  value :  and  it  is  therefore  merely  a  constant,  say  a„.     Then 

f{z)  =  a^z'^  +  a^z'^-^  + +  an-iZ  +  a„, 

a  poljTiomial  of  degree   equal  to  the  multiplicity  of  the  pole  at  infinity, 
supposed  to  be  the  only  pole  of  the  function. 

The  above  result  may  be  obtained  also  in  the  following  manner. 
Since  z  =  ao  is  a  pole  of  multiplicity  n,  the  limit  of  z-'^f{z)  is  not  infinite 

when  0  =  00  . 

Now  in  any  finite  part  of  the  plane  the  function  is  everywhere  finite,  so 

that  we  can  use  the  expansion 

/(^)=/(0)  +  ^/'(0)  + +^^/('^)(0)  +  E, 

^«+i  r/(0  dt 


where  ^=27rvU-+^^ 

the  integral  being  taken  round  a  circle  of  any  radius  r  enclosing  the  point  z 
and  having  its  centre  at  the  origin.  As  the  subject  of  integration  is  finite 
everywhere  along  the  circumference,  we  have,  by  Darboux's  expression  in 
(IV.)  §  15, 

6—2 


84  TRANSCENDENTAL   AND  [45. 

where  t  is  some  point  on  the  circumference  and  X,  is  a  quantity  of  modulus 
not  greater  than  unity. 

Let  T  =  re^°- ;  then 

r 

/"(t) 
By  definition,  the  limit  of   -—■  as  t  (and  therefore  r)  becomes  infinitely 

large  is  not  infinite ;  in  the  same  case,  the  limit  of  (1  —  e~"  j      is  unity. 

Since  |  \  |  is  not  greater  than  unity,  the  limit  of  Xjr  in  the  same  case  is  zero ; 
hence  with  indefinite  increase  of  r,  the  limit  of  R  is  zero,  and  so 

f{z)^f{0)  +  zf'{0)  + H-f;/<'nO), 

shewing  as  before  that  f{z)  is  a  polynomial  in  z. 

46.  As  the  quantity  n  is  necessarily  a  positive  integer*,  there  are  two 
distinct  classes  of  functions  discriminated  by  the  magnitude  of  n. 

The  first  (and  the  simpler)  is  that  for  which  n  has  a  finite  value.  The 
function  then  contains  only  a  finite  number  of  terms,  each  with  a  positive 
integral  index;  it  is  a  polynomial  or  a  rational  integral  function  of  z,  of 
degree  n. 

The  second  (and  the  more  extensive,  as  significant  functions)  is  that 
for  which  n  has  an  infinite  value.  The  point  ^^  =  oc  is  not  a  pole,  for  then 
the  function  does  not  satisfy  the  test  of  §  42 :  it  is  an  essential  singularity 
of  the  function,  which  is  expansible  in  an  infinite  converging  series 
of  positive  integral  powers.  To  functions  of  this  class  the  general  term 
transcendental  is  applied. 

The  number  of  zeros  of  a  function  of  the  former  class  is  known :  it  is 
equal  to  the  degree  of  the  function.  It  has  been  proved  that  the  zeros  of  a 
transcendental  function  are  isolated  points,  occurring  necessarily  in  finite 
number  in  any  finite  part  of  the  region  of  continuity  of  the  function,  no 
point  on  the  boundary  of  the  part  being  an  essential  singularity;  but  no 
test  has  been  assigned  for  the  determination  of  the  total  number  of  zeros  of 
a  function  in  an  infinite  part  of  the  region  of  continuity  f. 

Again,  when  the  zeros  of  a  polynomial  are  given,  a  product-expression  can 
at  once  be  obtained  that  will  represent  its  analytical  value.     Also  we  know 

*  It  is  unnecessary  to  consider  the  zero  value  of  n,  for  the  function  is  then  a  polynomial  of 
order  zero,  that  is,  it  is  a  constant. 

t  In  connection  with  the  zeros  of  a  transcendental  function,  as  expressed  in  a  Taylor's  series, 
a  paper  by  Hadamard,  Liouville,  4"«  S^r.,  t.  viii,  (1892),  pp.  101—186,  may  be  consulted  with 


46.]  EATIONAL  UNIFORM  FUNCTIONS  85 

that,  if  a  be  a  zero  of  any  uniform  analytic  function  of  multiplicity  n,  the 
function  can  be  represented  in  the  vicinity  of  a  by  the  expression 

where  (f)(2)  is  holomorphic  in  the  vicinity  of  a.  The  other  zeros  of  the 
function  are  zeros  of  (b(z);  this  process  of  modification  in  the  expression 
can  be  continued  for  successive  zeros  so  long  as  the  number  of  zeros  taken 
account  of  is  limited.  But  when  the  number  of  zeros  is  unlimited,  then  the 
inferred  product-expression  for  the  original  function  is  not  necessarily  a 
converging  product ;  and  thus  the  question  of  the  formal  factorisation  of  a 
transcendental  function  arises. 

47.  Theorem  IX.  A  function,  all  the  singularities  of  which  are  accid- 
ental, is  a  rational  nier amorphic  function. 

Since  all  the  singularities  are  accidental,  each  must  be  of  finite 
multiplicity ;  and  therefore  infinity,  if  an  accidental  singularity,  is  of  finite 
multiplicity.  All  the  other  poles  are  in  the  finite  part  of  the  plane;  they 
are  isolated  points  and  therefore  only  finite  in  number,  so  that  the  total 
number  of  distinct  poles  is  finite  and  each  is  of  finite  order.     Let  them  be 

«!,  eta, ,  a^  of  orders  mj,  m^,  ,  m^  respectively:    let  m  be  the  order 

of  the  pole  at  infinity :  and  let  the  poles  be  arranged  in  the  sequence  of 
decreasing  moduli  such  that  |  a^  |  >  |  a^_i  |  > >  |  a^  | . 

Then,  since  infinity  is  a  pole  of  order  m,  we  have 

f{z)  =  a^z'^  +  a^-.z'"'-^  + +a,z+f  (z), 

where  fo{z)  is  not  infinite  for  infinite  values  of  z.     Now  the  polynomial 

m 

2  aiZ'^  is  not  infinite  for  any  finite  value  of  z ;  hence  f  (z)  is  infinite  for  all 

the  finite  infinities  of  f{z)  and  in  the  same  way,  that  is,  the  function  f  (z) 
has  tti, ,  a^  for  its  poles  and  it  has  no  other  singularities. 

Again,  since  a^  is  a  finite  pole  of  multiplicity  m^,  we  have 

where  f  (z)  is  not  infinite  for  z^a^  and,  as  f  (z)  is  not  infinite  for  z=co , 
evidently  f  (z)  is  not  infinite  for  2^  =  00  .  Hence  the  singularities  of  f  (z)  are 
merely  the  poles  aj,  ,  a^_i;  and  these  are  all  its  singularities. 

Proceeding  in  this  manner  for  the  singularities  in  succession,  we  ultimately 
reach  a  function  f^  (z)  which  has  only  one  pole  aj  and  no  other  singularity, 
so  that 

/'(^)  =  (^+ +.-^.+^w. 

where  g{z)  is  not  infinite  for  z  =  ai.     But  the  function  f^,(z)  is  infinite  only 


86  UNIFORM  [47. 

for  z  =  aT^,  and  therefore  g{z)  has  no  infinity.     Hence  g{z)  is  only  a  constant, 
say  ko :  thus 

g{z)  =  K 

Combining  all  these  results  we  have  a  finite  number  of  finite  series  to  add 
together :  and  the  result  is  that 

where  g-i.{z)  is  the  series  A;o  +  aa^+ ■{■a.nZ^,  and  ^^-j-x  is  the  sum  of  the 

finite  number  of  fractions.     Evidently  g^  {z)  is  the  product 

{z  -  ai)""'  {z  -  ct^)'"' {z-  a^)'^'^ ; 

and  ^2  (z)  is  at  most  of  degree 

mi+  7n2+ +  m^  —  1- 

If  F{z)  denote  gi(z)g3(z)+g2{z),  the  form  of  f(z)  is 

l(z) 

9z{zy 

that  is,  f{z)  is  a  rational  meromorphic  function. 

It  is  evident  that,  when  the  function  is  thus  expressed  as  a  rational 
fraction,  the  degree  of  F{z)  is  the  sum  of  the  multiplicities  of  all  the  poles 
when  infinity  is  a  pole. 

Corollary  I.  A  function,  all  the  singularities  of  which  are  accidental, 
has  as  many  zeros  as  it  has  accidental  singularities  in  the  'plane. 

When  2  =  00  is  a  pole,  it  follows  that,  because  f{z)  can  be  expressed  in 
the  form 

F{z) 

9M' 
the  function  has  as  many  zeros  as  F{z),  unless  one  such  should  be  also  a  zero  of 
GTs  {z).  But  the  zeros  of  ^3  {z)  are  known,  and  no  one  of  them  is  a  zero  oiF{z),  on 
account  of  the  form  of  f{z)  when  it  is  expressed  in  partial  fractions.  Hence 
the  number  of  zeros  off(z)  is  equal  to  the  degree  of  F  (z),  that  is,  it  is  equal 
to  the  number  of  poles  of  f(z). 

When  2^  =  00  is  not  a  pole,  two  cases  are  possible ;  (i)  the  function  f(z)  may 
be  finite  for  ^  =  00  ,  or  (ii)  it  may  be  zero  for  z  =  00  .  In  the  former  case,  the 
number  of  zeros  is,  as  before,  equal  to  the  degree  of  F(z),  that  is,  it  is  equal 
to  the  number  of  infinities. 

In  the  latter  case,  if  the  degree  of  the  numerator  F{z)  be  k  less  than 
that  of  the  denominator  g^  (z),  then  z  =  cc  is  a  zero  of  multiplicity  k  ;  and  it 
follows  that  the  number  of  zeros  is.  equal  to  the  degree  of  the  numerator 
together  with  k,  so  that  their  number  is  the  same  as  the  number  of  accidental 
singularities. 


47:]i  RATIONAL  FUNCTIONS  ^1l 

GoROLLARY  II.  At  the  beginning  of  the  proof  of  the  theorem:  of '  the 
present  section,  it  is  proved  that  a  function,  all  the  singularities  of  which  are 
accidental,  has  only  a  finite  number  of  such  singularities. 

Hence,  by  the  preceding  Corollary,  such  a  function  can  have  only  a  finite 
number  of  zeros. 

If,  therefore,  the  number  of  zeros  of  a  function  be  infinite,  the  function 
must  have  at  least  one  essential  singularity. 

Corollary  III.  When  a  uniform  function  has  no  essential  singularity, 
if  the  (finite)  number  of  its  poles,  say  Ci,  ...,  c,„,  be  m,  no  one  of  them  being 
at  ^^=  00 ,  and  if  the  number  of  its  zeros,  say  aj,  ...,  a^y^,  be  also  m,  no  one  of 
them  being  at  ^-  =  00,  then  the  function  is 

-pr  I  ^         ar 


except  possibly  as  to  a  constant  factor. 

When  z=  CO  is  a  zero  of  order  n,  so  that  the  function  has  tn  —  n  zeros,  say 
tti,  a^,  ...,  in  the  finite  part  of  the  plane,  the  form  of  the  function  is 

111  -  n 

n  {z  —  a^ 

r=l . 

m  ' 

U{z-c,) 

and,  when  z  =  00  is  a  pole  of  order  p,  so  that  the  function  has  m  —  p  poles, 
say  Ci,  C2,  ...,  in  the  finite  part  of  the  plane,  the  form  of  the  function  is 

m 

U  (z  —  a,.) 

r=l 

m-p 
U(z-Cr) 
r=l 

Corollary  IV.     All  the  singularities  of  rational  meromorphic  functions 
are  accidental. 

48.     Some  properties  of  the  simplest  functions  thus  defined  may  con- 
veniently be  given  here*.     We  shall  begin  with  polynomials. 

(i)     Let  P  {z)  denote 

a.nz'^  +  a^_i^"*-^  + +  a^z-\-ao, 

where  the  coefficients  a  are  constants  which  may  be  complex;  it  is  con- 
tinuous, for  every  one  of  the  finite  number  of  terms  is  continuous  ;  it 
is  finite  for  all  finite  values  of  z  ;  and  \P{z)\  tends  to  become  infinite  as 
I  z  I  tends  to  become  infinite. 

*  For  these  and  other  properties,  reference  may  be  made  to  Jordan's  Cours  d' Analyse,  t.  i, 
p.  198. 


88  SOME   PROPERTIES   OF  [48. 

Further,  a  finite  value  oi  \z\  can  be  determined  which  will  make  \P{z)\ 
greater  than  any  assigned  finite  value,  say  A .     For  we  have 

I  P  (^)  I  >  I  a,^  1  I  ^  I'"  -  j  a„,_i  I  I  z  \^-^  -  I  a,n-2 1  |  ^  l"*"'  - -  |  o^  [  1  ^  ]  -  |  ao ! 

IiJivn 1  tZ"m 9  ^1  ^0 

■^'^'    ^"*"^l        Ul  1^1^        \z\^-^     \z\ 


so  that,  when  \z\  >  1, 


-  1 r  { 1  ttm-i  I  +  I  C^m-2  I  + +  I  «!  I  +  !  tto 

\z\ 


Now  take 


c  =  , r{|  a„^_l  l  +  l  (X^-sI  + +  I  (Xil  +  jao  1  +  ^} ; 


then  \P{z)\>A  +  \ar,n\{\z\  —  c). 

Hence  if  \z\,  already  supposed  greater  than  unity,  is  also  greater  than  c 
should  c  be  greater  than  unity,  we  have 

\P{z)\>A, 

for  values  of  z  such  that  |  ^^  |  >  1,  |  2^  I  >  c. 

(ii)  Next,  the  equation  P{z)  =  0  always  has  a  root.  The  quantity 
I  P  (2^)  I  is  continuous,  is  never  negative,  and  tends  to  become  infinite  as 
I  z  j  tends  to  become  infinite.  Hence,  if  it  cannot  be  zero,  there  must  be 
at  least  one  minimum  value  greater  than  zero  below  which  it  cannot  fall. 
Denote  this  value  by  fi ;  and  suppose  it  acquired  for  the  value  c  of  z,  so  that 

|P(c)|  =  ^. 

Construct  a  circle  of  radius  greater  than  \c\,  and  take  a  place  c  +  h  lying 
within  that  circle.     Then 

P{c  +  h)  =  P(c)+hP'{c)-^ +^£P^^-Kc), 

where  the  coefficient  of  h^  is  a^,  a  quantity  different  from  zero.  As 
(hypothetically)  P(c)  is  not  zero,  the  first  term  and  the  last  term  in 
P{c  +  h)  do  not  disappear;  but  intervening  terms  may  disappear,  and 
so  we  write 

P(c  +  h)  =  P{c)+  brh"-  +  br+yh'-+'  + +  a„,h'^, 

where  r  is  the  lowest  index  of  the  powers  of  h  that  survive.  Now  choose  h 
in  such  a  way  that  |  A.  |  is  small  enough  to  secure  the  inequality 

|P^|.|/i|'-<|P(c)|<^, 
while  at  the  same  time 

r  {arg.  h]  +  {arg.  Br]  =  {arg.  P  (c)}  +  (2?i  +  1)  tt, 


48.]  RATIONAL   FUNCTIONS  .    89 

SO  that  the  arguments  of  Brh''  and  P  {c)  differ  by  an  odd  multiple  of  tt. 
Hence,  if 

P{c)  =  \P{c)\e^\ 

then  Brhr  =  -  \  Brlf  \  e^\ 

so  that  P  (c)  +  Brh'  =  { |  P  (c)  |  -  |  Brh^'  \  ]  e'\ 

and  therefore  i  -P  (c)  +  Brh''  \  =  \P  {c)\-\Brlr  \. 

Now  P{c->rli)  =  P{c)^  h?-Br  +  /l^'+^^.+i  +  . . . ; 

consequently 

I  P(c  +  /i)  I  <  I  P(c)  +/i'-5,.  I  + 1  A'^+i^^+i  i  +  ... 

^  I  P  (c)  I  -  I  /^'•P,  I  +  I  /i'-+i  1 1  P,+i  I  +  . . . 

As  I  P^  I  differs  from  zero,  the  coefficient  of  —\'h  I*"  on  the  right-hand  side  is 
positive  when  |  A  |  is  quite  small ;  consequently,  for  such  values  of  A, 

|P(c  +  /i)|<|P(c)|, 

that  is,  the  modulus  of  P  {£)  in  the  immediate  vicinity  of  c  can  be  made  less 
than  I  P(c)  |,  contrary  to  the  hypothesis  that  \P  {g)\  is  a  minimum  different 
from  zero.  Thus  there  cannot  be  a  minimum  different  from  zero,  and  |  P  (^)  | 
can  always  be  diminished  so  long  as  it  is  different  from  zero.  Hence  there 
must  be  a  value  of  z  which  makes  P  iz)  zero. 

It  now  follows,  by  the  customary  argument,  that  there  are  m,  such  values, 
(iii)     Any  rational  function  of  z,  say  w,  is  of  the  form 

w  = 

P(^)' 

where  Q  {z)  and  P  {£)  are  polynomials  in  z  of  degrees  m  and  n  respectively. 

Every  zero  of  Q  (z)  is  a  zero  of  w.  Every  zero  of  P  (z)  is  a  pole  of  w. 
The  place  z=  oo  is  a  pole  of  w  if  m  >  n,  and  it  is  of  order  m  —  n;  it  is  a  zero 
of  w  ii  m<n,  and  it  is  of  order  m  —  n;  it  is  neither  if  m  =  w.  The  number 
of  poles  is  equal  to  the  number  of  zeros,  being  the  greater  of  the  two 
integers  m  and  n. 

Two  results,   which  are  of  use  in  one  method  of  establishing  some  of  the  special 
cases  of  Abel's  theorem  concerning  integrals  of  algebraic  functions,  may  be  noted. 

Let  the  roots  of  P  {z)  be  simple,  say  aj, ... ,  a„.     Let  A  be  the  coeflficient  of  a"  in  P {z). 
Then 

(a)    when  m,  the  order  of  Q  (z),  is  less  than  n  —  l, 

(/3)    when  m  =  n-l,  and  Bi  is  the  coefficient  of  0™-Mn  $  {z), 

rtiP'iar)       A' 


CHAPTER   V. 

Transcendental  Integral  Functions. 

49.  We  now  proceed  to  consider  the  properties  of  uniform  functions 
which  have  essential  singularities. 

The  simplest  instance  of  the  occurrence  of  such  a  function  has  already 
been  referred  to  in  §  42  ;  the  function  has  no  singularity  except  at  ^=00, 
and  that  value  is  an  essential  singularity  solely  through  the  failure  of  the 
limitation  to  finiteness  that  would  render  the  singularity  accidental.  The 
function  is  then  an  integral  function  of  transcendental  character ;  and  it  is 
analytically  represented  (§  26)  by  G  (z),  an  infinite  series  in  positive  powers  of 
z,  which  converges  everywhere  in  the  finite  part  of  the  plane  and  acquires 
an  infinite  value  at  infinity  alone. 

The  preceding  investigations  shew  that  uniform  functions,  all  the  singu- 
larities of  which  are  accidental,  are  rational  functions  of  the  variable — their 
character  being  completely  determined  by  their  uniformity  and  the  accidental 
nature  of  their  singularities,  and  that  among  such  functions  having  the  same 
accidental  singularities  the  discrimination  is  made,  save  as  to  a  constant 
factor,  by  means  of  their  zeros. 

Hence  the  zeros  and  the  accidental  singularities  of  a  rational  function 
determine,  save  as  to  a  constant  factor,  an  expression  of  the  function  which 
is  valid  for  the  whole  plane.  A  question  therefore  arises  how  far  the  zeros 
and  the  singularities  of  a  transcendental  function  determine  the  analytical 
expression  of  the  function  for  the  whole  plane. 

We  have  to  deal  with  converging  products ;  it  is  therefore  convenient  to  state,  as  for 
converging  series,  the  definitions  of  the  terms  used.  For  proofs  of  the  statements, 
developments,  and  applications,  as  well  as  the  various  tests  of  convergence,  the  references 
which  were  given  at  the  beginning  (p.  21)  of  Chapter  II.  may  be  consulted. 

When  a  series  of  quantities 

Ui,  u^i  U3, ...  ad  inf. 
is  given,  the  infinite  product 

n  (i+«s) 

s=0 


49.]  DEFINITIONS   AS   TO    CONVERGENCE  '&i 

is  said  to  converge  when  the  limit  of  n„ ,  where 

n„=n(l+z(,), 
s=o 

as  n  increases  indefinitely,  is  a  unique  finite  quantity  P  different  from  zero.  (The  last 
condition,  that  P  should  not  be  zero,  is  omitted  by  some  writers :  as  our  products  arise 
through  quantities  involving  z  and  do  not  vanish  for  every  value  of  z,  no  difficulty 
is  caused.  See  also  Pringsheim,  Math.  Ann.,  t.  xxxiii,  p.  125.)  When,  in  the  same 
circumstances,  the  limit  of  n„  either  is  infinite,  or  is  zero,  or  if  finite  is  not  unique 
(that  is,  may  be  one  of  several  quantities),  the  infinite  product  is  said  to  diverge. 

The  necessary  and  sufficient  conditions  that  the  product  should  converge  are :  that  n„ 
is  finite  and  different  from  zero,  however  large  n  may  be ;  and  that,  corresponding  to 
every  finite  positive  quantity  e  taken  as  small  as  we  please,  an  integer  m  can  be  found 
such  that 

n„+,.       I 
li —    n"' 

for  all  integers  n  such  that  n  ^  m  and  for  every  integer  r. 
When  the  product 

n  (1  +  I  M«  I ) 

S=0 

converges,  the  product 

n(i+?0 

also  converges ;  and  it  is  said  to  converge  absolutely.  In  an  absolutely  converging  product, 
the  factors  may  be  arranged  in  any  order  without  affecting  the  convergence  or  the  value 
of  the  product.  The  convergence  is  sometimes  called  unconditional.  The  necessary  and 
sufficient  condition  for  the  absolute  convergence  of  the  joroduct  is  that  the  series 

Ml,    U2,    U^,...  , 

should  converge  absolutely. 

When  the  series  %,  ^2,  «3, ...  does  not  converge  absolutely,  while  the  product  11  (l  +  iCs) 

converges,  the  convergence  of  the  infinite  product  is  called  coiiditional.  The  tests  differ 
according  as  the  quantities  u  are  real  or  complex :  we  shall  not  be  concerned  with 
conditionally  converging  infinite  products. 

The  instances,  which  we  shall  have  to  consider,  are  those  where  the  quantities  m 
depend  upon  a  variable  (complex)  quantity  s.  TJhe;  convergence  is  required  as  z  varies,  the 
quantities  u  being  regular  functions  throughout '  the  region  in  which  z  varies.  When  any 
small  quantity  S  has  been  chosen,  and  a  positive  integer  m  can  be  determined,  such 
that 

5z^_i  !<s, 

for  every  value  of  n^m,  for  all  positive  integers  r,  and  for  all  values  of  z  within  the 
region,  the  convergence  of  the  infinite  product  is  said  to  be  uniform  within  the  region. 

Convergence  of  an  infinite  product  may  be  uniform  without  being  unconditional; 
it  may  be  unconditional  without  being  uniform. 

When  an  infinite  product  converges  uniformly  and  unconditionally  within  a  given 
region,  then  every  partial  product,  which  is  formed  by  taking  any  number  of  factors 
in  the  original  product,  also  converges  uniformly  and  unconditionally  within  that  region. 

When  an  infinite  product  converges  uniformly  and  unconditionally  within  a  region, 
the  series  constituted  by  the  logarithms   of  the  factors  (that  is,  taking  the  principal 


92  CONVEEGING  [49. 

logarithms,  whose  imaginary  part  is  ia,  where  tt  ^  a  ^  -tt)  also  converges  uniformly 
and  unconditionally  at  all  points  within  the  region  except  the  zeros  of  the  factors :  and 
the  logarithmic  series  can  be  differentiated,  if  the  series  of  the  derivatives  of  the  terms  in 
this  logarithmic  series  itself  converges  uniformly.  In  other  words,  we  can  (under  the 
condition  stated)  take  logarithmic  derivatives  of  an  infinite  product,  which  converges 
uniformly  and  unconditionally  within  a  region ;  and  the  infinite  series  is  equal  to  the 
logarithmic  derivative  of  the  value  of  the  product. 

50.  We  shall  consider  first  how  far  the  discrimination  of  transcendental 
integral  functions,  vs^hich  have  no  infinite  value  except  for  ^  =  oo  ,  is  effected 
by  means  of  their  zeros*. 

Let  the  zeros  «!,  a^,  a^,  ...  be  arranged  in  order  of  increasing  moduli;  a 
filnite  number  of  terms  in  the  series  may  have  the  same  value  so  as  to  allow 
for  the  existence  of  a  multiple  zero  at  any  point.  After  the  results  stated 
in  §  46,  it  will  be  assumed  that  the  number  of  zeros  is  infinite ;  that, 
subject  to  limited  repetition,  they  are  isolated  points ;  and,  in  the  present 
chapter,  that,  as  n  increases  indefinitely,  the  limit  of  |  a„  ]  is  infinity.  And  it 
will  be  assumed  that  |  aj :  >  0,  so  that  the  origin  is  temporarily  excluded  from 
the  set  of  zeros. 

Let  z  be  any  point  in  the  finite  part  of  the  plane.  Then  only  a  limited 
number  of  the  zeros  can  lie  within  and  on  a  circle  centre  the  origin  and 
radius  equal  to  j^ | ;  let  these  be  aj,  ag,  ... ,  a^-i,  and  let  a^  denote  any  one  of 
the  other  zeros.  We  proceed  to  form  the  infinite  product  of  quantities  u^, 
where  Ur  denotes 

and  gr  is  a  rational  integral  function  of  z  which,  being  subject  to  choice,  will 
be  chosen  so  as  to  make  the  infinite  product  converge  everywhere  in  the 
plane.     We  have 

a  series  which  converges  because  ]  ^  |  <  |  a,,  j .     Now  let 

gr=  2,  -[  —  ]    , 

then  \ogUr=-  t  -  (^)   , 

n=s  "'  \^r' 

and  therefore  w,.  =  e   "~* "  ^"' 

*  The  following  investigations  are  based  upon  the  famous  memoir  by  Weierstrass,  "  Zur 
Theorie  der  eindeutigen  analytisehen  Punctionen,"  published  in  1876:  see  his  Ges.  Werke,  t.  ii, 
pp.  77—124. 

In  connection  with  the  product-expression  of  a  transcendental  function,  Cayley,  "Memoire  sur 
les  fonctions  doublement  periodiques,"  Liouville,  t.  x,  (1845),  pp.  385—420,  or  Collected  Mathe- 
matical Papers,  vol.  i,  pp.  156 — 182,  should  be  consulted. 


50.] 


INFINITE   PRODUCTS 

1  f  z  \» 


93 


Hence 


if  the  expression  on  the  right-hand  side  is  finite,  that  is,  if  the  series 

2    2  -  (- ) 

r=k  n=s  ^  \^rJ 

converges.     Denoting  the  modulus  of  this  series  by  M,  we  have 


so  that 


00  00         1 

i/<  2    2  - 

r='k  n=s  ^ 

CO  00 

sM<  2     2 


<  2 

r=k 


1-    - 


whence,  since  1  — 
sum,  we  have 


is  the  smallest  of  the  denominators  in  terms  of  the  last 


sM  \l- 


2^ 
— 


<  \z\'  2 


'=]c    dr 


If,  as  is  not  infrequently  the  case,  there  be  any  finite  integer  5  for  which 
(and  therefore  for  all  greater  indices)  the  series 

00  1 

2 


7»=1   1  W'^ 

00 

and  therefore  the  series   2  |  a^  |~*,  converges,  we  choose  s  to  be  that  least 

r  =  k 

integer.     The  value  of  M  then  is  finite  for  all  finite  values  of  z ;  the  series 

1  f  z 


2    2  -(-) 
converges  unconditionally,  and  therefore 


n  Ur 

r=k 


is  a  product,  which  converges  unconditionally,  when 


Mr  =  I   1 I  e  ' 

ar 


9'4  WEIERSTRASS'S    CONVERGING 

Moreover,  it  converges  uniformly.     We  have 


i+v 

n    Ur 

r=k 

n  w,. 

r=k 


-1 


l+V 

-  S 


l+V    00     1   I     y    I  m 


L      I  2;  li    r=i\ai\' 

<e     (      l^jlJ  -1. 


[50. 


Now  the  series    2 


:1       tlj- 


converges ;   hence  when   any  finite  quantity  e  is 


assigned,  we  can  choose  an  integer  I  such  that,  for  all  integers  l"  ^  I, 


r=l"    ^r 


<  e. 


Denoting  by  p  any  positive  quantity  which  is  less  than  |  a^  ] ,  consider  a  region 
in  the  ^r-plane  given  by  \2\^  p.  Let  B  denote  any  assigned  finite  quantity, 
however  small ;  and,  after  8  is  assigned,  choose  a  quantity  e  so  that 


s\l 


ai 


e<  Log(l  +  8), 


taking  the  principal  logarithm.     Then 


l+V 
n   Ur 

r=k 
I 
n    Ur 

r=k 

-1 

<  e 


l<(l  +  g)-l<S; 


shewing  that  the  product  converges  uniformly  for  all  values  of  2;  such  that 
\z\^p.     But  I  can  be  taken  as  large  as  we  please :   so  that  the  product 
converges  uniformly  for  all  finite  values  of  z. 
Let  the  finite  product 


}  If  z  y 


m  =  l  [\  (^mJ  j 


be  associated  as  a  factor  with  the  foregoing  infinite  converging  product.    Then 

the  expression 

f(z)=U    '-       ^  .-'-«Uv 


a. 


is  an  infinite  ■prvduct,  converging  uniformly  and  unconditionally  for  all  finite 

00 

values  of  z,  -provided  the  finite  integer  s  he  such  as  to  make  the  series  S  |  a^  |~* 

r=\ 

converge. 


51.] 


INFINITE   PRODUCT 


95 


51.     But  it  may  happen  that  no  finite  integer  s  can  be  found  which  will 
make  the  series 


2  \ar\- 

?•  =  ! 


converge*.     We  then  proceed  as  follows. 

Instead  of  having  the  same  index  s  throughout  the  series,  we  associate 
with  every  zero  a^  an  integer  nir,  chosen  so  as  to  make  the  series 


2 

n  =  l 


converge.  To  obtain  these  integers,  we  take  any  series  of  decreasing  real 
positive  quantities  e,  ej,  eg,  ...,  such  that  (i)  e  is  less  than  unity  and  (ii)  they 
form  a  converging  series ;  and  we  choose  integers  irir  such  that 

These  integers  make  the  foregoing  series  of  moduli  converge.  For, 
neglecting  the  limited  number  of  terms  for  which  |  ^r  |  ^  |  a  |  e,  and  taking  the 
first  term  for  a^  such  that 

I  z 


ak 


<e, 


we  have  for  all  succeeding  terms  {r  =  k-^\,  k+ '2,  ...) 

z 


and  therefore 


|m_+i 


<e, 


^  gm^+i  ^  ^^ 


Hence,  except  for  the  first  k—1  terms,  the  sum  of  which  is  finite,  we  have 


n=k 


1    [  Z\M  1 

an  \aj     I      \z\ 

\  z\ 


which  is  finite  because  the  series  e  +  ej  +  eg  +  . . .  converges.     Hence  the  series 


S 


Gin  \aJ 


IS  a  converging  series. 

Just  as  in  the  preceding  case  a  special  expression  was  formed  to  serve  as 
a  typical  factor  in  the  infinite  -  product,  we  now  form  a  similar  expression 
for  the  same  purpose.     Evidently 


l—x  = 


log(l-a;) 


-   2 

—   o      r=0 


r  +  1 


*  For  instance,  there  is  no  finite  integer  s  that  can  make  the  infinite  series 
(log  2)-«  +  (log  3)-^  +  (log  4)-«  +  . . . 
conyerge.     This  series  is  given  in  illustration  by  Hermite,  Cours  a  lafaculte  des  Sciences,  (4"^  ed«, 
1891),  p.  86. 


96  WEIERSTRASS'S   CONVERGING  [51. 

if  I  a;  I  <  1.     Forming  a  function  E{x,  m)  defined  by  the  equation 


E(x,  m)  =  (1  -  x)e'' 


=1  r 


CO  r/m+r 
•2  ^- 
r=i  m  +  r 


we  have  E  {x,  m)  =  e 

In  the  preceding  case  it  was  possible  to  choose  the  integer  m  so  that 
it  should  be  the  same  for  all  the  factors  of  the  infinite  product,  which  was 

ultimately  proved  to  converge.     Now,  we  take  x  =  —  and  associate  m„  as 
the  corresponding  value  of  m.     Hence,  if 

/(^)=n^(-, 

n=k        x"^?! 

where  I  a^-i  I  <  |  ^^  |  <  |  a^;  |,  we  have 


/(^) 


-  s  s 


_^     n=kr=l'>'  +  m„ 


2  \r+mn 


n  W/ 


The  infinite  product  represented  by  f{z)  will  converge,  if  the  double  series  in 
the  exponential  be  a  converging  series. 

Denoting  the  double  series  by  S,  we  have 


^|<   2    t 


n=k  r=\  ^  '   ^»i 


r+mn 


00       cx)     1    ~    r+m„ 

<   2    2    -i 

n=k r=l  I  ^n  : 


<  2 

M  =  & 


1- 


on  effecting  the  summation  for  r.     Let  A  be  the  value  of  1 
all  the  remaining  values  of  n,  we  have 

>A, 


z 


then  for 


1- 


and  so 


1^|<^  2     - 


|i+?n.„ 


< 


This  series  converges ;  hence  for  finite  values  of  \z\,  the  value  of  |  ^S  |  is 
finite,  so  that  *S  is  an  unconditionally  converging  series.  Hence  it  follows 
that  f{z)  is  an  unconditionally  converging  product.  We  now  associate  with 
f{z)  as  factors  the  h—\  functions 


E 


,  wi  J  , 


51.]  INFINITE   PRODUCT  97 

for  1  =  1,  2,  ...,  ^—  1 ;  their  number  being  finite,  their  product  is  finite  and 
therefore  the  modified  infinite  product  still  converges.     We  thus  have 


it  is  an  unconditionally  converging  product. 

In  the  same  way  as  for  the  simpler  case,  we  prove  that  the  infinite 
product  converges  uniformly  for  finite  values  of  z. 

Denoting  the  series  in  the  exponential  by  gn  (z),  so  that 

^'  1  /^ 


771,1    1     /    C'  \r 

r=ir\aj 


we  have  E  ( — ,  m,i )  =  ( 1  -  —  ]  es^nfz)  • 

\an        J      \       ay, 

and  therefore  the  function  obtained  is 


«w=n{(^-|;)^'-4 


The  series  gn  usually  contains  only  a  limited  number  of  terms ;  when  the 
number  of  terms  increases  without  limit,  it  is  only  with  indefinite  increase 
of  j  an  |,  and  the  series  is  then  a  converging  series. 

Since  the  product  G{z)  converges  uniformly  and  unconditionally,  no 
product  constructed  from  its  factors  E,  say  from  all  but  one  of  them,  can 
be  infinite.     The  factor 


\an         J      \       a  J 


vanishes  for  the  value  z  =  an  and  only  for  this  value  ;  hence  O  {z)  vanishes  for 
z  =  an.  It  therefore  appears  that  G{z)  has  the  assigned  points  a-^,  a^,  a^,  ... 
for  its  zeros. 

Further,  take  any  finite  quantity,  say  p ;  and  let  a^  be  such  that 

/J  <  I  a^  I  <  1  ttm+i  I  <  .  • .  . 


G{z)=Il  E(  —  ,mn]     n       1  +  —  e*^^ 


Then 


But  n     \(l-^)e^='-  '"^'  \  =  e 

The  double  sum  in  the  index  is  a  series,  which  converges  unconditionally  for 
values  of  z  such  that  \z\<p;  and  therefore  it  is  expressible  in  the  form 
P{z,m  +  1),  which  is  a  power-series  converging  absolutely  for  those  values. 
Hence  e~^^^'^^  can  be  expressed  in  the  form 

1  +  niiZ  +  mzZ-  +  ..., 

F.   F.  • 


9^  TRANSCENDENTAL   INTEGRAL   FUNCTION  [51. 

converging-  absolutely   for   values   of   z   such   that    \z\<  p.     Also    each   of 

the  finite  number  of  factors  ^f  — ,  mn\,  for  ?i  =  l,  ...,  m,  is  expressible  in  a 

series  of  the  form 

1  +  niZ  +  n2Z^+  ..., 

which  converges  absolutely  for  finite  values  of  z  and  therefore  for  values  of  z 
such  that  I  z-\  <  p.  The  product  of  all  these  n  +  1  series  is  also  an  absolutely 
converging  series,  of  the  form 

l  +  giz  +  g2z^+.,., 
which  is  an  expression  for  G(z)  representing  it  as  a  holomorphic  uniform 
function.     Clearly  we  can  take  p  as  large  as  we  please  without  affecting  the 
foregoing  argument. 

In  the  first  place,  since  G  (z)  is  a  uniform  analytic  function  which  has  no 
singularity  in  any  finite  part  of  the  plane  and  which  clearly  is  transcendental, 
the  value  z=  ao  is  an  essential  singularity  of  G  {z). 

In  the  second  place,  G  (z)  has  no  zero  other  than  the  assigned  zeros.  For 
let  a  be  a  value  of  z ;  and  choose  m  sufficiently  large  to  secure  that  a  lies 
within  the  region  of  convergence  of  P{z,  m  +  1);  hence  e-^<2.'«'+i)  is  finite  for 
z  =  a.     No  one  of  the  factors 

e(—,  mnj  (n=l,  ...,  m) 

can  vanish,  if  a  is  not  included  in  the  set  ai,  dg,  ...,  a^-  Therefore  G  could 
not  vanish  for  a,  proving  the  statement. 

It  should  be  noted  that  the  factors  of  the  infinite  product  G  (z)  are  the 
expressions  E.  No  one  of  these  expressions,  for  the  purposes  of  the  product, 
is  resoluble  into  factors  that  can  be  distributed  and  recombined  with  similarly 
obtained  factors  from  other"  expressions  B ;  for  there  is  no  guarantee  that 
the  product  of  the  factors,  when  so  modified,  would  converge  uniformly  and 
unconditionally..  It  is  to  secure  such  convergence  that  the  expressions 
E  have  been  constructed. 

It  was  assumed,  merely  for  temporary  convenience,  that  the  origin  was 
not  a  zero  of  the  required  function ;  there  obviously  could  not  be  a  factor  of 
exactly  the  same  form  as  the  factors  E,  if  a  were  the  origin. 

If,  however,  the  origin  were  a  zero  of  order  X,  we  should  have  merely 
to  associate  a  factor  z'^  with  the  function  already  constructed. 

We  thus  obtain  Weierstrass's  theorem  : — 

It  is  possible  to  construct  a  transcendental  integral  function  such  that  it 
shall  have  infinity  as  its  only  essential  singularity  and  have  the  origin  {of 
multiplicity  A,),  a^,  a,,  a^,   ...   as  zeros;    and  such  a  function  is 


ggjz) 


51]  AS   AN   INFINITE   PRODUCT  99 

whej-e  gn(^)  'is  a  rational,  integral  function  of  z,  the  form  of  which  is  dependent 
upon  the  law  of  succession  of  the  zeros. 

52.  But,  unlike  uniform  functions  with  only  accidental  singularities,  the 
function  is  not  unique :  there  are  an  unlimited  number  of  transcendental 
integral  functions  with  the  same  series  of  zeros  and  infinity  as  the  sole  essential 
singularity,  a  theorem  also  due  to  Weierstrass. 

For,-  if  G^i  {z)  and  G  {z)  be  two  transcendental,  integral  functions  with  the 
same  series  of  zeros  in  the  same  multiplicity,  and  z=oo  as,  their  only  essential 
singularity,  then 

GAz) 

G{z) 

is  a  function  with  no  zeros  and  no  infinities  in  the  finite  part  of  the  plane. 
Denoting  it  by  G2,  then 

1^  dG^ 

(t2  dz 

is  a  function  which,  in  the  finite  part  of  the  plane,  has  no  infinities ;   and 
therefore  it  can  be  expanded  in  the  form 

C,  +  2C.z  +  dGsz'^+..., 

a  series  converging  everywhere  in  the  finite  part  of  the  plane.  Choosing  a 
constant  Cq  so  that  G2(0)  =  e^'>,  we  have  on  integration 

where  g  (z)  =  Cq  +  C^z  +  C^z^  +  ..., 

and  g  (z)  is  finite  everywhere  in  the  finite  part  of  the  plane.  Hence  it  follows 
that,  ifg  (z)  denote  any  integral  function  of  z  which  is  finite  everywhere  in  the 
finite  part  of  the  plane,  and  if  G  {z)  he  some  transcendental  integral  function 
with  a  given  series  of  zeros  and  z  =  co  as  its  sole  essential  singularity,  all 
transcendental  integral  functions  ivith  that  series  of  zeros  and  z  =  00  as  the 
sole  essential  singularity  are  included  in  the  form 

G(z)e^^'K 

Corollary  I.  A  function  which  has  no  zeros  in  the  finite  part  of  the 
plane,  no  accidental  singularities,  and  z=cc  for  its  sole  essential  singularity, 
is  necessarily  of  the  form 

where  g  {z)  is  an  integral  function  of  z  finite  everywhere  in  the  finite  part 
of  the  plane. 

Corollary  II.  Every  transcendental  function,  which  has  the  same  zeros 
in  the  same  multiplicity  as  a  polynomial  A  (z) — the  number,  therefore,  being 
necessarily  finite — ,  which  has  no  accidental  singularities,  and  has  z=co  for  its 
sole  essential  singularity,  can  be  expressed  in  the  form 

A{z)e9^'K  ■      ■ 

7—2 


100  INFINITE   PRODUCTS  [52. 

Corollary  III.  Every  function,  which  has  an  assigned  set  of  zeros 
and  an  assigned  set  of  poles,  and  has  z=  oc  for  its  sole  essential  singularity, 
is  of  the  form 

where  the  zeros  of  Go  {z)  are  the  assigned  zeros  and  the  zeros  of  Gp  (z)  are  the 
assigned  ijoles. 

For  if  Gp  {z)  be  any  transcendental  integral  function,  constructed  as  in 
the  proposition,  which  has  as  its  zeros  the  poles  of  the  required  function  in 
the  assigned  multiplicity,  the  most  general  form  of  that  function  is 

Gp{z)e^^'K 

where  h  {z)  is  integral.  Hence,  if  the  most  general  form  of  function  which 
has  those  zeros  for  its  poles  be  denoted  hy  f{z), 

f{z)Gp{z)e^i^^ 

is  a  function  with  no  poles,  with  infinity  as  its  sole  essential  singularity,  and 
with  the  assigned  series  of  zeros.  But  if  Gq  {z)  be  any  transcendental  integral 
function  with  the  assigned  zeros  as  its  zeros,  the  most  general  form  of  function 

with  those  zeros  is 

Go{z)e9^^); 

and  so  f{z)  Gp  (z)  e^  (^'  =  G,  {z)  es  (^), 

whence  /(^)  =  |ii^e^<^), 

in  which  g  {z)  denotes  g  (z)  —  h  (z). 

If  the  number  of  zeros  be  finite,  we  evidently  may  take  Go{z)  as  the 
polynomial  in  z  with  those  zeros  as  its  only  zeros. 

If  the  number  of  poles  be  finite,  we  evidently  may  take  Gp{z)  as  the 
polynomial  in  z  with  those  poles  as  its  only  zeros. 

And,  lastly,  if  a  function  has  a  finite  number  of  zeros,  a  finite  number 
of  accidental  singularities,  and  ^;  =  oo  as  its  sole  essential  singularity,  it  can 
be  expressed  in  the  form 

e' 


P(^).,i. 


where  P  and  Q  are  polynomials.  This  is  valid,  even  though  the  number  of 
assigned  zeros  be  not  the  same  as  the  number  of  assigned  poles;  the  sole 
effect  of  the  inequality  of  these  numbers  is  to  complicate  the  character  of  the 
essential  singularity  at  infinity. 

53.     It  follows  from  what  has  been  proved  that  any  uniform  function, 
having  z=oo  for  its  sole  essential  singularity  and  any  number  of  assigned 


53.]  PRIMARY   FACTORS  101 

zeros,  can  be  expressed  as  a  product  of  expressions  of  the  form 

1  —  — ^  e9n(^) 

aj 

Such  a  quantity  is  called*  a  primary  factor  of  the  function. 
It  has  also  been  proved  that : — 

(i)     If  there  be  no  zero  an,  the  primary  factor  has  the  form 

e^n  (2). 

(ii)  The  exponential  index  gn  {z)  may  be  zero  for  individual  primary 
factors,  though  the  number  of  such  factors  must,  at  the  utmost, 
be  finite  f. 

(iii)     The  factor  takes  the  form  z  when  the  origin  is  a  zero. 

Hence  we  have  the  theorem,  due  to  Weierstrass : — 

Every  uniform  integral  function  of  z  can  he  expressed  as  a  product  of 
primary  factors,  each  of  the  form 

{kz  +  l)e9^^>, 

where  g  (z)  is  an  appropriate  polynomial  in  z  vanishing  with  z,  and  where  k,  I 
are  constants.  In  particular  factors,  g  {z)  may  vanish ;  and  either  k  or  I,  hut 
not  hoth  k  and  I,  may  vanish  with  or  without  a  non-vanishing  exponential 
index  g  {z). 

54.  It  thus  appears  that  an  essential  distinction  between  transcendental 
integral  functions  is  constituted  by  the  aggregate  of  their  zeros  :  and  we  may 
conveniently  consider  that  all  such  functions  are  substantially  the  same  when 
they  have  the  same  zeros. 

There  are  a  few  very  simple  sets  of  functions,  thus  discriminated  by  their 
zeros :  of  each  set  only  one  member  will  be  given,  and  the  factor  e^'^',  which 
makes  the  variation  among  the  members  of  the  same  set,  will  be  neglected 
for  the  present.  Moreover,  it  will  be  assumed  that  the  zeros  are  isolated 
points. 

I.  There  may  be  a  finite  number  of  zeros ;  the  simplest  function  is  then 
a  polynomial. 

II.  There  may  be  a  singly-infinite  set  of  zeros.  Various  functions  will 
be  obtained,  according  to  the  law  of  distribution  of  the  zeros. 

Thus  let  them  be  distributed  according  to  a  law  of  simple  arithmetic 

progression  along  a  given  line.     If  a  be  a  zero,  (o  a  quantity  such  that  |  co  \ 

is  the  distance  between  two  zeros  and  arg.  w  is  the  inclination  of  the  line, 

we  have 

a  +  mto, 

*  Weierstrass's  term  is  Primfunction ;  see  Ges.  Werke,  t.  ii,  p.  91. 

t  Unless  the  class  (§  59)  be  zero,  when  the  index  is  zero  for  all  the  factors. 


102  PRIMARY  [54. 

for  integer  values  of  m  from  -  oo  to  +  oo ,  as  the  expression  of  the  set  of 
the  zeros.  Without  loss  of  generality,  we  may  take  a  at  the  origin — this 
is  merely  a  change  of  origin  of  coordinates — and  the  origin  is  then  a 
simple  zero :   the  zeros  are    given   by  mco,  for  integer  values  of  m   from 

—  00    to    +  00  . 

111. 

Now  S  —  =  -  5^  —  is  a  diverging  series :  but  an  integer  s — the  lowest 

value  is  s  =  2 — can  be  found  for  which  the  series  2  f  —  |  converges  uncon- 
ditionally.    Taking  s  =  2,  we  have 

«-i  1 

=1  n  \am/        mco 

so  that  the  primary  factor  of  the  present  function  is 


gm  {z)=  2  -    - 


( 


1--^^." 


met)/ 
and  therefore,  by  §  52,  the  product 

z  \    -- 


/(^>=^n{(i-£) 


fflO) 


converges  uniformly  and  unconditionally  for  all  finite  values  of  z. 

The  term  corresponding  to  m  =  0  is  to  be  omitted  from  the  product ;  and 
it  is  unnecessary  to  assume  that  the  numerical  value  of  the  positive  infinity 
for  m  is  the  same  as  that  of  the  negative  infinity  for  m.  If,  however,  the 
latter  assumption  be  adopted,  the  expression  can  be  changed  into  the  ordinary 
product-expression  for  a  sine,  by  combining  the  primary  factors  due  to  values 
of  m  that  are  equal  and  opposite.     In  any  case,  we  have 

J.,    .  (O      .       TTZ 

f{z)  =  —  syn  —  . 

-'    ^    -^         TT  CO 

This  example  is  sufficient  to  shew  the  importance  of  the  exponential  term  in  the 
primary  factor.    If  the  product  be  formed  exactly  as  for  a  polynomial,  then  the  function  is 

zU    (1 ) 

m=-q\        incoj 

in  the  limit  when  both  p  and  q  are  infinite.     But  this  is  known*  to  be 

Another  illustration  is  afforded  by  Gauss's  n-function,  which  is  the  limit  when  k  is 

infinite  of 

1.2.3 k 

{z+l){z  +  2) {s  +  k) 

*  Hobson's  Trigonometry,  §  287. 


54.]  FACTORS  108 

This  is  transformed  by  Gauss*  into  the  reciprocal  of  the  expression 

that  is,  of  (1  +  z)   n  [(\  +  ,^)  e  "'  ^°^  (;^i)] 


the  primary  factors  of  which  have  the  same  characteristic  form  as  in  the  preceding 
investigation,  though  not  the  same  literal  form.  This  is  associated  with  the  Gamma 
Function  t. 

It  is  chiefly  for  convenience  that  the  index  of  the  exponential  j)art  of  the  primary 

S-l    1    /0  \» 

factor  is  taken,  in  §  50,  in  the  form   2    -  ( — )  .     With  equal  eSfectiveness  it  may  be 

s-\  1 

taken  in  the  form   2   -hr^^z"^,  provided  the  series 


2    2  ji  (6,.  „-«,.-»)  ^4 


converges  uniformly  and  unconditionally. 
Ex.  1.     Prove  that  each  of  the  products 

"{0-,-£>"}-  (i^-';?)„"£[{'-(2^}*~]> 

for  w=±l,  ±3,  ±5, to  infinity,  the  term  for  »  =  0  being  excluded  from  the  latter 

product,  converges  vmiformly  and  unconditionally,  and  that  each  of  them  is  equal  to 
cos  2.  (Hermite  and  Weyr.) 

Ex.  2.  Prove  that,  if  the  zeros  of  a  transcendental  integral  function  be  given  by  the 
series 

0,    ±co,   ±4cB,    ±9a), to  infinity, 

the  simplest  of  the  set  of  functions  thereby  determined  can  be  expressed  in  the  form 

sin{zv(l)* 

Ex.  3.  Construct  the  set  of  transcendental  integral  functions  which  have  in  common 
the  series  of  zeros  determined  by  the  law  m^wi  +  2m&)2  +  W3  for  all  integral  values  of  m 
between  -  qo  and  +  qo  ;  and  express  the  simplest  of  the  set  in  terms  of  circular  functions. 

Ex.  4.     A  one-valued  analytical  function  satisfies  the  equation 

f{x)=-xf{ax), 

where  |  a  |  7-^  1  ;  it  has  a  simple  zero  at  each  of  the  points  x  =  a™-(m=0,  ±1,...)  and  no 
other  zero,  and  it  is  finite  for  all  values  of  x  which  are  neither  zero  nor  infinite.  Shew 
that  it  has  essential  singularities  aX  x=0,  x=<xi  ;  and  resolve  it  into  primary  factors. 

(Math.  Trip.,  Part  II.,  1898.) 

*  Ges.  Werke,  t.  ill,  p.  145;  the  example  is  quoted  in  this  connection  by  Weierstrass,  Ges. 
Werke,  t.  ii,  p.  15. 

t  On  the  theory  of  the  Gamma  Function,  a  paper  by  Barnes,  Messenger  of  Mathematics,  t.  xxix, 
(1900),  pp.  64—128,  may  be  consulted.  Eeferences  to  later  memoirs  on  the  subject  are  to  be  found 
in  Whittaker  and  Watson's  Modern  Analysis  (2nd  ed.). 


104'  PRIMARY  [54. 

Ex.  5.  Three  straight  lines  are  drawn  through  a  point  equally  inclined  to  one 
another;  and  by  means  of  three  infinite  series  of  lines,  respectively  parallel  to  these  three 
lines,  the  plane  is  divided  into  an  infinite  number  of  equilateral  triangles.  Construct  an 
integral  uniform  function  which  vanishes  at  the  centre  of  each  of  the  triangles. 

(Math.  Trip.,  Part  II.,  1894.) 

Ex.  6.     Take  a  series  of  concentric  circles 

^2  +  ^2=^  (%=1,  2,  3,...). 

in  the  plane  ;  and  four  common  radii 

^  =  0,    6  =  \n,    6  =  7r,    ^  =  §77. 

Construct  a  function  which  shall  vanish  at  every  one  of  these  radial  points  on  the 
circumferences  :  and  express  it  by  means  of  circular  functions. 

55.  The  law  of  distribution  of  the  zeros,  next  in  importance  and  sub- 
stantially next  in  point  of  simplicity,  is  ^at  in  which  the  zeros  form  a 
doubly-infinite  double  arithmetic  progression,  the  points  being  the  qo  ^ 
intersections  of  one  infinite  system  of  equidistant  parallel  straight  lines 
with  another  infinite  system  of  equidistant  parallel  straight  lines. 

The  origin  may,  without  loss  of  generality,  be  taken  as  one  of  the  zeros. 

If  ft)  be  the  coordinate  of  the  nearest  zero  along  the  line  of  one  system 

passing  through  the  origin,  and  o'  be  the  coordinate  of  the  nearest  zero  along 

the  line  of  the  other  system  passing  through  the  origin,  then  the  complete 

series  of  zeros  is  given  by 

ft  =  mo)  +  m'ft)', 

for  all  integral  values  of  m  and  all  integral  values  of  m  between  —  oo  and 
+  00 .  The  system  of  points  may  be  regarded  as  doubly-periodic,  having  « 
and  ft)'  for  periods. 

It  must  be  assumed  that  the  two  systems  of  lines  intersect.  Other- 
wise, ft)  and  ft)'  would  have  "the  same  argument,  and  their  ratio  would  be  a 
real  quantity,  say  a ;  and  then 

n 

—  —  in  +  m  a. 
ft) 

If  a  be  commensurable,  let  -  denote  its  value,  where  p  and  q  are  positive 

integers  having  no  common  factor;  also  let  -  be  expressed  as  a  continued 

fraction,  and  let  — ,  denote  the  convergent  next  before  the  last  (which,  of 
q  V 

p\ 

-  =  ^,     pq-pq=±l; 
ft)       q 


course,  is  -   .     Then 


ft)      ft)  /  /   /  \        // 

and  therefore  —  =  -=±(5'«  -  q^o)  =  co  , 

p      q 


55.]  FACTORS  105 

that  is,  o)'  and  <w  are  integral  multiples  of  a  single  period  (o"  \  and  the 
apparently  double  system  of  points  would  be  singly-periodic. 

When  a  is  incommensurable,  the  number  of  pairs  of  integers  for  which 
?n  +  m'a  may  be  made  less  than  any  assigned  small  quantity  S  is  infinite ; 
and  then  the  function  would  have  an  unlimited  number  of  zeros  in  any 
assigned  small  region  round  the  origin.  This  would  make  the  origin  an 
essential  singularity  instead  of,  as  required,  an  ordinary  point  of  the  tran- 
scendental integral  function.  Hence  the  ratio  of  the  quantities  co  and  co'  is 
not  real. 

56.  For  the  construction  of  the  primary  factor,  it  is  necessary  to  render 
the  series 

SO"*™'"*' 

converging,  by  appropriate   choice    of  integers  s^,  m'-      It  is   found   to  be 
possible  to  choose  an  integer  s  to  be  the  same  for  every  term  of  the  series, 
corresponding  to  the  simpler  case  of  the  general  investigation,  given  in  §  50. 
As  a  matter  of  fact,  the  series 

diverges  for  5=1  (we  have  not  made  any  assumption  that  the  positive  and 
the  negative  infinities  for  m  are  numerically  equal,  nor  similarly  as  to  m') ; 
the  series  tends  to  a  finite  value  for  s  =  2,  but  the  value  depends  upon  the 
relative  values  of  the  infinities  for  m  and  m' ;  and  s  =  3  is  the  lowest  integral 
value  for  which,  as  for  all  greater  values,  the  series  converges  uncon- 
ditionally. 

There  are  various  ways  of  proving  the  unconditional  convergence  of  the 
series  20~'^  when  yu.  >  2  :  the  following  proof  is  based  upon  a  general  method 
due  to  Eisenstein*. 


m  =  a>        ?i  =  oo 


First,  the  series     2         %    (m^  +  n^)~i^  converges  unconditionally,  if  yu.  >  1. 


•  00    n=  — «> 


Let  the  whole  series  be  arranged  in  partial  series :   for  this  purpose,  we 
choose  integers  k  and  I,  and  include  in  each  such  partial  series  all  the  terms 

which  satisfy  the  inequalities 

2*  <  m  ^  2^+1, 

2^  <  n  <  2^+1, 
so  that  the  number  of  values  of  m  is  2*^  and  the  number  of  values  of  n  is  2^ 
Then,  if  A;  +  ^  =  2/c,  we  have 

2-2/c  <  2-^+1  ^  ^k  j^  2fl  <  w^  -1-  ?^^ 

so  that  each  term  in  the  partial  series  <  ^^  •     "^^^  number  of  terms  in  the 

*  Crelle,  t.  xxxv,  (1847),  p.  161.  A  geometrical  exposition  is  given  by  Halphen,  Traite  des 
fonctions  elliptiques,  t.  i,  pp.  358 — 362 ;  and  another  by  Goursat,  Cours  d' Analyse  MatMmatique, 
t.  ii,  §  324. 


106  WEIERSTRASS'S   FUNCTION   AS  [56. 

partial  series  is  2* .  2^,  that  is,   2^" :   so  that  the  sum  of  the  terms  in  the 
partial  series  is 

1 

Expressing  the  latter  in  the  form 

1  1 


2*  ^~^^  *  2^  ^~'^i ' 

and  taking  the  upper  limit  of  k  and  I  to  be  p,  ultimately  to  be  made  infinite, 
we  have  the  sum  of  all  the  partial  series 

P     M       1  1 


1  _  2-(K-i) 

which,  when  j9  =  oo  ,  is  a  finite  quantity  if  yu-  >  1. 
Next,  let  ft)  =  a  +  /3t,  tu'  =  7  +  8*,  so  that 

n  =  mci)  -f  wco'  =  ma.  +  ny  +  i  (m^  +  nS) ; 
hence,  if  6  =  ma  +  ny,     cf)  =  m/3  +  wS, 

we  have  |  H  |^  =  ^^  +  </)l  •  ■ 

Now  take  integers  r  and  s  such  that 

The  number  of  terms  11  satisfying  these  conditions  is  definitely  finite  and  is 
independent  of  m  and  n.     For  since 

m  (aB  —  ^y)  =  08  —  <py, 

n(a8-^y)=-d/3+<lia, 

and  aS  —  ^y  does  not  vanish  because  w' /(o  is  not  purely  real,  the  number  of 
values  of  m  is  the  integral  part  of 

(r+l)S-sy 
aS  —  /S7 
less  the  integral  part  of 

rS—  (s  4-  1)  7 

otS  —  ^87      ' 

that  is,  it  is  the  integral  part  of  (7  +  S)l(aB  —  ^y),  or  is  greater  than  it  by 
unity.     Similarly,  the  number  of  values  of  n  is  the  integral  part  of 

(a  +  ^)/(aB-M> 

or  is  greater  than  it  by  unity.     Let  the  product  of  the  two  numbers  be  g ; 
then  the  number  of  terms  D.  satisfying  the  inequalities  is  q. 


56.]  A    DOUBLY-INFINITE   PRODUCT  IQT 

Then  SS  |  O  j-^^  =  22  (^^  +  c^^)"'* 

which,  by  the  preceding  result,  is  finite  when  yu,  >  1.     Hence 

converges  unconditionally  when  /a  >  1 ;  and  therefore  the  least  integer  s,  for 
which 

22  {mw  +m'ft)')~* 

converges  unconditionally,  is  3.     But  this  series  converges  unconditionally  for 
any  real  value  of  s  which  is  definitely  greater  than  2. 

The  series  22  (wo)  +  m'<a')~2  has  a  finite  sum,  the  value  of  which  depends*  upon 
the  infinite  Hmits  for  the  summation  with  regard  to  m  and  m'.  This  dependence  is 
inconvenient,  and  it  is  therefore  excluded  in  view  of  the  present  purpose. 

Ex.     Prove  in  the  same  manner  that  the  series 

22 2(?V  +  >^2^  + +'m'n)~'^, 

the  multiple  summation  extending  over  all  integers  mi,  m2,  ,  m^  between  -  oo  and 

+  00,  converges  unconditionally  if  2iJL>n.  (Eisenstein.) 

57.  Returning  now  to  the  construction  of  the  transcendental  integral 
function  the  zeros  of  which  are  the  various  points  fi,  we  use  the  preceding 
result  in  connection  with  §  50  to  form  the  general  primary  factor.  Since 
s=  S,  we  have 


_  z      ^  z^ 


and  therefore  the  primary  factor  is 


z 
pi. 


Moreover,  the  origin  is  a  simple  zero.     Hence,  denoting  the  required  function 
by  or  (z),  we  have 


-^«=4n{(i-n) 


'.'^^a^ 


as  a  transcendental  integral  function  which,  since  the  product  converges  uni- 
formly and  unconditionally  for  all  finite  values  of  z,  exists  and  has  a  finite 
value  everywhere  in  the  finite  part  of  the  plane;  the  quantity  D.  denotes 
m&)  +  mco',  and  the  double  product  is  taken  for  all  values  of  m  and  of  m' 
between  -  oo  and  +  oo ,  simultaneous  zero  values  alone  being  excluded. 

This  function  will  be  called  Weierstrass's  o--function ;  it  is  of  import- 
ance in  the  theory  of  doubly-periodic  functions  which  will  be  discussed  in 
Chapter  XL 

*  See  a  paper  by  the  author,  Quart.  Journ.  of  Math.,  vol.  xxi,  (1886),  pp.  261—280. 


108  PRIMARY    FACTORS  [57. 

Ex.     If  the  doubly-infinite  series  of  zeros  be  the  points  given  by 

Q.  =  m^coj  +  2m7ia>2  +  n^oa^ , 

0)1,  0)2,  COS  being  complex  constants  such  that  Q  does  not  vanish  for  real  values  of  m  and  n, 
then  the  series 

2    2   i2-» 
converges  for  s  =  2  but  not  for  s  =  l.     The  primary  factor  is  thus 

1  -  -  V^ , 
and  the  simplest  transcendental  integral  function  having  the  assigned  zeros  is 

The  actual  points  that  are  the  zeros  are  the  intersections  of  two  infinite  systems  of 
parabolas. 

58.  One  other  result — of  a  negative  character — will  be  adduced  in  this 
connection.  We  have  dealt  with  the  case  in  which  the  system  of  zeros  is  a 
singly-infinite  arithmetical  progression  of  points  along  one  straight  line,  and 
with  the  case  in  which  the  system  of  zeros  is  a  doubly-infinite  arithmetical 
progression  of  points  along  two  different  straight  lines.  We  proceed  to  prove 
that  a  uniform  transcendental  integral  function  cannot  exist  with  a  triply- 
infinite  arithmetical  progression  of  points  for  zeros. 

A  triply-infinite  arithmetical  progression  of  points  would  be  represented 
by  all  the  possible  values  of 

for  all  possible  integer  values  for  pi,  p^,  Pz  between  —  oo  and  +  oo  ,  where  no 
two  of  the  arguments  of  the  complex  constants  Oi,  Ho,  Ho  are  equal.     Let 

O^  —  air  +  iwr,  (r  =  1,  2,  3) ; 

then,  as  will  be  proved  (§  107)  in  connection  with  a  later  proposition,  it  is 
possible* — and  possible  in  an  unlimited  number  of  ways — to  determine 
integers  ^i,  |>2,  p^  so  that,  save  as  to  infinitesimal  quantities, 

j3i  Vi  _  Pz 

all  the  denominators  in  which  equations  differ  from  zero  on  account  of  the 
fact  that  no  two  arguments  of  the  three  quantities  fli,  fl.,,  Dg  are  equal.  For 
each  such  set  of  determined  integers,  the  quantit}^ 

PiHi  +  i?2i^2+i?3^3 

is  zero  or  infinitesimal.  If  it  is  zero,  then  (as  in  §  107  for  periods)  the  triple 
infinitude  is  really  only  a  double  infinitude.  If  it  is  infinitesimal,  then  (as 
at  the  end  of  §  55)  the  origin  is  an  essential  singularity,  contrary  to  the 

*  Jacobi,  Ges.  Werke,  t.  ii,  p.  27. 


58.]  CLASS    OF   A    FUNCTION  109 

hypothesis  that  the  only  essential  singularity  is  for  ^  =  oo  .  Hence  a  uniform 
transcendental  function  cannot  exist  having  a  triply-infinite  arithmetical 
succession  of  zeros. 

59.  In  effecting  the  formation  of  a  transcendental  integral  function  by 
means  of  its  primary  factors,  it  has  been  proved  that  the  expression  of  the 
primary  factor  depends  upon  the  values  of  the  integers  which  make 

a  converging  series.  Moreover,  the  primary  factors  are  not  unique  in  form, 
because  any  finite  number  of  terms  of  the  proper  form  can  be  added  to  the 
exponential  index  in 

mn-l  1    ^r 

aj 

the  added  terms  will  only  the  more  effectively  secure  the  convergence  of  the 
infinite  product.  But  there  is  a  lower  limit  to  the  removal  of  terms  with  the 
highest  exponents  from  the  index  of  the  exponential ;  for  there  are,  in  general, 
least  values  for  the  integers  mi,  m^,  ...,  below  which  these  integers  cannot  be 
reduced,  if  the  convergence  of  the  product  is  to  be  secured. 

The  simplest  case,  in  which  the  exponential  must  be  retained  in  the 
primary  factor  in  order  to  secure  the  convergence  of  the  infinite  product,  is 
that  discussed  in  §  50,  viz.,  when  the  integers  wij,  m^,  ...  are  equal  to  one 
another.  Let  m  denote  this  common  value  for  a  given  function,  and  let 
m  be  the  least  integer  effective  for  the  purpose:  the  function  is  then  said* 
to  be  of  class  m,  and  the  condition  that  it  should  be  of  class  m  is,  that  the 
integer  m  be  the  least  integer  to  make  the  series 

00 

y    I  f,    I— TO— 1 

M  =  l 

converge,  the  constants  a„  being  the  zeros  of  the  function. 

Thus  algebraical  polynomials  are  of  class  0 ;  the  circular  functions  sin  z 
and  cos  z  are  of  class  1 ;  Weierstrass's  o--function  and  the  Jacobian  elliptic 
function  sn  2  are  of  class  2,  and  so  on :  but  for  no  one  of  these  classes  do  the 
functions  mentioned  constitute  the  whole  of  the  functions  of  that  class. 

60.  One  or  two  of  the  simpler  properties  of  an  aggregate  of  transcendental 
integral  functions  of  the  same  class  can  easily  be  obtained. 

Let  a  function  f{z),  of  class  n,  have  a  zero  of  order  r  at  the  origin  and 
have  a,,  a^,  ...  for  its  other  zeros,  arranged  in  order  of  increasing  moduli. 
Then,  by  §  50,  the  function  f{z)  can  be  expressed  in  the  form 


*  The  French  word  is  genre  ;   the  Italian  is  genere.     Laguerre  (see  references  on  p.  113) 
appears  to  have  been  the  first  to  discuss  the  class  of  transcendental  integral  functions. 


110 


CLASS-PROPERTIES   OF 


[60. 


where  gi{z)  denotes  the  series  2  -  ( —  ]  and  0{z)  must  be  properly  determined 

s—X  ^  \(^i' 

to  secure  the  equality. 

Now  consider  the  series 

f=i  at""  {ai  -  z) 

for  all  values  of  z  that  lie  outside  circles  round  the  points  a,  taken  as  small 
as  we  please.     The  sum  of  the  series  of  the  moduli  of  its  terms  is 

^11 


t=i   ai 


Let  d  be  the  least  of  the  quantities 


1- 


1-^ 


,  necessarily  non-evanescent 


because  z  lies  outside  the  specified  circles ;    then  the  sum  of  the  series 


1  '^ 


which  is  a  converging  series  since  the  function  is  of  class  n.     Hence  the 
series   of  moduli    converges,   and   therefore   the    original   series  converges. 

Moreover,  the  series  2  j  cij  |  """^  converges.     Denoting  by  e  any  real  positive 

quantity,  as  small  as  we  please,  we  can  choose  an  integer  m  such  that 

lj.  +  r 

S  I  a,- 1 -'*-!<  e, 

for  all  integers  yu^m  and  for  all  positive  integers  r.     Accordingly,  for  the 
values  of  z  considered,  we  have 


)  =  u.     Oji 


1- 


dl 


d' 


for  all  integers  fx'^m,  for  all  positive  integers  r,  and  for  all  the  values  of  z. 
Hence  the  series  converges  unconditionally  and  unifot-mly  within  the  specified 
region  of  variation  of  z ;  let  it  be  denoted  by  S  (z),  so  that 

S(z)  =  t-—^ .. 

i=iai''{ai-z) 

We  have 

f(z)      ^,,  .      r      "    1   r,       ^    ,  ,    5r«-i  1     ] 

f{z)  z     i=i  ai  {        ai  ai" 


1-1 

ai 


^  00  1 

G'(z)+~-z^^ 


z         i=iai''{ai- z) 


GUz)  +  --z''S(z\ 
z  ^  ^ 


60.]  TRANSCENDENTAL   INTEGRAL    FUNCTIONS  111 

Each  step  of  this  process  is  reversible  in  all  cases  in  which  the  original  product 
converges.  If,  therefore,  it  can  be  shewn  of  a  function  f(z)  that  "Ct^^  takes 
this  form,  the  function  is  thereby  proved  to  be  of  class  n. 

If  there  be  no  zero  at  the  origin,  the  term  -  is  absent. 

z 

If  the  exponential  factor  G  {z)  be  a  constant  so  that  G'  {z)  is  zero,  the 
function  f{z)  is  said  to  be  a  simple  function  of  class  n. 

^  61.  There  are  several  criteria,  used  to  determine  the  class  of  a  function : 
the  simplest  of  them  is  contained  in  the  following  proposition,  due  to 
Laguerre*. 

If,  as  z  tends  to  the  value  oo ,  a  very  great  value  of  \z\  can  he  found  for 

f  (z) 
which  the  limit  of  z~'^y-~,  where  f(z)  is  a  transcendental  integral  function, 

tends  uniformly  to  the  value  zero,  thenf{z)  is  of  class  n. 

Take  a  circle,  centre  the  origin  and  of  radius  R  equal  to  this  value  of  |  ^  j ; 
then,  by  §  24,  II.,  the  integral 

1    {lf{t)    dt 


2iri\  t^  fit)  t-z' 

taken  round  the  circle,  is  zero  when  R  becomes  indefinitely  great.      But  the 
value  of  the  integral  is,  by  the  Corollary  in  §  20, 


lirij      t""  f{t)t-z'^  27riJ       f'  f{t)  t-z^  ^iri  ^=i  j       V"  f{t)t- 


taken  round  small  circles  enclosing  the  origin,  the  point  z,  and  the  points 
tti,  which  are  the  infinities  of  the  subject  of  integration;  the  origin  being 
supposed  a  zero  of  f{f)  of  multiplicity  r.     Now 

1   r'^)  i/'(0  dt      \f'{z) 


27nJ      t""  fit)  t-z     z'"f{z)' 
1     r<«i'  l/'(0    dt    __1       1 


2iri]       f^  f{t)  t  - z      at  at-z' 

and  1     r'°'  Ifjt)    dt    ^      ct>{z)        r 

27ri]       f^  f{t)  t-z  z^        ^^+1' 

where  ^  {z)  denotes  the  polynomial 

l/l(Q_*^U.ii/l(i)_^l4-        ,      ^''-'      d-^  {fit)     r 
\f{t)      ~t]^'dt\f{t)      t]''-''{n-l)\dt-^\f{t)      t 

when  t  is  made  zero.     Hence 

z""  f{z)       itiai^ia-i-z)       z>'        z'^^^        ' 
*  Comptes  Rendus,  i.  xciv,  (1882),  p.  636  ;  (Euvres  Completes,  t.  i,  p.  172. 


112  '  CLASS-PEOPERTIES   OF  [61. 

and  therefore 

which,  by  §  60,  shews  that/ (2^)  is  of  class  n. 

Corollary.  The  product  of  any  finite  number  of  functions  of  the  same 
class  n  is  a  function  of  class  not  higher  than  n  ;  and  the  class  of  the  product 
of  any  finite  number  of  functions  of  different  classes  is  not  greater  than  the 
highest  class  of  the  component  functions. 

Note  1.  In  connection  with  Weierstrass's  theorem  in  §  52,  one  remark 
may  be  made  as  to  its  influence  upon  the  class  of  a  function;  it  will  be 
sufficiently  illustrated  by  taking  e^'  sin  z  as  an  example.  Laguerre's  test 
shews  that  the  class  is  two,  whereas  by  the  test  of  §  60  the  class  apparently 
is  unity.  The  explanation  of  the  difference  is  that,  in  §  60,  the  zeros  of  the 
generalising  factor  e^<^'  of  §  52  are  not  taken  into  account.  It  is  true  that  all 
these  zeros  are  at  infinity ;  but  their  existence  may  affect  the  integer,  which 
is  the  least  that  secures  the  convergence  of  the  series  2  |  ai\~'^~^.  Thus  the 
zeros  of  the  function  e^' sin  ^  are  mir,  where  m  =  0,  ±1,  ...,  ±  oo  ,  arising 
from  sin  z :    and 

^p^     -ip^, 

each  occurring  p  times,  where  p  is  an  infinite  positive  integer :  the  latter 
arising  from  e^\  by  regarding  it  as  the  limit  of 

when  p  is  an  infinite  positive  integer.  In  order  that  the  critical  series  may 
converge,  it  is  necessary  that,  as  these  new  zeros  are  at  infinity,  the  integer  n 
should  be  chosen  so  as  to  make 

p  I  (ip^y-^  \+p\{-  ip^y-^  I 
vanish.     The  lowest  value  of  n  is  two ;  and  therefore  the  function  really  is  of 
class  two,  agreeing  with  the  result  of  Laguerre's  test. 

More  generally,  consider  a  function 

F(z)=e<^i^^f{z), 

where  f{z)  is  of  class  n,  and  G  (z)  is  itself  an  integral  function.  On  the 
application  of  Laguerre's  test,  the  limit  of 

when  \z\  increases  indefinitely,  is  the  limit  of  z~''^  G'  (z).  Thus  F(z)  is  not 
of  class  n,  unless  G  {z)  is  a  polynomial  in  z  of  degree  <  n.  If  G  (z)  is  a 
polynomial  of  degree  m>n,  then  F(z)  is  of  class  m.  If  G{z)  is  a  transcen- 
dental integral  function,  F(z)  is  of  infinite  class. 


61.]  TEANSCENDENTAL   INTEGRAL   FUNCTIONS  llS 

Of  course,  this  is  not  the  only  manner  in  which  functions  of  infinite  class 
can  arise.  Thus  consider  an  integral  function  having  log  2,  log  8,  log  4,  . . .  for 
its  infinite  succession  of  zeros.    It  has  been  noted  (p.  95,  foot-note)  that  no  finite 

00 

integer  s  exists  such  that  the  series   2   (log  n)~^  converges ;  consequently  the 

11  =  2 

class  of  the  series  is  infinite*. 

Note  2.  Borelf  introduces  the  notion  of  the  order  of  an  integral  function 
as  distinct  from  the  class  of  the  function.  In  the  preceding  investigation  (§  59), 
the  class  of  the  equation  is  taken  to  be  the  lowest  integer  s  (if  any)  for  which 
the  series 

(where  aj,  a.2,  ...  are  the  zeros  arranged  in  non-descending  magnitude  of 
moduli)  converges  absolutely.  Borel  takes  the  order  of  the  function  to  be  the 
lowest  real  quantity  for  which  the  same  series  converges  absolutely ;  so  that, 
if  /jl  be  the  class  and  /n'  the  order  of  a  function, 

/x'  ^  /Jb<  fl   +1. 

Thus  the  class  of  the  product 


is  unity,  because  2  is  the  lowest  integer  which  makes  the  series  2  n~^  converge; 

n=l 

its  order  is  1  -I-  k,  where  k  is  any  quantity  greater  than  zero  but  as  small  as 

00 

we  please,  because  the  series  2  w~^~*  converges. 

n=l 

The  following  are  the  chief  references  to  memoirs  discussing  the  class  of  functions  :— 

Laguerre,  Comptes  Rendus,  t.  xciv,  (1882),  pp.  160—163,  pp.  6.35—638,  ib.  t.  xcv,  (1882), 
pp.  828—831,  ib.  t.  xcviii,  (1884),  pp.  79 — 81 J  ;  Poincare,  Bull,  des  Sciences  Math.,  t.  xi, 
(1883),  pp.  136—144;  Ceskro,  Comptes  Rendus,  t.  xcix,  (1884),  pp.  26—27  (followed 
(p.  27)  by  a  note  by  Hermite),  Giornale  di  Battaglini,  t.  xxii,  (1884),  pp.  191 — 200 ; 
Vivanti,  Giornale  di  Battaglini,  t.  xxii,  (1884),  pp.  243 — 261,  pp.  378— 380,  ib.  t.  xxiii, 
(1885),  pp.  96—122,  ib.  t.  xxvi,  (1888),  pp.  303—314;  Hermite,  Cours  d  la  faculte 
des  Sciences  (4™«  ed.,  1891),  pp.  91—93;  ll&d&ma.v&,  Liouville,  ^^^  Ser.,  t.  ix,  (1893), 
pp.  171 — 214;  Borel,  Acta  Math.,  t.  xx,  (1897),  pp.  357 — 396,  Lecons  sur  les  fonctions 
entteres,  (1900),  ch.  ii. 

Ex.  1.     Prove  that  the  class  of  the  functions  sin0,  1+2  sin  2  is  unity. 

Ex.  2.     The  function 

n 

2  eV/i(2), 
i=l 

where  the  quantities  c  are  constants,  n  is  a,  finite  integer,  and  the  functions  fi{z)  are 
polynomials,  is  of  class  unity. 

*  For  functions  of  infinite  class,  reference  may  be  made  to  Blumenthal's  monograph 
Principes  de  la  theorie  des  fonctions  entieres  d'ordre  infini  (1910). 

t  Lecons  sur  les  fonctions  entieres,  p.  26. 

X  All  these  are  included  in  the  first  volume  of  the  CEuvres  de  Laguerre,  (1898,  Gauthier- 
Villars). 

F.  P.  S 


114  EXAMPLES  [61, 

Ex.  3.     If  a  simple  function  be  of  class  n,  its  derivative  is  also  of  class  n. 

Ex.  4.     Discuss  the  conditions  under  which  the  sum  of  two  functions,  each  of  class  w, 
is  also  of  class  n, 

Ex.  5.     Examine  the  following  test  for  the  class  of  a  function,  due  to  Poincare. 

Let  a  be  any  number,  no  matter  how  small  provided  its  argument  be  such  that  e"^ 
vanishes  when  z  tends  towards  infinity.     Then  f{z)  is  of  class  ?i,  if  the  limit  of 

vanish  with  indefinite  increase  of  z. 

A  possible  value  of  a  is  2  Cia,"""^,  where  Cj  is  a  constant  of  modulus  unity. 

Ex.  6.     Verify  the  following  test  for  the  class  of  a  function,  due  to  de  Sparre*. 

Let  X  be  any  positive  non-infinitesimal  quantity ;  then  the  function  f(z)  is  of  class  n, 
if  the  liinit,  for  ?w  =  oo ,  of 

I  <^7»  I  (i'^m  +  ll       \^m\f 

be  not  less  than  X.     Thus  sin  z  is  of  class  unity. 

Ex.  7.     Let  the  roots  of  (9"^  +  i=l  be  1,  a,  a^,   ,  a'^;  and  let  f{z)  be  a  function 

of  class  n.     Then  forming  the  product 

n 

n  f{a>z\ 
s=o 

we  evidently  have  an  integral  fimction  of  2"  +  i;  let  it  be  denoted  by  F{z'^*'^).     The  roots 

of  i^(2''"^i)  =  0  are  a^a*,  for^  =  l,  2, ,  and  5  =  0,  1, ,  n;  and  therefore,  replacing  2"+ ^ 

by  z,  the  roots  of  F{z)=0  are  «»«■  +  !,  for  z  =  l,  2,  

0 

Since  f{z)  is  of  class  n,  the  series 

00       1 

converges  unconditionally.     This  series  is  the  sum  of  the  first  powers  of  the  reciprocals  of 
the  roots  of  F{z)  =  0;  hence,  according  to  the  definition  (p.  109),  F{z)  is  of  class  zero. 

It  therefore  follows  that  from  a  function  of  any  class,  a  function  of  class  zero  with  a 
modified  variable  can  he  deduced.  Conversely,  hy  appropriately  modifying  the  variable  of 
a  given  function  of  class  zero,  it  is  possible  to  deduce  functions  of  any  required  class. 

Ex.  8.     If  all  the  zeros  of  the  function 


be  real,  then  all  the  zeros  of  its  derivative  are  also  real.  (Witting.) 

*  Gomptes  Rendus,  t.  cii,  (1886),  p.  741. 


CHAPTER   YL 

Functions  with  a  Limited  Number  of  Essential  Singularities. 

62.  Some  indications  regarding  the  character  of  a  function  at  an 
essential  singularity  have  already  been  given.  Thus,  though  the  function 
is  regular  in  the  vicinity  of  such  a  point  a,  it  may,  like  sn  {Ijz)  at  the  origin, 
have  a  zero  of  unlimited  multiplicity  or  an  infinity  of  unlimited  multiplicity 
at  the  point ;  and  in  either  case  the  point  is  such  that  there  is  no  factor  of 
the  form  {z  —  a)\  which  can  be  associated  with  the  function  so  as  to  make  the 
point  an  ordinary  point  for  the  modified  function.  Moreover,  even  when  the 
path  of  approach  to  the  essential  singularity  is  specified,  the  value  acquired 
may  not  be  definite :  thus,  as  z  approaches  the  origin  along  the  axis  of  x, 
so  that  its  value  may  be  taken  to  be  1  -=-  {4)mK  +  x),  the  value  of  sn  (Ijz)  is  not 
definite  in  the  limit  when  w  is  made  infinite.  One  characteristic  of  the 
point  is  the  indefiniteness  of  value  of  the  function  at  the  essential  singu- 
larity, though  in  the  vicinity  the  function  is  uniform. 

A  brief  statement  and  a  proof  of  this  characteristic  were  given  in  §  32 ; 
the  theorem  there  proved — that  a  uniform  analytical  function  can  assume 
any  value  at  an  essential  singularity — may  also  be  proved  as  follows.  The 
essential  singularity  will  be  taken  at  infinity — a  supposition  that  does  not 
detract  from  generality. 

Let  f{z)  be  a  function  having  any  number  of  zeros  and  any  number 
of  accidental  singularities  and  z  —  cc  for  its  sole  essential  singularity ;  then 
it  can  be  expressed  in  the  form 

w^here  Gi  (z)  is  polynomial  or  transcendental  according  as  the  number  of  zeros 
is  finite  or  infinite,  and  G2  (z)  is  polynomial  or  transcendental  according  as 
the  number  of  accidental  singularities  is  finite  or  infinite. 

If  Gsiz)  be  transcendental,  we  can  omit  the  generalising  factor  e^^^K 
Then/(3;)  has  an  infinite  number  of  accidental  singularities;  each  of  them 

8—2 


116  FORM   OF   A   FUNCTION   NEAR  [62. 

in  the  finite  part  of  the  plane  is  of  only  finite  multiplicity  and  therefore  some 
of  them  must  be  at  infinity.  At  each  such  point,  the  function  0^  {z)  vanishes 
and  (ti  {z)  does  not  vanish  ;  and  Qof{z)  has  infinite  values  for  ^  =  cso  . 

If  Gz  {z)  be  polynomial  and  G^  {z)  be  also  polynomial,  then  the  factor  e^  '** 
may  not  be  omitted,  for  its  omission  would  make  f{z)  a  rational  function. 
Now  ^^  =  00  is  either  an  ordinary  point  or  an  accidental  singularity  of 

G,{z)IGAz); 

hence  as  g  {z)  is  integral,  there  are  infinite  values  of  z  which  make 

G,{z) 

G,(^) 
infinite. 


gg  (2) 


If  G2  (z)  be  polynomial  and  Gi  (z)  be  transcendental,  the  factor  e^  '^'  may 
be  omitted.     Let  a^ ,  ag,  . . . ,  a„  be  the  roots  of  G^  (z)  :  then  taking 

n  A 

f{z)=    S       ^+G,,{z), 

1  G-i,  {ar) 

we  have  ^ ,.  =  j^ — r , 

(jTo  yO^r) 

a  non-vanishing  constant ;  and  so 

where  Gn  (z)  is  a  transcendental  integral  function.     When  z=  cc  ,  the  value 
of  G^{z)IG2{z)  is  zero,  but  Gn{z)  is  infinite  ;  hence /(^)  has  infinite  values  for 

z  =  cc  . 

Similarly  it  may  be  shewn,  as  follows,  that/(^)  has  zero  values  for  ^•^  00  . 

In  the  first  of  the  preceding  cases,  if  (tj  {z)  be  transcendental,  so  that  f{z) 
has  an  infinite  number  of  zeros,  then  some  of  them  must  be  at  an  infinite 
distance ;  f{z)  has  a  zero  value  for  each  such  point.  And  if  (zi  {z)  be 
polynomial,  then  there  are  infinite  values  of  z  which,  not  being  zeros  of 
(t2  {z),  make  f{z)  vanish. 

In  the  second  case,  when  z  is  made  infinite  with  such  an  argument  as  to 
make  the  highest  term  in  g{z)  a  real  negative  quantity,  then  f  {z)  vanishes 
for  that  infinite  value  of  z. 

In  the  third  case,  f{z)  vanishes^  for  a  zero  of  Gi  {z)  that  is  at  infinity. 

Hence  the  value  of  f{z)  for  ^r  =  00  is  not  definite.  If,  moreover,  there 
be  any  value  neither  zero  nor  infinity,  say  G,  which  f{z)  cannot  acquire 
for  z  =  <X) ,  then 

f{z)-G 

is  a  function  which  cannot  be  zero  at  infinity,  and  therefore  all  its  zeros  are 
in  the  finite  part  of  the  plane  :  no  one  of  them  is  an  essential  singularity,  for 


62.]  AN  ESSENTIAL  SINGULARITY  117 

f{z)  has  only  a  single  value  at  any  point  in  the  finite  part  of  the  plane ; 
hence  they  are  finite  in  number  and  are  isolated  points.  Let  il^{z)  be 
the  polynomial  having  them  for  its  zeros.  The  accidental  singularities  of 
f{z)—G  are  the  accidental  singularities  oi  f{z);   hence 

where,  if  Gz{z)  be  polynomial,  the  exponential  h(z)  must  occur,  since  f{z), 
and  therefore /(^)  —  C,  is  transcendental.     The  function 

F  (z\  -  ^  -  ^^^^^  p-h  (z) 

evidently  has  .3^=00  for  an  essential  singularity,  so  that,  by  the  second  or 
the  third  case  above,  it  certainly  has  an  infinite  value  for  ^r  =  go  ,  that  is, 
f{z)  certainly  acquires  the  value  (7  for  2^  =  oo  . 

Hence  the  function  can  acquire  any  value  at  an  essential  singularity. 

63.  We  now  proceed  to  obtain  the  character  of  the  expression  of  a 
function  at  a  point  z  which,  lying  in  the  region  of  continuity,  is  in  the 
vicinity  of  an  essential  singularity  h  in  the  finite  part  of  the  plane. 

With  h  as  centre  describe  two  circles,  so  that  their  circumferences  and 
the  whole  area  between  them  lie  entirely  within  the  region  of  continuity. 
The  radius  of  the  inner  circle  is  to  be  as  small  as  possible  consistent  with 
this  condition ;  and  therefore,  as  it  will  be  assumed  that  6  is  the  only 
singularity  in  its  own  immediate  vicinity,  this  radius  may  be  made  very 
small. 

The  ordinary  point  z  of  the  function  may  be  taken  as  lying  within  the 
circular  ring-formed  part  of  the  region  of  continuity.  At  all  such  points  in 
this  band,  the  function  is  holomorphic ;  and  therefore,  by  Laurent's  Theorem 
(§  28),  it  can  be  expanded  in  a  converging  series  of  positive  and  negative 
integral  powers  of  z  —  h,  in  the  form 

Wo  +  ih  {z  —  h)-Vu^X^-hy'^... 
+  vXz-h)-^-\-vXz-h)-^+...; 
the  coefiicients  Un  are  determined  by  the  equation 


h\jBr^'''        (»=o.i.2,...). 


'^      27rt  j  {t  -  hY+' 

the  integrals  being  taken  positively  round  the  outer  circle,  and  the  coefiicients 
Vn  are  determined  by  the  equation 

the  integrals  being  taken  positively  round  the  inner  circle. 


118  FOEM   OF   A   FUNCTION   NEAR  [63. 

The  series  of  positive  powers  converges  everywhere  within  the  outer  circle 
of  centre  b,  and  so  (§  26)  it  may  be  denoted  by  P (z—h);  and  the  function  P 
may  be  either  polynomial  or  transcendental. 

The  series  of  negative  powers  converges  everywhere  without  the  inner 
circle  of  centre  b  ;  and,  since  b  is  not  an  accidental  but  an  essential  singularity 
of  the  function,  the  series  of  negative  powers  contains  an  infinite  number  of 

terms.     It  may  be  denoted  by  G  I        ,  ) ,  a  series  converging  for  all  points 

in  the  plane  except  z=b,  and  vanishing  when  z  —  b=  co . 


Thus  f(^)  =  G{^^+P('-b) 


is  the  analytical  representation  of  the  function  in  the  vicinity  of  its  essential 
singularity  b ;  the  function  G  is  transcendental  and  converges  everywhere  in 
the  plane  outside  an  infinitesimal  circle  round  b,  and  the  function  P,  if 
transcendental,  converges  for  sufficiently  small  values  of  \z  —  b\. 

Had  the  singularity  at  b  been  accidental,  the  function  G  would  have  been 
polynomial. 

Corollary  I.  If  the  function  have  any  essential  singularity  other  than 
b,  it  is  an  essential  singularity  of  P(^  —  6)  continued  outside  the  outer  circle  ; 

but  it  is  not  an  essential  singularity  of  G  ( y  ] ,  for  the  latter  function 

converges  everywhere  in  the  plane  outside  the  inner  circle. 

Corollary  II.  Suppose  the  function  has  no  singularity  in  the  plane 
except  at  the  point  b ;  then  the  outer  circle  can  have  its  radius  made  infinite. 
In  that  case,  all  positive  powers  except  the  constant  term  u^  disappear : 
and  even  this  term  survives  only  in  case  the  function  have  a  finite  value  at 
infinity.     The  expression  for  the  function  is 

z-b      {z—bf 

and  the  transcendental  series  converges  everywhere  outside  the  infinitesimal 

circle  round  b,  that  is,  at  every  point  in  the  plane  for  which  | j-.  remains 

less  than  any  assigned  quantity,  however  large.  Hence  the  function  can  be 
represented  by 

1   A 


G 


-b) 


This   special   result   is   deduced    by   Weierstrass    from   the    earlier    in- 
vestigations*, as   follows.     If  f{z)  be   such  a  function   with   an   essential 

*  Weierstrass,  Ges.  Werke,  t.  ii,  p.  102. 


63.]  AN   ESSENTIAL   SINGULARITY  11^ 

singularity    at    b,    and    if    we    change    the    independent   variable    by   the 
relation 

1 


2   = 


Z-h' 


then  f{z)  changes  into  a  function  of  z',  the  only  essential  singularity  of  which 
is  at  /  =  00  .  It  has  no  other  singularity  in  the  plane ;  and  the  form  of  the 
function  is  therefore  0 (z),  that  is,  a  function  having  an  essential  singularity 
at  b,  but  no  other  singularity  in  the  plane,  is 

G 


z-b 


Corollary  III.  The  most  general  expression  of  a  function  having  its 
sole  essential  singularity  at  b,  a  point  in  the  finite  part  of  the  plane,  g,nd  any 
number  of  accidental  singularities,  is 


0,, 


i~ 


where  the  zeros  of  the  function  are  the  zeros  of  Gi,  the  accidental  singularities 
of  the  function  are  the  zeros  of  G2,  and  the  function  g  in  the  exponential  is  a 

function  which  is  finite  for  all  finite  values  of ^ . 

This  can  be  derived  in  the  same  way  as  before ;  or  it  can  be  deduced 
from  the  corresponding  theorem  relating  to  transcendental  integral  functions, 
as  above.  It  would  be  necessary  to  construct  an  integral  function  G^iz), 
having  as  its  zeros 


a^  —  b'    a^  —  b''"' 

and  then  to  replace  z  by  r ;    and  G^  is  polynomial  or  transcendental, 

according  as  the  number  of  zeros  is  finite  or  infinite. 

Similarly  we  obtain  the  following  result : — 

Corollary  IV.  A  uniform  function  of  z,  which  has  its  sole  essential 
singularity  at  b,  a  point  in  the  finite  part  of  the  plane,  and  no  accidental 
singularities,  can  be  represented  in  the  form  of  an  infinite  product  of  primary 
factors  of  the  form 

which  converges  uniformly  and  unconditionally  everywhere  in  the  plane  outside 
an  infinitesimal  circle  drawn  round  the  point  b. 


1^0  FUNCTIONS   WITH   A   LIMITED   NUMBER  [63. 

The  function  g  i r )   is  an  integral  function  of  — -r    vanishing  when 

J  vanishes ;   and  h  and  I  are  constants.     In  particular  factors,  9  i—^i 

may  vanish ;  and  either  Jc  or  I  (but  not  both  k  and  I)  may  vanish,  with  or 
without  a  vanishing  exponent  g  i  — y  j . 

If  o-i  be   any  zero,  the   corresponding  primary  factor  may  evidently  be 
expressed  in  the  form 


Similarly,  for  a  uniform  function  of  z  with  its  sole  essential  singularity  at  b 
and  any  number  of  accidental  singularities,  the  product-form  is  at  once 
derivable  by  applying  the  result  of  the  present  Corollary  to  the  result  given 
in  Corollary  III. 

These  results,  combined  with  the  results  of  Chapter  V.,  give  the  general 
theory  of  uniform  functions  with  only  one  essential  singularity. 

64.  We  now  proceed  to  the  consideration  of  functions,  which  have  a 
limited  number  of  assigned  essential  singularities. 

The  theorem  of  §  63  gives  an  expression  for  the  function  at  any  point  in 
the  band  between  the  two  circles  there  drawn. 

Let  c  be  such  a  point,  which  is  thus  an  ordinary  point  for  the  function ; 
then  in  the  domain  of  c,  the  function  is  expansible  in  a  form  Pj  (2  —  c). 
This  domain  may  extend  as  far  as  an  infinitesimal  circle  round  an  essential 
singularity  b,  or  it  may  be  limited  by  a  pole  d  which  is  nearer  to  c  than  b  is, 
or  it  may  be  limited  by  an  essential  singularity  /  which  is  nearer  to  c  than  b 
is.  In  the  first  case,  we  form  -a  coTitinuation  of  the  function  in  a  direction 
away  from  b ;  in  the  second  case,  we  continue  the  function  by  associating 
with  the  function  a  factor  (z  -  d)"-  which  takes  account  of  the  accidental 
singularity ;  in  the  third  case,  we  form  a  continuation  of  the  function 
towards  /.  Taking  the  continuations  for  successive  domains  of  points  in  the 
vicinity  of  /,  we  can  obtain  the  value  of  the  function  for  points  on  two  circles 
that  have  /  for  their  common  centre.  Using  these  values,  as  in  §  63,  to 
obtain  coefficients,  we  ultimately  construct  a  series  of  positive  and  negative 
powers  converging  outside  an  infinitesimal  circle  round  /.  Different  express- 
ions in  different  parts  of  the  plane  will  thus  be  obtained,  each  being  valid 
only  in  a  particular  portion :  the  aggregate  of  all  of  them  is  the  analytical 
expression  of  the  function  for  the  whole  of  the  region  of  the  plane  where  the 
function  exists. 

We  thus  have  one  mode  of  representation  of  the  function;  its  chief 
advantage  is  that  it  indicates  the  form  in  the  vicinity  of  any  point,  though  it 


64.]  OF   ESSENTIAL  SINGULARITIES  121 

gives  no  suggestion  of  the  possible  modification  of  character  elsewhere.  This 
deficiency  renders  the  representation  insufficiently  precise  and  complete  ;  and 
it  is  therefore  necessary  to  have  another  mode  of  representation. 

65.  Suppose  that  the  function  has  n  essential  singularities  a^,  a^,  ...,  a„, 
and  that  it  has  no  other  singularity.  Let  a  circle,  or  any  simple  closed 
curve,  be  drawn  enclosing  them  all,  every  point  of  the  boundary  as  well 
as  the  included  area  (with  the  exception  of  the  n  singularities*)  lying  in 
the  region  of  continuity  of  the  function.  ~~ 

Let  z  be  any  ordinary  point  in  the  interior  of  the  circle  or  curve;  and 
consider  the  integral 

IB-- 

taken  round  the  curve.  If  we  surround  z  and  each  of  the  n  singularities  by 
small  circles  with  the  respective  points  for  centres,  then  the  integral  round 
the  outer  curve  is  equal  to  the  sum  of  the  values  of  the  integral  taken  round 
the  n  +  1  circles.     Thus 


2iri  <  gt  —  z  27^^  J  zt  —  z  27ri     J a^t  —  z 

and  therefore 


^Tri'zt  —  z  2771]  gt  —  z  27ri     Jart~'^ 

The  left-hand  side  of  the  equation  is  f(z). 
Evaluating  the  integrals,  we  have 

27n.'a,.t  —  z                  \z  —  a. 
where   Gr  is,  as  before,  a  transcendental  function  of  :  vanishing  when 

1       . 

IS  zero. 

z  —  ttr 

Now,  of  these  functions,  (r,.  ( i  converges  everywhere  in  the  plane 

\Z  fty/ 

outside  the  infinitesimal  circle  round  a,.,  (say  except  at  a^) :  and  therefore,  as 
n  is  finite, 

V     Gr 


r  =  l 


z.—  a. 


is  a  function  which  converges  everywhere  in  the  plane  except  at  the  n  points 

<Ii,  ...,  (In- 

Because  2^  =  oo  is  not  an  essential  singularity  oi  f{z),  the  radius  of  the 
circle  in  the  integral  - — .  /  '^^  dt  may  be  indefinitely  increased.     The  value 

.     *  This  phrase  will  frequently  be  used  as  an  abbreviation  for  "the  infinitesimal  regions 
enclosed  by  infinitesimal  circles  round  the  singularities." 


122'  FUNCTIONS   WITH   A   LIMITED   NUMBER  [65. 

of  f(t)  tends,  with  unlimited  increase  of  t,  to  some  determinate  value  G  which 
is  not  infinite ;  hence,  as  in  §  24,  II.,  Corollary,  the  value  of  the  integral  is 
C.     We  therefore  have  the  result  that  /(^)  can  be  expressed  in  the  form 

n  /       1 

C+    X    Or 


J*  r=  1  \^    ~"~    (J/'j* 

or,  absorbing  the  constant  G  into  the  functions  G  and  replacing  the  limitation 

that  the  function  Gr  i 1  shall  vanish  for  =0,  by  the  limitation 

V^  —  aj  z  —  Or  "^ 

that,  for  the  same  value =  0,  it  shall  be  finite,  we  have  the  theorem*: — 

z  —  ar 

If  a  given  function  f{z)  have  n  singularities  ftj,  ...,  a„,  all  of  which  are  in 
the  finite  part  of  the  plane  and  are  essential  singularities,  it  can  he  expressed, 
in  the  form 

L    Gr 


r=\        \z  —  a 

where  Gr  is  a  transcendental  function,  converging  everywhere  in  the  plane 

outside  an  infinitesimal  circle   round   Ur,  and   having  a  determinate  finite 

1  '^         .  . 

value  gr  for =  0,  such  that  2  gr  is  the  finite  value  of  the  given  func- 

Z  ~~  Cv^'  J" = 1 

tion  at  infinity.  ~ 

Corollary.  If  the  given  function  have  a  singularity  at  qo  ,  and  n  singu- 
larities in  the  finite  part  of  the  plane,  then  the  function  can  be  expressed  in 
the  form 


*^         /     1     \ 
G{z)+  S  Grl^—), 

r  =  l  \Z  —  ar/ 


where  Gr  is  a  transcendental  or  a  polynomial  function,  according  as  <x^  is  an 
essential  or  an  accidental  singularity :  and  so  also  for  G  (z),  according  to  the 
character  of  the  singularity  at  infinity. 

66.     Any  uniform  function,  which  has  an  essential  singularity  at  z  =  a, 
can  (§  63)  be  expressed  in  the  form 


^1^^)+^^"-^^' 


for  points  z  in  the  vicinity  of  a.  Suppose  that,  for  points  in  this  vicinity, 
the  function  f{z)  has  no  zero,  and  that  it  has  no  accidental  singularity. 
Therefore,  among  such  points  z,  the  function 

1     df{z) 
.    f{z)    dz 

*  The  method  of  proof,  by  an  integration,  is  used  for  brevity  :  the  theorem  can  be  established 
by  purely  algebraical  reasoning. 


66.]  OF   ESSENTIAL  SINGULARITIES  123 

has  no  pole,  and  therefore  no  singularity  except  that  at  a  which  is  essential. 
Hence  it  can  be  expanded  in  the  form 


'^(j^)+-P(^-«)' 


where  G  converges  everywhere  in  the  plane  except  at  a,  and  vanishes  for 

1  r.  -r 

=  0.     Let 

z  —  a 


^U^-^a  +  d-zf'L-r 

where  G,  [j:^]  converges  everywhere  in  the  plane  except  at  a,  and  vanishes 

for-^  =  0. 

z  —  a 

Then  c,  evidently  not  an  infinite  quantity,  is  an  integer.     To  prove  this, 
describe  a  small  circle  of  radius  p  round  a :  then  taking  z-a  =  pe^\  so  that 

=  idd,  we  have 

z  —  a 

and  therefore 

Now  JP(z  -a)dz  is  a  uniform  function  :  and  so  is  f(z).     But  a  change 
of  6  into  6  +  27r  does  not  alter  z  or  any  of  the  functions :  thus 

and  therefore  c  is  an  integer. 

67.     If  the  function  / (2')  have  essential  singularities  ai,...,a^  and  no 
others,  then  it  can  be  expressed  in  the  form 


r=l        KZ—ClrJ 


.  If  there  be  no  zeros  for  this  function  f(z)  anywhere  (except  of  course  such 
as  may  enter  through  the  indeterminateness  at  the  essential  singularities), 
then 

J_  df(z) 
f(z)    dz 

has  n  essential  singularities  a^, ...,  an  and  no  other  singularities  of  any  kind. 
Hence  it  can  be  expressed  in  the  form 


G+  2  Grl  — 


124  EXPRESSION   OF  [67. 

where  the  function  Gr  vanishes  with .     Let 

Z  —  ttr 

\2  —  firJ      z  —  a-r     dz  (      \z  —  a, 

where  Gr  ( ]  is  a  function  of  the  same  kind  as  G,, 

\z  —  art 


Then  all  the  coefficients  c,.,  evidently  not  infinite  quantities,  are  integers. 
For,  let  a  small  circle  of  radius  p  be  drawn  round  a^ :  then,  \i  z  —  a,.  =  /^e^^  we 
have 

=  Crldo, 

z  —  a,. 

and  — =  dPs  (z  —  aA 

z~as 

We  proceed  as  before :  the  expression  for  the  function  in  the  former 
case  is  changed  so  that  now  the  sum  SPg  (^  —  Qr)  ^or  s  =  l,...,  r—1, 
r  +  1,  ...,nis  a  uniform  function;  there  is  no  other  change.  In  exactly  the 
same  way  as  before,  we  shew  that  every  one  of  the  coefficients  c,.  is  an 
integer. 

Hence  it  appears  that  if  a  given  function  f{z)  have,  in  the  finite  part  of 
the  plane,  n  essential  singularities  ai,  ...,  a^  and  no  other  singularities,  and  if 
it  have  no  zeros  anywhere  in  the  plane,  then 


1  <'/(-)  =  P+|^^+l^k 


f{z)    dz  j=i  z-  ai     i=i  dz  {    ''\z-ai 

where  all  the  coefficients  Ci  are  integers,  the  functions  G  converge  everywhere 
in  the  plane  except  at  the  essential  singularities,  and  Gi  vanishes  for 

z  -  ai 
Now,  since /(^■)  has  no  singularity  at  oo  ,  we  have  for  very  large  values  of  z 

/(^)  =  Wo  +  ^+;5+..., 

and  /  (^)=--.--r--"' 

and  therefore,  for  very  large  values  of  z, 

1     df{z)  ^_ih  I  ,  ^1  , 
/  {z)    dz  Uo  z^      z^ 

Thus  there  is  no  constant  term  in  tt^^   "^ }     ,  and  there  is  no  term  in  - .     But 

f{z)    dz  _  z 

the  above  expression  for  it  gives  G  as  the  constant  term,  which  must  therefore 


67.]  A  FUNCTION  125 

vanish  ;  and  it  gives  Xci  as  the  coefficient  of  -  ,  for  ~  \Gi  f—^")}  will  begin 
with  -  at  least ;  thus  Ici  must  therefore  also  vanish, 

Z~  ' 

Hence  for  a  function  f{z),  which  has  no  singularity  at  \0  =  oo  and  no 
zeros  anywhere  in  the  plane,  and  of  which  the  only  singularities  are  the  n 
essential  singularities  at  ttj,  a^,  ...,  cin,  we  have 


f(z)     dz        .^^z-ai     i=^dz\    '  U  -  a^ 
where  the  coefficients  Cj  are  integers  subject  to  the  condition 

n 

If  Un  =  00  ,  so  that  ^  =  00  is  an  essential  singularity  in  addition  to  a^,  a^, 

... ,  cin-i,  there  is  a  term  G(z)  instead  of  Gn  ( ) ;  there  is  no  term,  that 

\z  -  aj 

c 
corresponds  to  — ^'—^ ,  but  there  may  be  a  constant  C.     Writing 

0+G{z)  =  ^JGiz)], 
with  the  condition  that  G(z)  vanishes  when  z  =  0,  we  then  have 

f{z)    dz        {ZxZ-ai      dz'^     ^    ^      i=\dz\    'U-ai/j' 

where  the  coefficients  Ci  are  integers,  but  are  no  longer  subject  to  the 
condition  that  their  sum  vanishes. 

Let  jR*  {£)  denote  the  function 

the  product  extending  •  over  th^  factors  associated  with  the  essential 
singularities  of  f{£)  that  lie  in  the  finite  part  of  the  plane;  thus  R*  {z) 
is  a  rational  meromorphic  function.     Since 

1      t^E*  {z)  ^  ^      Cj 
R*  (z)      dz  i=i  z  —  tti' 

we  have 

1     df(z)  1       dR'^jz)  ^  ^  ±{^_  (    1     M 


f{z)    dz        R*  {z)      dz  ,;=i  dz  {    ^  \z-  ai 

where  Gn  i J  is  to  be  replaced  by  G  (z)  if  a„  =  oo  ,  that  is,  if  ^  =  oo  be  an 

\Z  —  an' 
essential  singularity  of  f(z).     Hence,  except  as  to  an  undetermined  constant 

factor,  we  have 

f{z)  =  R*(z)U  e    ^^-*^ 

^  =  l 

which  is  therefore  an  analytical  representation  of  a  function  with  n  essential 


126  PRODUCT-EXPRESSION   OF  [67. 

singularities,  no    accidental   singularities,  and   no   zeros :    and   the   rational 
function  R*  (z)  becomes  zero  or  oo  only  at  the  singidarities  of  f{z). 

If  ^  =  00  be  not  an  essential  singularity,  then  i^*  {£)  for  ^  =  oo  is  equal  to 

n 

unity  because  S  c^  =  0. 

^  =  l 

Corollary.     It  is  easy  to  see,  from  §  43,  that,  if  the  point  a^  be  only  an 

accidental  singularity,  then  c;  is  a  negative  integer  and  Gi  [ 1  is  zero  :  so 

- —  \z  —  ai) 

that  the  polar  property  at  ai  is  determined  by  the  occurrence  of  a  factor 
{z  —  aifi  solely  in  the  denominator  of  the  rational  meromorphic  function  i2*  {£). 

And,  in  general,  each  of  the  integral  coefficients  Ci  is  determined  from  the 
expansion  of  the  function  /'  {z)  -^fiz)  in  the  vicinity  of  the  singularity 
with  which  it  is  associated. 

68.  Another  form  of- expression  for  the  function  can  be  obtained  from 
the  preceding;  and  it  is  valid  even  when  the  function  possesses  zeros 
not  absorbed  into  the  essential  singularities f. 

Consider  a  function  with  one  essential  singularity,  and  let  a  be  the 
point.  Suppose  that,  within  a  finite  circle  of  centre  a  (or  within  a  finite 
simple  curve  which  encloses  a),  there  are  m  simple  zeros  a,  &,  ...,\  of  the 
function  /  {z) ;  assume  m  to  be  finite,  and  also  assume  that  there  are  no 
accidental  singularities  within  or  on  the  circle,  or  at  a  merely  iufinitesimal 
distance  from  its  circumference.     Then,  if 

f{z)  =  {z-a){z-^)...{z-\)F{z), 

the  function  F  {z)  has  a  for  an  essential  singularity  and  has  no  zeros  within 
the  circle.     Hence,  for  points  z  within  the  circle, 

F{z)      z-a     dz[      \z-aj) 

where   G.{  ]  converges  uniformly  everywhere  in  the  plane   outside  a 

\z  —  a) 

small  circle  round  a  and  vanishes  with  — — ,  and  P  {z-a)  is  an  integral 

function  converging  uniformly  within  the  circle ;   moreover,  c  is  an  integer. 

Thus 

F{z)  =  A{z-  a)«/'(~^)  ^!Pi^-a)ciz^ 

Let  (^-«)(^-^)...(^-^)  =  (^-«)'"{l  +  ^+---  +  (JZ^} 

t  See  Guichard,  Theorie  des  points  singuliers  essentiels,  (These,  Gauthier-Villars,  Paris,  1883), 
especially  the  first  part. 


68.]  A   FUNCTION  127 

then  f{z)  =  {z-ay-g,(^-^^F{z) 

=  A(z-  a)"»+«  g^  [^ZTaJ  ^  ^ 

Now  of  this  product-expression  for/(2^)  it  should  be  noted  : — 

(i)     That  ?w  +  c  is  an  integer,  finite  because  m  and  c  are  finite : 

(ii)     The  function  e   '  ^^-»^  can  be  expressed  in  the  form  of  a  series  con- 
verging uniformly  everywhere  outside  a  small  circle  round  a,  and  proceeding 

in  powers  of  in  the  form 


z  —  a 


1+— ^  +  , — H^  +  .... 
z  —  a      [z  ~  af 

It  has  no  zero  within  the  circle  considered,  for  F(z)  has  no  zero.    Also  ^^  [ ] 

is  a  polynomial  in ,  beginning  with  unity  and  containing  only  a  finite 

number  of  terms :  hence,  multiplying  the  two  series  together,  we  have  as  the 
product  a  series  proceeding  in  powers  of in  the  form 

Ai  ho 

1+  ^-  +  7 -^,+  ---> 

Z—a     {z  —  af 

which  converges  uniformly  everywhere  outside  any  small  circle  round  a.     Let 

this  series  be  denoted  by  H  ( ] ;  it  has  an  essential  singularity  at  a  and 

its  only  zeros  are  the  points  a,  /3,  ...,  X,  because  the  series  multiplied  by 

g^  ( )  has  no  zeros  :  - 


(iii)  The  function  f  P  (z  —  a)  dz  is  a  series  of  positive  powers  of  z  —  a, 
converging  uniformly  in  the  vicinity  of  a;  and  therefore  e/-P(«-«)c^«  ^g^j^  j^g 
expanded  in  a  series  of  positive  integral  powers  of  ^  -  a,  which  converges 
in  the  vicinity  of  a.  Let  it  be  denoted  by  Q(z  —  a)  which,  since  it  is  a 
factor  of  F{z),  has  no  zeros  within  the  circle. 

Hence  we  have 

f{z)^A{z-ayQiz-a)H(^^y 

where  fj,  is  an  integer ;    H  ( J  is  a  series  that  converges  everywhere 

outside  an  infinitesimal  circle  round  a,  is  equal  to  unity  when vanishes, 

and  has  as  its  zeros  the  (finite)  number  of  zeros  assigned  to  f(z)  within  a 


128  GENERAL  FORM  OF  A  FUNCTION  [68. 

finite  circle  of  centre  a ;  and  Q{z  —  a)  is  a  series  of  positive  powers  of  ^  -  a. 
beginning  with  unity  which  converges  (but  has  no  zero)  within  the  circle. 

The  foregoing  function  f{z)  is  supposed  to  have  no  essential  singularity 
except  at  a.  If,  however,  a  given  function  have  singularities  at  points 
other  than  a,  then  the  circle  would  be  taken  of  radius  less  than  the  distance 
of  a  from  the  nearest  essential  singularity. 

Introducing  a  new  function /i  {z)  defined  by  the  equation 

f{z)=^A{z-aYH(^j^\Mz), 

the  value  of  /i  {z)  is  Q  {z  —  a)  within  the  circle,  but  it  is  not  determined  by 
the  foregoing  analysis  for  points  without  the  circle.     Moreover,  as  {z  —  aY 

and  also  H  ( )  are  finite  everywhere  except  in  the  immediate  vicinity  of 

the  isolated  singularity  at  a,  it  follows  that  essential  singularities  of  f{z) 

other  than  a  must  be  essential  singularities  of  /j  {z).  Also  since  /i  {z)  is 

Q{z  -  a)  in   the   immediate   vicinity   of  a,    this  point  is    not  an   essential 
singularity  of  /i  {z). 

Thus  /i  {z)  is  a  function  of  the  same  kind  SiS,  f{z) ;  it  has  all  the  essential 
singularities  of  f{z)  except  a,1)ut  it  has  fewer  zeros,  on  account  of  the  m 

zeros  o{  f{z)  possessed  by  H  (-^— )  •     The  foregoing  expression  ior  f{z)  is 

the  one  referred  to  at  the  beginning  of  the  section. 

If  we  choose  to  absorb  into  f^{z)  the  factors  e  '  ^^-a'  and  e/-P(a^-a)d«^ 
which  occur  in 

A  (z-ay^+'g,  ( ")  e^'  ^^  eJP(«-a)f/^. 

an  expression  that  is  valid  within  the  circle  considered,  then  we  obtain  a 
result  that  is  otherwise  obvious,  by  taking 

f{z)  =  iz-aYg,(^^^f,{z), 

where  now  g-i  ( )  is  polynomial  in ,  and  has  for  its  zeros  all  the 

zeros  within  the  circle ;  /x,  is  an  integer ;  and  f^  (z)  is  a  function  of  the  same 
kind  as  f{z),  which  now  possesses  all  the  essential  singularities  of /(^),  but 

its  zeros  are  fewer  by  the  m  zeros  that  are  possessed  by  g^  ( 

69.  Next,  consider  a  function  f(z)  with  n  essential  singularities  (Xj, 
a.j, ...,  a^  but  without  accidental  singularities;  and  let  it  have  any  number 
of  zeros. 


69.]  WITH   ESSENTIAL   SINGULARITIES  129 

When  the  zeros  are  limited  in  number,  they  may  be  taken  to  be  isolated 
points,  distinct  in  position  from  the  essential  singularities. 

When  the  zeros  are  unlimited  in  number,  then  at  least  one  of  the 
singularities  must  be  such  that  the  zeros  in  infinite  number  lie  within 
a  circle  of  finite  radius,  described  round  it  as  centre  and  containing  no  other 
singularity.  For  if  there  be  not  an  infinite  number  in  such  a  vicinity  of 
some  one  point  (which  must  be  an  essential  singularity :  the  only  alternative 
is  that  the  zeros  should  form  a  continuous  aggregate,  and  then  the  function 
would  be  zero  everywhere),  the  points  are  isolated  and  there  must  be  an 
infinite  number  outside  a  circle  \z\  =  R,  where  i^  is  a  finite  quantity  that 
can  be  made  as  large  as  we  please,  say  an  infinite  number  at  5  =  oo  .  If 
^  =  00  be  an  essential  singularity,  the  above  alternative  is  satisfied :  if  not, 
the  function,  as  in  the  preceding  alternative,  must  be  zero  at  all  other  parts 
of  the  plane.  Hence  it  follows  that,  if  a  uniform  function  have  a  finite  number 
of  essential  singularities  and  an  infinite  number  of  zeros,  all  but  a  finite 
number  of  the  zeros  lie  within  circles  of  finite  radii  described  round  the 
essential  singularities  as  centres  ;  at  least  one  of  the  circles  contains  an 
infinite  number  of  the  zeros,  and  some  of  the  circles  may  contain  only  a  finite 
number  of  them. 

We  divide  the  whole  plane  into  regions,  each  containing  one  but  only  one 
singularity  and  containing  also  the  circle  round  the  singularity ;  let  the 
region  containing  ai  be  denoted  by  C,-,  and  let  the  region  C„  be  the  part  of 
the  plane  other  than  (7,,  Co,  ...,  Gn-i- 

If  the  region  C^  contain  only  a  limited  number  of  the  zeros,  then,  by  §  68, 
we  can  choose  a  new  function  /j  {z)  such  that,  if 

the  function  f-^  (z)  has  a^  for  an  ordinary  point,  has  no  zeros  within  the  region 
Cj,  and  has  a.,,  ctg,  ...,  an  for  its  essential  singularities. 

If  the  region  C-^  contain  an  unlimited  number  of  the  zeros,  then,  as  in 
Corollaries   II.  and   III.  of  §63,  we  construct  any  transcendental   function 

(ti  I )  ,  having  aj  for  its  sole  essential  singularity  and  the  zeros  in  Oi  for 

all  its  zeros.     When  we  introduce  a  function  ^i  {z),  defined  by  the  equation 

the  function  gi{z)  has  no  zeros  in  G^  and  certainly  has  tta,  Cis,  ••■,  a^  for 
essential  singularities;  in  the  absence  of  the  generalising  factor  of  (tj,  it  can 
have  tti  for  an  essential  singularity.     By  §  67,  the  function  g^  {z),  defined  by 

g,{z)  =  {z-a,y^e    ^^-^^ , 

F.  P.  9 


130  GENERAL  FORM  OF  A  FUNCTION  [69. 

has  no  zero  and  no  accidental  singularity,  and  it  has  a^  as  its  sole  essential 
singularity  :  hence,  properly  choosing  c^  and  h^ ,  we  may  take 

so  that  /i  (2)  does  not  have  a^  as  an  essential  singularity,  but  it  has  all  the 
remaining  singularities  of  ^^  (z),  and  it  has  no  zeros  within  C^. 

In  either  case,  we  have  a  new  function  /j  (z)  given  by 


/(^)  =  (^-a,)'^^G,(-j^^-j/,(^), 


where  /Xj  is  an  integer.  The  zeros  o{f(z)  that  lie  in  Cj  are  the  zeros  of  Gi]  the 
function /i  (2:)  has  ttg,  as,  ...,  a„  (but  not  (Xj)  for  its  essential  singularities, 
and  it  has  the  zeros  off{z)  in  the  remaining  regions  for  its  zeros. 

Similarly,  considering  Cg,  we  obtain  a  function /a  (z),  such  that    . 


/,(^)  =  (^-a,^(?,(-_^)/,(4 


where  yu-g  is  an  integer,  G^  is  a  transcendental  function  finite  everywhere  except 
at  tta  and  has  for  its  zeros  all  the  zeros  of/i  (z) — and  therefore  all  the  zeros  of 
f(z) — that  lie  in  C^.  Then/g  (-s^)  possesses  all  the  zeros  of /(^)  in  the  regions 
other  than  Cj  and  Co,  and  has  as,  a^,  ...,  an  for  its  essential  singularities. 

Proceeding  in  this  manner,  we  ultimately  obtain  a  function  /„  (z)  which 
has  none  of  the  zeros  of /(2^)  in  any  of  the  n  regions  C^,  G^,  ...,  Cn,  that  is, 
has  no  zeros  in  the  plane,  and  it  has  no  essential  singularities ;  it  has  no 
accidental  singularities,  and  therefore  /„  {z)  is  a  constant.     Hence,  when  we 

n 

substitute,  and  denote  by  8*  {z)  the  product  11  {z  —  aiYi,  we  have 


f(.)  =  8*i.)nG,{^). 


which  is  the  most  general  form  of  a  function  with  n  essential  singularities,  no 
accidental  singularities,  and  any  number  of  zeros.  The  function  S^  (z)  is  a 
rational  function  of  z,  usually  meromorphic  in  form,  and  it  has  the  essential 
singularities  of  f{z)  as  its  zeros  and  poles;  and  the  zeros  of  f{z)  are  dis- 
tributed among  the  functions  Gi. 

As  however  the  distribution  of  the  zeros  by  the  regions  G  and  therefore 
the  functions  G  i ]  are  somewhat  arbitrary,  the  above  form  though  general 

is  not  unique. 

If  any  one  of  the  singularities,  say  a,„,  had  been  accidental  and  not 

essential,  then  in  the  corresponding  form  the  function  Gm  ( )  would  be 

polynomial  and  not  transcendental. 


TO.]  WITH   ESSENTIAL   SINGULARITIES  131 

70.  A  function  fiz),  which  has  any  -finite  number  of  accidental  singu- 
larities in  addition  to  n  assigned  essential  singularities  and  any  number  of 
assigned  zeros,  can  be  constructed  as  follows. 

Let  A  (z)  be  the  polynomial  which  has,  for  its  zeros,  the  accidental 
singularities  oif{z),  each  in  its  proper  multiplicity.     Then  the  product 

f(z)A{z) 
is  a  function  which  has  no  accidental  singularities ;  its  zeros  and  its  essential 
singularities  are  the  assigned  zeros  and  the  assigned  essential  singularities  of 
f{z),  and  therefore  it  is  included  in  the  form 


^*«n{«'G-^)} 


where  S*  (z)  is  a  rational  meromorphic  function  having  the  points  ai,a2,  .. 
for  zeros  and  poles.     The  form  of  the  function  / (2)  is  therefore 


A  (z)  ,-=1  I     '\z-  tti 

71.  A  function  f(z),  which  has  an  unlimited  number  of  accidental  singu- 
larities in  addition  to  n  assigned  essential  singularities  and  any  number  of 
assigned  zeros,  can  be  constructed  as  follows. 

Let  the  accidental  singularities  be  a',  /3', ....  Construct  a  function  f  (z), 
having  the  n  essential  singularities  assigned  to  f{z),  no  accidental  singu- 
larities, and  the  series  a',  /3',  ...  of  zeros.  It  will,  by  §69,  be  of  the  form  of  a 
product  of  n  transcendental  functions  Gn+i,  ...,  G2n,  which  are  such  that  a 
function  G  has  for  its  zeros  the  zeros  oi  f  (z)  lying  within  a  region  of  the  plane, 
divided  as  in  §  69 ;  and  the  function  G^+i  is  associated  with  the  point  a^. 
Thus 


f{z)  =  T*{z)hG..^J~^), 
i=\  \^  —  ay 


where  T*  {z)  is  a  rational  meromorphic  function  having  its  zeros  and   its 
poles,  each  of  finite  multiplicity,  at  the  essential  singularities  oif{z). 

Because  the  accidental  singularities  of/(^;)  are  the  same  points  and  have 
the  same  multiplicity  as  the  zeros  oi  f{z),  the  innction  f  {z)  f-^  (z)  has  no 
accidental  singularities.  This  new  function  has  all  the  zeros  of  f  {z),  and 
Oi,  ...,  a^  are  its  essential  singularities;  moreover,  it  has  no  accidental  singu- 
larities.    Hence  the  product/ (2^) /i  {z)  can  be  represented  in  the  form 

8*  {z)  U  Gi^     ^ 

i=l 

and  therefore  we  have 


f/  ^  ^*  (^)  ft 


ft 


\2  -  aj 


Gn+i 


as  an  expression  of  the  function. 


z  —  a. 


9—2 


132  GENERAL  FORM  OF  A  FUNCTION  [71. 

But,  as  by  their  distribution  through  the  n  selected  regions  of  the  plane 
in  §  69,  the  zeros  can  to  some  extent  be  arbitrarily  associated  with  the 
functions  G^,  G^,  ■■■,  Gn  and  likewise  the  accidental  singularities  can  to  some 
extent  be  arbitrarily  associated  with  the  functions  Gn-\-i,  Gn+2,  •■■,  G^n,  the 
product-expression  just  obtained,  though  definite  in  character  and  general, 
is  not  unique  in  the  detailed  form  of  the  functions  which  occur. 

The  fraction  T*  {  \ 

is  rational,  neither  S*'  nor  T*  being  transcendental ;  it  vanishes  or  becomes 
infinite  only  at  the  essential  singularities  ai,  a^,  ...,  a^,  being  the  product 
of  factors  of  the  form  {z  —  aij^i,  for  ^  =  1 ,  2,  . . . ,  n.  Let  the  power  {z  —  ciiy'h 
be  absorbed  into  the  function  GijGn+i  for  each  of  the  n  values  of  ^;  no 
substantial  change  in  the  transcendental  character  of  Gi  and  of  Gn^i  is 
thereby  caused,  and  we  may  therefore  use  the  same  symbol  to  denote  the 
modified  function  after  the  absorption.  Hence  f  the  most  general  ■product- 
expression  of  a  uniform  function  of  z,  which  has  n  essential  singularities 
a^,  tta,  ...,  an,  ciny  unlimited  number  of  assigned  zeros,  and  any  unlimited 
number  of  assigned  accidental  singularities,  is 

n  M  ^  — 

n  — 


j=i 


G. 


ck 


The  resolution  of  a  transcendental  function  with  one  essential  singularity 
into  its  primary  factors,  each  of  which  gives  only  a  single  zero  of  the  function, 
has  been  obtained  in  §  63,  Corollary  IV. 

We  therefore  resolve  each  of  the  functions  Gi,  ...,  G^n  into  its  primary 
factors.  Each  factor  of  the  first  n  functions  will  contain  one  and  only  one  zero 
of  the  original  functions  f{z) ;  and  each  factor  of  the  second  ??  functions  will 
contain  one  and  only  one  of  the  poles  oi  f{z).  The  sole  essential  singularity 
of  each  primary  factor  is  one  of  the  essential  singularities  oif{z).  Hence  we 
have  a  method  of  constructing  a  uniform  function  with  any  finite  number  of 
essential  singularities  as  a  product  of  any  number  of  primary  factors,  each 
of  which  has  one  of  the  essential  singularities  as  its  sole  essential  singularity 
and  either  (i)  has  as  its  sole  zero  either  one  of  the  zeros  or  one  of  the 
accidental  singularities  oi  f{z),  so  that  it  is  of  the  form 


or  (ii)  it  has  no  zero  and  then  it  is  of  the  form 


t  Weierstrass,  Ges.  Werke,  t.  ii,  p.  121. 


71.]  WITH   ESSENTIAL   SINGULAEITIES  133 

When  all  the  primary  factors  of  the  latter  form  are  combined,  they  constitute 
a  generalising  factor  in  exactly  the  same  way  as  in  §  52  and  in  §  63, 
Cor.  III.,  except  that  now  the  number  of  essential  singularities  is  not 
limited  to  unity.  The  product  converges  uniformly  for  all  finite  values  of  z 
that  lie  outside  small  circles  round  the  singularities ;  and  similarly  for  infinite 
values,  if  the  function  is  regular  for  ^  =  oo  . 

Two  forms  of  expression  of  a  function  with  a  limited  number  of  essential 
singularities  have  been  obtained :  one  (§  65)  as  a  sum,  the  other  (§  69)  as  a 
product,  of  functions  each  of  which  has  only  one  essential  singularity.  Inter- 
mediate expressions,  partly  product  and  partly  sum,  can  be  derived,  e.g. 
expressions  of  the  form 

n 


,•=1        \z-c. 


2    (r    ,  ■ 


But  the  pure  product-expression  is  the  most  general,  in  that  it  brings  into 
evidence  not  merely  the  n  essential  singularities  but  also  the  zeros  and  the 
accidental  singularities,  whereas  the  expression  as  a  sum  tacitly  requires  that 
the  function  shall  have  no  singularities  other  than  the  n  which  are  essential. 

Note.  The  formation  of  the  various  elements,  tlie  aggregate  of  which  is  the  complete 
representation  of  the  function  with  a  limited  number  of  essential  singularities,  can  be 
carried  out  in  the  same  manner  as  in  §  34 ;  each  element  is  associated  with  a  particular 
domain,  the  range  of  the  domain  is  limited  by  the  nearest  singularities,  and  the  aggregate 
of  the  singularities  determines  the  boundary  of  the  region  of  continuity. 

To  avoid  the  practical  difficulty  of  the  gradual  formation  of  the  region  of  continuity 
by  the  construction  of  the  successive  domains  when  there  is  a  limited  number  of 
singularities  (and  also,  if  desirable  to  be  considered,  of  branch-points),  Fuchs  devised 
a  method  which  simplifies  the  process.  The  basis  of  the  method  is  an  appropriate  change 
of  the  independent  variable.  The  result  of  that  change  is  to  divide  the  plane  of  the 
modified  variable  ^  into  two  portions,  one  of  which,  (rg?  i^  finite  in  area  and  the  other  of 
which,  6^1,  occupies  the  rest  of  the  plane;  and  the  boundary,  common  to  Q-^  and  O^,  is 
a  circle  of  finite  radius,  called  the  discriminating  circle'^  of  the  function.  In  0<i  the 
modified  function  is  holomorphic;  in  Gx  the  function  is  holomorphic  except  at  f=GO  ; 
and  all  the  singularities  (and  the  branch-points,  if  any)  lie  on  the  discriminating  circle. 

The  theory  is  given  in  Fuchs's  memoir  "  Ueber  die  Darstellung  der  Functionen  com- 

plexer  Variabeln,  ,"  Crelle,  t.  Ixxv,  (1872),  pp.  176—223.     It  is  corrected  in  details 

and  is  amplified  in  Crelle,  t.  cvi,  (1890),  pp.  1 — 4,  and  in  Grdle.,  t.  cviii,  (1891), 
pp.  181 — 192;  see  also  Nekrassoff^  Math.  Ami.^  t.  xxxviii,  (1891),  pp.  82 — 90,  and 
AnissimoflF,  Math.  Ann.,  t.  xl,  (1892),  pp.  145—148. 

*  Fuchs  calls  it  Grenzkreis. 


CHAPTER   VII. 

Functions  with  unlimited  Essential  Singularities,  and  Expansion 
IN  Series  of  Functions. 

In  addition  to  the  memoirs  mentioned  below,  a.s  being  the  basis  of  the  present  chapter, 
there  are  several  others  (alluded  to  at  the  end  of  §  35)  of  the  greatest  importance,  dealing 
with  the  general  theory  of  uniform  analytic  functions  and  particularly  with  their  analytical 
representation  by  an  infinite  series  of  polynomials  in  the  variable.  Among  these,  specially 
worthy  of  note,  are : — 

Runge,  Acta  Math.,  t.  vi,  (1885),  pp.  229—248  ; 

Hilbert,  Gott.  Nachr.,  (1897),  pp.  63—70; 

Painleve,  Comptes  Rendus,  t.  cxxvi,  (1898),  pp.  200—202,  318—321,  385—388,  459—461, 

ib.  t.  cxxviii,  (1899),  pp.  1277—1280,  ib.  t.  cxxix,  (1899),  pp.  27—31 ;  see  also  his 

thesis,  quoted  in  §  86 ; 
Phragmen,  Comptes  Rendus,  t.  cxxviii,  (1899),  pp.  1434 — 1437; 
Mittag-Leffler,  Acta  Math.,  t.  xxiii,  (1900),  pp.  43 — 62,   where  references  are  given 

to  earlier  records  of  the  investigations;  also  Camh.  Phil.  Trans.,  (Stokes  Jubilee 

volume),  t.  xviii,  (1900),  pp.  1—11;  and  Acta  Math.,  t.  xxiv,  (1901),  pp.  183—244. 
See  also   Borel,   Legons  sur   la  theorie  des  fonctions,  (Gauthier-Villars,  Paris,   1898), 

ch.  vi. 
A  comprehensive  reference  may  here  be  given  to  the  Collection  de  monographies  sur  la 

theorie  des  fonctions,  puhliee  sous  la  direction  de  M.  Emile  Borel.     The  earliest  of 

them  is  the  monograph  by  Borel  just  quoted ;  and  some  of  them  deal  solely  with 

functions  of  real  variables. 

72.  It  now  remains  to  consider  functions  which  have  an  infinite  number 
of  essential  singularities*.  It  will,  in  the  first  place,  be  assumed  that  the 
essential  singularities  are  isolated  points,  that  is,  that  they  do  not  form  a 
continuous  line,  however  short,  and  that  they  do  not  constitute  a  continuous 

*  The  results  in  the  present  chapter  are  founded,  except  where  other  particular  references  are 
given,  upon  the  researches  of  Mittag-Leffler  and  Weierstrass.  The  most  important  investigations 
of  Mittag-Leffler  are  contained  in  a  series  of  short  notes,  constituting  the  memoir  "  Sur  la  th6orie 
des  fonctions  uuiformes  d'une  variable,"  Comptes  Rendus,  t.  xciv,  (1882),  pp.  414,  511,  713,  781, 
938,  1040,  1105,  1163,  t.  xev,  (1882),  p.  335;  and  in  a  memoir  "  Sur  la  representation  analytique 
des  fonctions  monog^nes  uniformes,"  Acta  Math.,  t.  iv,  (1884),  pp.  1 — 79.  The  investigations  of 
Weierstrass  referred  to  are  contained  in  his  two  memoirs  "  Ueber  eineu  funetionentheoretischen 
Satz  des  Herrn  G.  Mittag-Leffler,"  (1880),  and  "  Zur  Functionenlehre,"  (1880),  both  included  in 
the  volume  Ahhandlimgen  aus  der  Functionenlehre,  pp.  53 — 66,  67 — 101,  102 — 104,  Ges.  Werke, 
t.  ii,  pp.  189 — 199,  201 — 233.  A  memoir  by  Hermite,  "  Sur  quelques  points  de  la  theorie  des 
fonctions,"^ eta  Soc.  Fenn.,  t.  xii,  pp.  67 — 94,  Crelle,  t.  xci,  (1881),  pp.  54 — 78,  maybe  consulted 
with  great  advantage. 


72.]  mittag-leffler's  theorem  135 

area,  however  small,  in  the  plane.  Since  their  number  is  unlimited  and 
their  distance  from  one  another  is  finite,  there  must  be  at  least  one  point  in 
the  plane  (it  may  be  at  2:  =  oo  )  where  there  is  an  infinite  aggregate  of  such 
points.  But  no  special  note  need  be  taken  of  this  fact,  for  the  character  of  an 
essential  singularity  does  not  enter  into  the  question  at  this  stage ;  the 
essential  singularity  at  such  a  point  would  merely  be  of  a  nature  different 
from  the  essential  singularity  at  some  other  point. 

We  take,  therefore,  an  infinite  series  of  quantities  tti,  0-2,  ^s, ...  arranged  in 
order  of  increasing  moduli,  and  such  that  no  two  are  the  same :  and  so  we 
have  infinity  as  the  limit  of  j  a^  |  when  v  —  co  . 

Let  there  be  an  associated  series  of  uniform  functions  of  z  such   that 

for  all  values  of  i,  the  function  Gi  f 1 ,  vanishing  with  ,  has  (Xj  as 

its  sole  singularity ;  the  singularity  is  essential  or  accidental  according  as 
Gi  is  transcendental  or  polynomial.  These  functions  can  be  constructed 
by  theorems  already  proved.  Then  we  have  the  theorem,  due  to  Mittag- 
Leffler: — It  is  always  possible  to  construct  a  uniform  analytic  function  F{z), 
having  no  singularities  other   than    a^,  a2,  a^,  ...    and  such  that  for   each 

determinate  value  of  v,  the  difference  F{z)—  G^  ( )  is  finite  for  z  =  a^ 

and  therefore,  in  the  vicinity  of  a^,  is  expressible  in  the  form  P{z—  a^). 

73.     To  prove  Mittag-Leffler's  theorem,  we  first  form  subsidiary  functions 

F^  (z),  derived  from  the  functions  G  as  follows.     The  function  G„  ( — 

\z     ai- 

converges  everjrvvhere  in  the  plane  except  within  an  infinitesimal  circle  round 
the  point  a^ ;  hence  within  a  circle  \z\=  p,  where  p  is  less  than  |  a^  | ,  it  is  a 
monogenic  analytic  function  of  z,  and  can  therefore  be  expanded  in  a  series 
of  positive  powers  of  z  which  converges  uniformly  within  the  circle  \z\  =  p, 
say 

GJ^—)  =  iv,z'^, 
\z  —  a^l     ^=0 

for  values  of  z  such  that  \z\^p<\a^\.  If  a^  be  zero,  there  is  evidently  no 
expansion. 

Let  e  be  a  positive  quantity  less  than  1,  and  let  e^,  e^,  €3,  ...  be  arbitrarily 
chosen  positive  decreasing  quantities,  subject  to  the  single  condition  that  2e 
is  a  converging  series,  say  of  sum  A  :  and  let  eo  be  a  positive  quantity  inter- 
mediate between  1  and  e.     Let  g  be  the  greatest  value  of    G^  ( — 

\z  —  a^ 

points  on  or  within  the  circumference  |  ^  |  =  eo  |  a  J  ;  then,  because  the  series 
2  v^zi^  is  a  converging  series,  we  have,  by  §  29,  : 

\v^z'^\<g, 


for 


136  MIXTA  g-leffler's  [73. 

9 


or  ^M  <: 


to'     I  W,^  I' 

Hence,  with  values  of  z  satisfying  the  condition  \z\^e\a^\,  we  have,  for 
any  value  of  tn, 

2     I'^^'^l^     2      \Vy:\\z\l^ 


since  e  <  eo.     Take  the  smallest  integral  value  of  m  such  that 

9      (^^ 


it  will  be  finite  and  may  be  denoted  by  m^.     Thus  we  have 

for  values  of  z  satisfying  the  condition  \z\^€\a^,\. 

We  now  construct  a  subsidiary  function  F^  (z)  such  that,  for  all  values  of  z, 


I     1     \      '"""^ 
xz  —  a^,/      ^=0 


then,  for  values  of  |  ^r }  which  are  ^  e\a^\, 

I  F,  {z)  I  ^  e,. 

Moreover,  the  function   2  i^^^'^  is  finite  for  all  finite  values  oi  z;  so  that,  if  we 
take 


^^  {z)  =  z-'^'^'G, 2 


then  (f>^  (z)  is  zero  at  infinity,  because,  when  z=  cc  .  G^( )  is  finite  by 

hypothesis.     Evidently  cf)u{z)  is  infinite  only  at  z  =  a^,  and  its  singularity  is 
of  the  same  kind  as  that  of  G^, 


74.  Now  let  c  be  any  point  in  the  plane,  which  is  not  one  of  the  points 
«!,  ^2,  (h,  •••  ;  it  is  possible  to  choose  a  positive  quantity  p  such  that  all  the 
points  a  lie  without  the  circle  \z  —c\=  p.  Let  a^  be  the  singularity,  which 
is  the  point  nearest  to  the  origin  satisfying  the  condition  \a^\>\c\+  p;  then, 
for  points  within  or  on  the  circle,  we  have 


<e, 


*^4i.]  THEOREM  137 

when  s  has  the  values  v,v  +  l,v +  2,  ....     Introducing  the  subsidiary  functions 
F^  (z),  we  have,  for  such  vahies  of  z, 

and  therefore  S  Fs(z)\^  t\Fs (z) \ 


a  finite  quantity.  Also  let  8  denote  any  assigned  finite  positive  quantity, 
however  small ;  an  integer  yu,'  can  be  chosen  so  that  2  eg  <  S,  for  all  integers 
IX  ^  //.',  and  for  all  positive  integers  r.     For  these  same  integers,  we  have 

tF,{z)\^  t  \F,(z)\^  2  es^S. 

00 

It    therefore   follows   that   the   series    S  Fg  (z)   converges   uniformly   for    all 

values  of  z  which  satisfy  the  condition  \z  -  c\^p.  Moreover,  all  the  functions 
Ft_{z),  Fziz), ... ,  Fr-i{z)  are  finite  for  such  values  of  z,  because  their  singularities 
lie  without  the  circle  \z  —  c\  =  p]  and  therefore  the  series 

i  F^z) 

)•  =  ! 

converges  uniformly  for  all  points  z  within  or  on  the  circle  \z  —  c\=  p,  where 
p  is  chosen  so  that  all  the  points  a  lie  without  the  circle. 

The  function,  represented  by  the  series,  can  therefore  be  expanded  in  the 
form  P{z  —  c),  in  the  domain  of  the  point  c. 

If  a^n  denote  any  one  of  the  points  a^,  a^,  ...,  and  we  take  p  so  small  that 
all  the  points,  other  than  a,,,,,  lie  without  the  circle 

I  I / 

I  ^        ^m  \  —  P  > 

then,  since  F^^i  {z)  is  the  only  one  of  the  functions  F  which  has  a  singularity 
at  a^,  the  series 

%^[Fr{z)], 
,.  =  1 

where  2"*  implies  that  F,^  {z)  is  omitted,  converges  uniformly  in  the  vicinity 
of  a,  and  therefore  it  can  be  expressed  in  the  form  P  (z  —  a^).     Hence 


^Fr{z)  =  F,,{z)+P{z-a,n) 

r=l 

-=G<n{-^)-^P,{z-a^\ 
\z  -  a^n  / 


138  mittag-leffler's  [74. 

the  difference  of  F^^  and  0^.  being  absorbed  into  the  series  P  to  make  Pj .     It 

00 

thus  appears  that  the  series  S  F^  {z)  is  a  function  which  has  infinities  only 
at  the  points  a^,  a^,  ...,  and  is  such  that 

tFAz)-G,J-^)  -         ■ 

r==l  \2  —  am/ 

00 

can  be  expressed  in  the  vicinity  of  a,^  in  the  form  P  (z  —  cim)-    Hence  S  Fr  {z) 

r=l 

is  a  function  of  the  required  kind. 

75.  It  may  be  remarked  that  the  function  is  not  unique.  As  the 
positive  quantities  e  were  subjected  to  merely  the  single  condition  that  they 
form  a  converging  series,  there  is  the  possibility  of  wide  variation  in  their 
choice :  and  a  difference  of  choice  might  easily  lead  to  a  difference  in  the 
ultimate  expression  of  the  function. 

This  latitude  of  ultimate  expression  is  not,  however,  entirely  unlimited. 
For,  suppose  there  are  two  functions  F{z)  and  F(z),  enjoying  all  the  assigned 
properties.  Then  as  any  point  c,  other  than  aj,  (Xa,  •  •  • ,  is  an  ordinary  point  for 
both  F  (z)  and  F  (z),  it  is  an  ordinary  point  for  their  difference :  and  so 

F(z)-F{z)^P{z-c) 

for  points  in  the  immediate  vicinity  of  c.  The  points  a  are,  however, 
singularities  for  each  of  the  functions:  in  the  vicinity  of  such  a  point  a^,. 
we  have 

\Z        ili' 

since  the  functions  are  of  the  required  form  :  hence 

F(z)-F  (z)  =  P{z-ai)-P{z-  ai), 

or  the  point  ai  is  an  ordinary  point  for  the  difference  of  the  functions.  Hence 
every  finite  point  in  the  plane,  whether  an  ordinary  point  or  a  singularity 
for  each  of  the  functions,  is  an  ordinary  point  for  the  difference  of  the 
functions :  and  therefore  that  difference  is  a  uniform  integral  function  of  z. 
It  thus  appears  that,  if  F  (z)  be  a  function  with  the  required  properties,  then 
every  other  function  with  those  properties  is  oftheforin 

F{z)+G{z), 

tvhere  G  {z)  is  a  uniform  integral  function  of  z  either  transcendental  or 
polynomial. 

The  converse  of  this  theorem  is  also  true.    • 


75.]  THEOREM  139 

00 

Moreover,  the  function  G  {z)  can  always  be  expressed  in  a  form  2  g^,  {z),  if 

v  =  l 

it  be  desirable  to  do  so :    and  therefore  it  follows  that  any  function  with  the 
assigned  characteristics  can  be  expressed  in  the  form 

i  [F„{z)+g,{z)]. 

v  =  l 

Note.  In  the  preceding  investigation,  the  integers  lUv  have  not  been  limited  to  be  the 
same  for  each  of  the  functions  G.  The  simplest  sets  of  functions  evidently  arise  when  a 
common  value  can  be  assigned  to  the  integers;  they  then  correspond  to  Weierstrass's 
converging  infinite  products  (§§  50,  59 — 61),  arranged  according  to  their  class.  But  as 
with  the  converging  infinite  products  (§  51),  it  may  happen  that  no  common  value  can 
be  assigned  :  and  then  the  preceding  investigation,  in  its  most  general  form,  establishes 
the  existence  of  the  functions. 

It  does  not,  however,  indicate  that  the  expression  is  unique.  If,  for  instance,  the 
series  of  functions  G  be 

G  ^ 

'^     X  —  loge  n  ' 

for  n=\,  2,...,  the  function  formed  by  the  preceding  method  is 

i    [      ^'"'^        I         1        j. 
71=1  1  (loge '0'"™     x-\og^n) 

and   there   is   no   finite    integer    which,    when   assigned   as    the   common   value   of    the 

integers  ?«„,  will  make  the  series  converge. 

But  we  may  use  the  function 

2    

71=1  nlog^n{x-\ogeny 

which  satisfies  all  the  conditions  and  is  a  converging  series*. 

76.  The  following  applications,  due  to  Weierstrass,  can  be  made  so  as 
to  give  a  new  expression  for  functions,  already  considered  in  Chapter  VI., 
having  5  =  oo  as  their  sole  essential  singularity  and  an  unlimited  number 
of  poles  at  points  «!,  a^,  — 

If  the  pole  at  a^  be  of  multiplicity  lUi,  then  {z  —  a^'^ifiz)  is  regular  at 
the  point  a^  and  can  therefore  be  expressed  in  the  form 

00 

.  2  Cy.{z-a>)f^. 

(1  =  0 

mi  - 1 

Hence,  if  we  take  ft  (z)  =    1  c^(z-  a^)-»^^+^ 

M.  =  0 

we  have  f(z)  =  fi  (z)  +P(z-  ai). 

Now  deduce  from  fi{z)  a  function  Fi  (z)  as  in  §  73,  and  let  this  deduction  be 

effected  for  each  of  the  functions /^  (ir).     Then  we  know  that 

^Fiiz) 
?:=i 

*  This  remark  was  made  to  me  by  Prof.  A.  C.  Dixon. 


140  FUNCTIONS   POSSESSING  [76. 

is  a  uniform  function  of  s  having  the  points  aj,  ag,  ...  for  poles  in  the  proper 
multiplicity  and  no  essential  singularity  except  z=  oo  .  The  most  general 
form  of  the  function  therefore  is 

"Hence  any  uniform  analytical  function  which  has  no  essential  singularity 
except  at  infinity  can  he  expressed  as  a  sum  of  functions  each  of  which  has  only 
one  singularity  in  the  finite  part  of  the  plane.     The  form  of  F^  {z)  is 

f.{z)-Gr{z), 

where  fr{z)  is  infinite  at  z  =  ay,  and  Gr{z)  is  a  properly  chosen  integral 
function. 

We  pass  to  the  case  of  a  function,  having  a  single  essential  singularity  at 
c  and  at  no  other  point,  and  any  number  of  accidental  singularities,  by  taking 

/  = as  in  5  63,  Cor.  II. :  and  so  we  obtain  the  theorem  : — 

z  —  c 

Any  uniform  function  which  has  only  one  essential  singularity ,  say  at  c, 
can  he  expressed  as  a  sum  of  uniform  functions  each  of  which  has  only  one 
singularity  different  from  c. 

Evidently  the  typical  summative  function  F^  (z)  for  the  present  case  is  of 
the  form 


y,.(,)  +  e,(_l_) 


.77.  The  results,  which  have  been  obtained  for  functions  possessed  of 
an  infinitude  of  singularities,  are  valid  on  the  supposition,  stated  in  §  72, 
that  the  limit  of  a\,  with  indefinite  increase  of  v  is  infinite ;  the  terms 
in  the  sequence  aj,  ttg,  ...  tend  to  one  definite  limiting  point  which  is 
^■=00  and,  by  the  substitution  z'  (z  —  g)  =  1,  can  be  made  any  point  c  in 
the  finite  part  of  the  plane. 

Such  a  sequence,  however,  does  not  necessarily  tend  to  one  definite  limiting 
point :  it  may,  for  instance,  tend  to  condensation  on  a  curve,  though  the 
condensation  does  not  imply  that  all  points  of  the  continuous  arc  of  the 
curve  must  be  included  in  the  sequence.  We  shall  not  enter  into  the  dis- 
cussion of  the  most  general  case,  but  shall  consider  that  case  in  which  the 
sequence  of  moduli  \aj\,  |  ttg  j;  •••  tends  to  one  definite  limiting  value  so  that, 
with  indefinite  increase  of  v,  the  limit  of  |  a^  \  is  finite  and  equal  to  It ; 
the  points  «!,  ag,  ...  tend  to  condense  on  the  circle  \z\=R. 


Such  a  sequence  is  given  by 


h'^] 


77.]  UNLIMITED    SINGULARITIES  141 

for  Z-  =  0,  1, ... ,  »,  and  n  =  l,  2,  ...ad  inf. ;  and  another*  by 

where  c  is  a  positive  proper  fraction. 

With  each  point  a^  we  associate  the  point  on  the  circumference  of  the 
circle,  say  6„,,,  to  which  a,„  is  nearest  :  let 

I  (^m       (^m  I  ^^  Pm  > 

so  that  p,„  approaches  the  limit  zero  with  indefinite  increase  of  m.  There 
cannot  be  an  infinitude  of  points  o^,  such  that  pp^  @,  any  assigned  positive 
quantity ;  for  then  either  there  would  be  an  infinitude  of  points  a  within  or 
on  the  circle  \z\  =  B,  —  ®,  or  there  would  be  an  infinitude  of  points  a  within 
or  on  the  circle  \z\  =  II  +  ®,  both  of  which  are  contrary  to  the  hypothesis 
that,  with  indefinite  increase  of  /',  the  limit  of  \a^\  is  R.  Hence  it  follows 
that  a  finite  integer  n  exists  for  every  assigned  positive  quantity  @,  such  that 

I  civi  -  K  !  <  ® 
when  m  ^  n. 

Then  the  theorem,  which  corresponds  to  Mittag-Leffler's  as  stated  in  §  72 
and  which  also  is  due  to  him,  is  as  follows : — 

It  is  always  possible  to  construct  a  uniform  analytical  function  of  z  which  is 
definite  over  the  whole  plane,  except  within  infinitesimal  circles  round  the  points 
a  and  b,  and  which,  in  the  immediate  vicinity  of  each  one  of  the  singularities  a, 
can  be  expressed  in  the  form 

where  the  functions  Gi  are  assigned  functions,  vanishing  luith ,  and  finite 

z  —  a^ 

everywhere  in  the  plane  except  at  the  single  points  a^  with  which  they  are 

respectively  associated. 

In  establishing  this  theorem,  we  shall  need  a  positive  quantity  e  less  than 
unity  and  a  converging  series  ej,  eg,  eg,  ...  of  positive  quantities,  all  less  than 
unity. 

Let  the  expression  of  the  function  Gn  be 

n     (       "^       ^    -      '^^.i        1  '^»,2         _|_        ^n,z 

Then,  since  z-an={z-  bn)  \  1  -  ^3^  p 

the  function  Gn  can  be  expressed!  in  the  form 

*  The  first  of  these  examples  is  given  by  Mittag-Leftler,  Acta  Math.,  t.  iv,  p.  11 ;  the  second 
was  stated  to  me  by  Prof.  Burnside. 

t  The  justification  of  this  statement  is  to  be  found  in  the  proposition  in  §  82. 


142  FUNCTIONS   POSSESSING  [77. 

for  values  of  z  such  that 

<  e 


z-h,_ 
and  the  coefficients  A  are  given  by  the  equations 

Now,  because  G^  is  finite  everywhere  in  the  plane  except  at  a„,  the  series 

I 


On.  ^  1  I  <^w,2  I  j^  I  ^71, 3  j 


has  a  finite  value,  say  g,  for  any  non-zero  value  of  the  positive  quantity  ^^ ; 
then 

Hence  | ^„ , < ^2^  ^^-^,  (^-.).(.-l)! 


^ii^\an-bnnfM-r)l{r-l)l 


Introducing  a  positive  quantity  a  such  that 

(l+a)e<  1, 
we  choose  ^n  so  that  ^n  <  «  |  ««  —  &n  I ; 

and  then  |  J.^^  ^  |  <  ^a  (1  +  ay~\ 

Because  (1  +  a)  e  is  less  than  unity,  a  quantity  6  exists  such  that 

(1  +a)e<^<l. 

Then  for  values  of  z  determined  by  the  condition    — — ^   <  e,  we  have 


^        1  -^n,  /x  1 


an  -  6,,  k        ga    6'^^+'^ 
z-hn\        1  +  a  1  -^ 

Let  the  integer  m,i  be  chosen  so  that 

ga     d^'^^+i 


it  will  be  a  finite  integer,  because  ^  <  1.     Then 

I  -^»i,fi  I 


1    •^         ^n 


<e.n 


We  now  construct,  as  in  §  73,  a  subsidiary  function  Fn  {z),  defining  it  by 
the  equation 

\z-aj      ^=0  \z-hnJ 


77.]  UNLIMITED   SINGULARITIES  143 

<  e,  we  have 


Z  —  K 


SO  that  for  points  z  determined  by  the  condition 

\Fn{z)\<en. 
A  function  with  the  required  properties  is 

F{z)=  I  F^{z). 

m=l 

To  prove  it,  let  c  be  any  point  in  the  plane  distinct  from  any  of  the  points 
a  and  h  ;  we  can  always  find  a  value  of  p  such  that  the  circle 

\z-c\=p 

contains  none  of  the  points  a  and  b.  Let  I  be  the  shortest  distance  between 
this  circle  and  the  circle  of  radius  R,  on  which  all  the  points  b  lie ;  then  for 
all  points  z  within  or  on  the  circle  \z  —  c\  =  p,  we  have 

Now  we  have  seen  that,  for  any  assigned  positive  quantity  ©,  there  is  a 
finite  integer  w  such  that 

\am-K\<  ®, 
when  m'^n.     Taking  ©  =  el,  we  have 

b    I 


^m       ^m 


when  'm'^n,'n  being  the  finite  integer  associated  with  the  positive  quantity  el. 
It  therefore  follows  that,  for  points  z  within  or  on  the  circle  \z~c\=p, 

I  Fm  {z)  I  <  e,„, 
when  m  is  not  less  than  the  finite  integer  n  ;  hence 

00 

2    i  Fm  {z)  I  <  e„  +  €n+i  +  e„+2  4- . . , . 

Now  the  series  of  positive  quantities  e^,  e^,  ...  converges;  and  therefore 

i  F^(z) 

is  a  series,  which  converges  uniformly  and  unconditionally.     Each  of  the 
functions  F^  {z),  Fo^ {z),  ... ,  Fn-i  {z)  is  finite  when  \z-c\^p;  and  therefore 

i  Fm{z) 

w  =  l 

is  a  series  which  converges  uniformly  and  unconditionally  for  all  values  of  z 
within  the  circle 

\z-c\  =  p. 


144  TRANSCENDENTAL    FUNCTION    AS  [77. 

Hence  the  function  represented  by  the  series  can  be  expressed  in  the  form 
P  {z  —  c)  for  all  such  values  of  z.  The  function  therefore  exists  over  the 
whole  plane  except  at  the  points  a  and  h. 

It  may  be  proved,  exactly  as  in  §  74,  that,  for  points  z  in  the  immediate 
vicinity  of  a  singularity  a^, 

F  {z)  =  G^  (— i— )  +  P(z-  a,,). 

\Z        (f"m' 

The  theorem  is  thus  completely  established. 

The  function  thus  obtained  is  not  unique,  for  a  wide  variation  of  choice  of 
the  converging  series  ej  +  63  -r  . . .  is  possible.  But,  in  the  same  way  as  in  the 
corresponding  case  in  §  75,  it  is  proved  that,  if  F{z)  he  a  function  with  the 
required  properties,  every  other  fu/nction  with  those  properties  is  of  the  fortn 

F(z)  +  G{z), 

where  Q  {z)  behaves  regularly  in  the  immediate  vicinity  of  every  point  in  the 
plane  except  the  points  h. 

Ex.     If  the  points  a  in  Mittag-Leffler's  theorem  are  given  by 

l+-jf^",  (>^  =  0,  1,  ...,n-l;  n=l,  2,  ...,  00), 

and  if  Om  { )  =      —    5  shew  that 

is  a  function  of  the  character  specified  in  the  theorem. 
Discuss  the  nature  of  the  function  defined  by 


-,._(i+l 


(Math.  Trip.,  Part  II.,  1899.) 


78.  The  theorem  just  given  regards  the  function  in  the  light  of  an 
infinite  converging  series  of  functions  of  the  variable  :  it  is  natural  to  suppose 
that  a  corresponding  theorem  holds  when  the  function  is  expressed  as  an 
infinite  converging  product.  With  the  same  series  of  singularities  as  in 
§  77,  when  the  limit  of  \a^\  with  indefinite  increase  of  v  is  finite  and 
equal  to  R,  the  theorem  *  is  : — 

It  is  always  possible  to  construct  a  uniform  analytic  function,  luhich 
behaves  regularly  everywhere  in  the  plane  except  within  infinitesimal  circles 

*  Mittag-Leffler,  Acta  Math.,  t.  iv,  p.  32 ;  it  may  be  compared  with  Weierstrass's  theorem 
in  §  67. 


78.] 


AN   INFINITE   PRODUCT   OF   FUNCTIONS 


145 


round  the  points  a  and  h,  and  which  in  the  vicinity  of  any  one  of  the  points  a^, 
can  be  expressed  in  the  form 

where  the  numbers  ?ii,  nQ,  ..;  ai^e  any  assigned  integers. 

The  proof  is  similar  in  details  to  proofs  of  other  propositions  and  it  will 
therefore  be  given  only  in  outline.     We  have 


7l„ 


riv  n^ 

+ 


z  —  a^     z  —  b^     z  —  b^  ^=i\z  —  b„J  ' 


dv  —  bS\i^ 


provided 


a^,—  b^, 


-r^\<e,  the  notation  being  the  same  as  in  S  77.     Hence,  for 

z       Oi,   I 

such  values  of  z, 

1  ((iv-i>v\l^ 


If  we  denote 


a„  —  6„ V*"  «r 


/        a^  -  6.A 
V         z~bj 


1  fa^-bv\iJ- 


^=1 M  \  z-K 


by  E^ {z),  we  have  E^ {z)  =  e    %=m;,+i ix\z-b^ 

Hence,  if  ^(^r)  denote  the  infinite  product 

H  E^(z), 


we  have 


-  S  <n„       S 

E  (z)  =  e   "^^      M=mi/+i 


1  /a„-6AM) 


and  E{z)  is  a  determinate  function  provided  the  double  series  in  the  index  of 
the  exponential  converges. 

Because  n^  is  a  finite  integer,  and  because 

1  /a^  -  b^Y 


2  1  /tzA-)* 

f,=i  fi\z-b^J 


is  a  converging  series,  it  is  possible  to  choose  an  integer  m^  so  that 

u  =  m..+lf^  \Z  —  Ov  J 


<  Vv, 


where  77^  is  any  assigned  positive  quantity.  We  take  a  converging  series  of 
positive  quantities  77^ ;  and  then  the  moduli  of  the  terms  in  the  double  series 
form  a  uniformly  converging  series.  The  double  series  itself  therefore 
converges  uniformly;  and  then  the  infinite  product  F{z)  converges  uniformly 
for  points  z  such  that 

a^  —  b, 

z  —  b^ 

10 


<  e. 


146  TRANSCENDENTAL  FUNCTION   HAVING  [78. 

As  in  §  77,  let  c  be  any  point  in  the  plane,  distinct  from  any  of  the 
points  a  and  h.  We  take  a  finite  value  of  p  such  that  all  the  points  a  and  h 
lie  outside  the  circle  \z  —  c\=  p;  and  then,  for  all  points  within  or  on  this 
circle, 

(f"m.  0,1 


z-K 


<  e, 


when  m  ^  n,  n  being  the  finite  integer  associated  with  the  positive  quantity 
el.     The  product 

n  E,(z) 

v—n 

is  therefore  finite,  for  its  modulus  is  less  than 


s 

■nv 

6"= 

n 

J 

«-l 

n 

E. 

(^) 

/=i 

the  product 

is  finite,  because  the  circle  \z  —  c\  =  p  contains  none  of  the  points  a  and  h ; 
and  therefore  the  function  F{z)  is  finite  for  all  points  within  or  on  the  circle. 
Hence  in  the  vicinity  of  c,  the  function  can  be  expanded  in  the  form  P  {z—  c); 
and  therefore  the  function  is  definite  everywhere  in  the  plane  except  within 
infinitesimal  circles  round  the  points  a  and  h. 

The  infinite  product  converges  uniformly  and  unconditionally.  As  in  1 51, 
it  can  be  zero  only  at  points  which  make  one  of  the  factors  zero  and,  from  the 
form  of  the  factors,  this  can  take  place  only  at  the  points  a^  with  positive 
integers  ?i„.  In  the  vicinity  of  a^,  all  the  factors  of  F  {z)  except  E,,{z)  are 
regular:  hence  F{z)/E^.{z)  can  be  expressed  as  a  function  of  ^^  —  a^  in  the 
vicinity.  But  the  function  has  no  zeros  there,  and  therefore  the  form  of  the 
function  is 

Hence,  in  the  vicinity  of  a^,  we  have 

F{z)=E.{z)eP^^^-''v) 

* 

on  combining  the  exponential  index  in  E^,  {z)  with  Pj  {z  —  a^).  This  is  the 
required  property. 

Other  general  theorems  will  be  found  in  Mittag-Leffler's  memoir  just 
quoted. 

79.  The  investigations  in  §§  72 — 75  have  led  to  the  construction  of  a 
function  with  assigned  properties.  It  is  important  to  be  able  to  change,  into 
the  chosen  form,  the  expression  of  a  given  function,  having  an  infinite  series 
of  singularities  tending  to  a  definite  limiting  point,  say  to  ,s=oo.     It  is 


79.]  AN   INFINITE    SET   OF   SINGULARITIES  147 

necessary  for  this  purpose  to  determine  (i)  the  functions  Fr  {z)  so  that  the 

series  2  Fr{z)  may  converge  uniformly  and  unconditional!}^,  and  (ii)  the 

fuiiction  G{z). 

Let  <l>  {z)  be  the  given  function,  and  let  >S'  be  a  simple  contour  embracing 
the  origin  and  yu,  of  the  singularities,  viz.,  a^,  ...,  a^:  then,  if  t  be  any 
point,  we  have 

■«"^(0  fzY\,^  ,    f^'^^jt)  fz\ 
t-  z\h 


^^(t)  fz 
t-z\t 


t  —  z  \t 


v  =  l  J  t    —  Z    \t/ 


where  I       implies  an  integral  taken  round  a  very  small  circle  centre  a. 

If  the  origin  be  one  of  the  points  aj,  as,  ...,  then  the  first  term  will  be 
included  in  the  summation. 

Assuming  that  z  is  neither  the  origin  nor  any  one  of  the  points  a-^,  ...,a 
we  have 


so 


Now  xr^('^  dt 


27ri  J      t  —  z  \t 
z 


(m  - 1) ! 


dV""-^  t  -  z 


^m-i    j-^(^)        ^0(^) 


{m  -  1)  !  Idt^-"-  {   z  z" 

z^ 


(to  —  1) 


-  ^'"-i  (0)  +  --^  ^"^-2 (0)  +  . . . 

z  z 


^  ^      1  (m  —  1) ! 


=  -G^(^), 


unless  ^  =  0  be  a  singularity  and  then  there  will  be  no  term  Q  {£).     Similarly, 
it  can  be  shewn" that 

2^1       t-z\tj 


is  equal  to 
where 


G^ 


\z  —  a^ 


-   lv,(^]  =F,(z), 
dt, 


2  —  ft,//  A  =  0  \^v 

1      \  1      fK^^(t) 


27^^ 


t-z 


10—2 


148  '  TRANSCENDENTAL   FUNCTION   AS  [79. 

and  the  subtractive  sum  of  m  terms  is  the  sum  of  the  first  m  terms  in  the 
development  of  G^  in  ascending  powers  of  z.     Hence 


^^,     dt. 

v  =  l 


If,  for  an  infinitely  large  contour,  m  can  be  chosen  so  that  the  integral 

1    f^(t)[^ 


I'm  J  t  —  z  \t/ 

diminishes  indefinitely  with  increasing  contours  enclosing  successive  singu- 
larities, then 

^{z)=G  {z)  +i  F,  (z). 

v  =  l 

The  integer  m  may  be  called  the  critical  integer. 
If  the  origin  be  a  singularity,  we  take 

1^ 


Fo(z)=Go 

and  there  is  then  no  term  G  (z) :  hence,  including  the  origin  in  the  summa- 
tion, we  have 

so  that  if,  for  this  case  also,  there  be  some  finite  value  of  m  which  makes 
the  integral  vanish,  then 

0(^)=  i  F^(z). 

Other  expressions  can  be  obtained  by  choosing  for  m  a  value  greater  than 
the  critical  integer;  but  it  is  usually  most  advantageous  to  take  m  equal  to 
its  lowest  effective  value. 

Ex.  1.     The  singularities  of  the  function  v  cot  ttz  are  given  by  z='\,  for  all  integer 
values  of  X  from  —  oo  to   +<x)  including  zero,  so  that  the  origin  is  a  singularity. 

The  integral  to  be  considered  is 

1    n»)^rcot^/_.x™ 

2Tri  J  t-z      \tj 

We  take  the  contour  to  be  a  circle  of  very  large  radius  R  chosen  so  that  the  circumference 
does  not  pass  infinitesimally  near  any  one  of  the  singularities  of  it  cot  nt  at  infinity ;  this 
is,  of  course,  possible  because  there  is  a  finite  distance  between  any  two  of  them.  Then, 
round  the  circumference  so  taken,  TrcotTr^  is  never  infinite:  hence  its  modulus  is  never 
greater  than  some  finite  quantity  M. 

Let  t—R^^,  so  that  ~  =  idd;  then 

1     /■2'r 
J=-r—   ;       IT  cot  irt  — 
Stt   /  0  t- 


79.]  AN   INFINITE   SERIES   OF   FUNCTIONS  149 


z 


I  TO—  1 


and  therefore  \J\^M- 

I  i-^\  \t\ 

for  some  point  t  on  the  circle.     Now,  as  the  circle  is  very  large,  we  have  \t-z\  infinite : 
hence  |  J  \  can  be  made  zero  merely  by  taking  m  unity. 

Thus,  for  the  function  tt  cot  n-s,  the  critical  integer  is  unity. 

Hence,  by  the  general  theorem,  we  have  the  equation 


J.  ^    ^  ['"'  cot  irt  z   ^ 


the  summation  extending  to  all  the  points  X  for  integer  values  of  X=  -  oo  to   +qo,  and 
each  integral  being  taken  round  a  small  circle  centre  X. 

^j  -o    ■  1        r(^^  TT  cot  7rt  z    - 

Now  if,  m  -— .  — _  dt, 

ZTTl   J  t—Z  t 

we  take  ^=X  +  f,  we  have 

7rCOt7r^  =  ^  +  P(t), 

where  P(^)  =  0  when  t=0;  *nd  therefore  the  value  of  the  integral  is 


Ini  }  {\-Z-{ 


In  the  limit  when  I  C|  is  infinitesimal,  this  integral 


(X- 

■z)\ 

1 

X-; 

1 

~z     X' 

and  therefore  F.  (z)  =  — -  +  - , 

-^         s-X     X 

if  X  be  not  zero. 

And  for  the  zero  of  X,  the  value  of  the  integral  is 


so  that  F(){z)  is  — .     In  fact,  in  the  notation  of  §  72,  we  have 


^. 


\z-\)      z-X' 
and  the  expansion  of  G^^  needs  to  be  carried  only  to  one  term. 


150  EEGION   OF   CONTINUITY  [79. 

1       A=<»    /I         1 
We  thus  have  n  cot  tts  =  -  4-     S'        r  +  r 

the  summation  not  including  the  zero  value  of  X. 
Bx.  2.     Obtain,  ab  initio,  the  relation 

1  A=co  1 


Ex.  3.     Shew  that,  if 


-^')-(-(?)l- {-(1)1 Hmi 


^,  TT  cot  TTS  1     ,    „       °0  1  1 

then  —  =  -  +  2s  2 


(Gylden,  Mittag-Leffler.) 
Ex.  4.     Obtain  an  expression,  in  the  form  of  a  sum,  for 

TT  cot  TTZ 


where  §(2)  denotes  (i_5)M_^y  (l-|)    (  1--)  . 


n) 


Ex.  5.     Construct  a  uniform  analytical  function  E{x),  which  is  finite  at  all  finite 
points  except  at  the  points  0,  1,  2,  3,  ...,  at  which  it  is  infinite  in  such  a  way  that 

F  (x)  —  e^  cot  ttx 
is  finite  at  each  of  the  points. 

(Math.  Trip.,  Part  II.,  1897.) 

80.  The  results  obtained  in  the  present  chapter  relating  to  functions 
which  have  an  unlimited  number  of  singularities,  whether  distributed  over 
the  whole  plane  or  distributed  over  only  a  finite  portion  of  it,  shew  that 
analytical  functions  can  be  represented,  not  merely  as  infinite  converging 
series  of  powers  of  the  variable,  but  also  as  infinite  converging  series  of 
functions  of  the  variable.  The  properties  of  functions  when  represented  by 
series  of  powers  of  the  variable  depended  in  their  proof  on  the  condition  that 
the  series  proceeded  in  powers  ;  and  it  is  therefore  necessary  at  least  to 
revise  those  properties  in  the  case  of  functions  when  represented  as  series 
of  functions  of  the  variable. 

Let  there  be  a  series  of  uniform  functions  f\  (z),  f^  {z), ...  ;  then  the 
aggregate  of  values  of  z,  for  wdiich  the  series 

i  =  l 

has  a  finite  value,  is  the  region  of  continuity  of  the  series.     If  a  positive 
quantity  p  can  be  determined  such  that,  for  all  points  2  within  the  circle 

{z  -a\  =  p, 
00 
the  series  X  fi  {z)   converges   uniformly*,    the  series   is   said   to   converge 

*  In  connection  with  most  of  the  investigations  in  the  remainder  of  this  chapter,  Weierstrass's 
memoir  "  Zur  Functionenlehre  "  already  quoted  (p.  134,  note)  should  be  consulted. 


80.]  '  OF   A    SERIES   OF   POWERS  151 

uniformly  in  the  vicinity  of  a.     If  R  be  the  greatest  value  of  p  for  which  this 
holds,  then  the  area  within  the  circle 

\z  —  a\=R 

is  called  the  domain  of  a ;  and  the  series  converges  uniformly  in  the  vicinity 
of  any  point  in  the  domain  of  a. 

It  will  be  proved  in  §  82  that  the  function,  represented  by  the  series  of 
functions,  can  be  represented  by  power-series,  each  such  series  being  equiva- 
lent to  the  function  within  the  domain  of  some  one  point.  In  order  to  be 
able  to  obtain  all  the  power-series,  it  is  necessary  to  distribute  the  region  of 
continuity  of  the  function  into  domains  of  points  where  it  has  a  uniform 
finite  value.  We  therefore  form  the  domain  of  a  point  h  in  the  domain  of  a 
from  a  knowledge  of  the  singularities  of  the  function,  then  the  domain  of 
a  point  c  in  the  domain  of  6,  and  so  on ;  the  aggregate  of  these  domains  is  a 
continuous  part  of  the  plane  which  has  isolated  points  and  which  has  one  or 
several  lines  for  its  boundaries.     Let  this  part  be  denoted  by  J-j. 

For  most  of  the  functions,  which  have  already  been  considered,  the  region 
-4],  thus  obtained,  is  the  complete  region  of  continuity.  But  examples  will 
be  adduced  almost  immediately  to  shew  that  J.^  does  not  necessarily  include 
all  the  region  of  continuity  of  the  series  under  consideration.  Let  a  be  a 
point  not  in  A^,  within  whose  vicinity  the  function  has  a  uniform  finite 
value;  then  a  second  portion  A.^  can  be  separated  from  the  whole  plane,  by 
proceeding  from  a  as  before  from  a.  The  limits  of  A^  and  A^  may  be  wholly 
or  partially  the  same,  or  may  be  independent  of  one  another :  but  no  point 
within  either  can  belong  to  the  other.  If  there  be  points  in  the  region  of 
continuity  which  belong  to  neither  ^i  nor  A^,,  then  there  must  be  at  least 

another  part  of  the  plane  J. 3  with  properties  similar  to  A-^  and  Ao.     And  so 

00 
on.      The  series  S  fi  {z)  converges  uniformly  in  the  vicinity  of  every  point 

1=1 
within  each  of  the  separate  portions  of  its  region  of  continuity. 

It  was  proved  that  a  function  represented  by  a  series  of  powers  has  a 
definite  finite  derivative  at  every  point  lying  actually  within  the  circle 
of  convergence  of  the  series,  but  that  this  result  cannot  be  affirmed  for  a 
point  on  the  boundary  of  the  circle  of  convergence  even  though  the  value  of 
the  series  itself  should  be  finite  at  the  point,  an  illustration  being  provided 
by  the  hypergeometric  series  at  a  point  on  the  circumference  of  its  circle  of 
convergence.  It  will  appear  that  a  function  represented  by  a  series  of 
functions  has  a  definite  finite  derivative  at  every  point  lying  actually  within 
its  region  of  continuity,  but  that  the  result  cannot  be  affirmed  for  a  point 
on  the  boundary ;  and  an  example  will  be  given  (§  83)  in  which  the  derivative 
is  indefinite. 

Again,  it  has  been  seen  that  a  function,  initially  defined  by  a  given  power- 
series,  is,  in  most  cases,  represented  by  different  analytical  expressions  in 


152  REGION   OF   CONTINUITY   OF  [80. 

different  parts  of  the  plane,  each  of  the  elements  being  a  valid  expression  of 
the  function  within  a  certain  region.  The  questions  arise  whether  a  given 
analytical  expression,  either  a  series  of  powers  or  a  series  of  functions : 
(i)  can  represent  different  functions  in  the  same  continuous  part  of  its  region 
of  continuity,  (ii)  can  represent  different  functions  in  distinct,  that  is,  non- 
continuous,  parts  of  its  region  of  continuity. 

81.     Consider  first  a  function  defined  by  a  given  series  of  powers. 

Let  there  be  a  region  A'  in  the  plane  and  let  the  region  of  continuity  of 
the  function,  say  g  (z),  have  parts  common  with  A'.  Then  if  Wo  be  any  point 
in  one  of  these  common  parts,  we  can  express  g  (z)  in  the  form  P(z  —  ao)  in 
the  domain  of  a-o- 

As  already  explained,  the  function  can  be  continued  from  the  domain  of 
tto  by  a  series  of  elements,  so  that  the  whole  region  of  continuity  is  gradually 
covered  by  domains  of  successive  points  ;  to  find  the  value  in  the  domain  of 
any  point  a,  it  is  sufficient  to  know  any  one  element,  say,  the  element  in  the 
domain  of  Uq.     The  function  is  the  same  through  its  region  of  continuity. 

Two  distinct  cases  may  occur  in  the  continuations. 

First,  it  may  happen  that  the  region  of  continuity  of  the  function  g  (z) 
extends  beyond  A'.  Then  we  can  obtain  elements  for  points  outside  A', 
their  aggregate  being  a  uniform  analytical  function.  The  aggregate  of 
elements  then  represents  within  A'  a  single  analytical  function  :  but  as  that 
function  has  elements  for  points  without  A',  the  aggregate  within  A'  does  not 
completely  represent  the  function.     Hence  : — 

If  a  function  he  defined  within  a  continuous  region  of  a  plane  by  an 
aggregate  of  elements  in  the  form  of  power-series,  which  are  continuations  of 
one  another,  the  aggregate  represents  in  that  'part  of  the  plane  one  {and  only 
one)  analytical  function :  but  if  the  power-series  can  be  continued  beyond  the 
boundary  of  the  region,  the  aggregate  of  elements  within  the  region  is  not  the 
complete  r^epresentation  of  the  analytical  function. 

This  is  the  more  common  case,  so  that  examples  need  not  be  given. 

Secondly,  it  may  happen  that  the  region  of  continuity  of  the  function  does 
not  extend  beyond  A'  in  any  direction.  There  are  then  no  elements  of  the 
function  for  points  outside  A'  and  the  function  cannot  be  continued  beyond 
the  boundary  of  A'.  The  aggregate  of  elements  is  then  the  complete  repre- 
sentation of  the  function  and  therefore  : — 

If  a  function  be  defined  within  a  continuous  region  of  a  plane  by  an 
aggregate  of  elements  in  the  form  of  power-series,  which  are  continuations  of 
one  another,  and  if  the  power-series  cannot  be  continued  across  the  boundary 
of  that  region,  the  aggregate  of  elements  in  the  region  is  the  complete  repre- 
sentation of  a  single  ^iniform  monogenic  function  which  exists  only  for  values 
of  the  variable  within  the  region. 


81.]  A   SERIES   OF   FUNCTIONS  153 

The  boundary  of  the  region  of  continuity  of  the  function  is,  in  the  latter 
case,  called  the  natural  limit  of  the  function*,  as  it  is  a  line  beyond  which 
the  function  cannot  be  continued.     Such  a  line  arises  for  the  series 

in  the  circle  [  ^r  |  =  1,  a  remark  due  to  Kronecker ;  other  illustrations  occur 
in  connection  with  the  modular  functions,  the  axis  of  real  variables  being 
the  natural  limit,  and  in  connection  with  the  automorphic  functions  (see 
Chapter  XXII.)  when  the  fundamental  circle  is  the  natural  limit.  A  few 
examples  will  be  given  at  the  end  of  the  present  chapter. 

It  appears  that  Weierstrass  was  the  first  to  announce  the  existence  of  natural  limits 
for  analytic  functions,  Berlin.  Monatsher.  (1866),  p.  617;  see  also  Schwarz,  Oes.  Werke, 
t.  ii,  pp.  240 — 242,  who  adduces  other  illustrations  and  gives  some  references ;  Klein  and 
Fricke,  Vorl.  iiher  die  Theorie  der  elliptisohen  Modulfunctionen,  t.  i,  (1890),  p.  110.  Some 
interesting  examples  and  discussions  of  functions,  which  have  the  axis  of  real  variables 
for  a  natural  limit,  are  given  by  Hankel,  "  Untersuchungen  iiber  die  unendlich  oft 
oscillirenden  und  unstetigen  Functionen,"  Math.  Ann.,  t.  xx,  (1870),  pp.  63 — 112. 

82.  Consider  next  a  series  of  functions  /i {z),  fo  {z),  /g  {z),  ...  of  the 
variable  z. 

In  the  first  place,  let  each  of  them  occur  in  the  form  of  power-series  in  z, 
with  (it  may  be)  positive  and  negative  indices,  say  in  the  form 

fs{z)  =  ^a,^zi^. 
Assume  that  the  power-series  for  the  separate  functions,  as  well  as  the  series 

s=l 

of  functions,  have  a  common  region  of  continuity  in  the  vicinity  of  the  origin 
such  that,  for  values  of  z  given  by 

R<\z\=r<R, 

the  function-series  and  each  of  the  power-series  converge  uniformly.  Then 
the  '^um 

00 

2j  agfj, 
has  a  definite  finite  value,  say  A^;  for  the  values  of  z  considered,  the  series 

l.Ay.Z" 

converges ;  and  we  have 

if{z)  =  ^A,z-. 

s  =  l  '^ 

*  Die  naturliche  Grenze,  according  to  German  mathematicians. 


154  REGION   OF    CONTINUITY   OF  [82. 

Let  k  denote  any  arbitrary  positive  quantity,  taken  as  small  as  we  please. 
In  consequence  of  the  uniform  convergence  of  the  function-series,  it  is  possible 
to  choose  an  integer  m,  such  that 


S    fs{z) 


< 


for  all  integers  n  ^  m,  and  for  all  values  of  z  such  that  R  <  i\<  r  <  VoK  R', 
where  i\  —  R  and  R'  —  r2  are  non-vanishing  quantities,  no  matter  how  small 
they  may  be  assigned ;  and  therefore  for  the  same  range  of  variables,  it  is 
possible  to  choose  an  integer  m  so  that,  for  all  integers  n  ^  m  and  for  all 
finite  positive  integers  p,  we  have 

oo  I 

-p+1  1 

n+p+1  I 

<  2'^  "I"  2'"'  ^  ^• 

Owing  to  the  finiteness  of  the  integer  p,  we  have 

n+p  /n+p 


n+p 
s=n 

fs{^) 

= 

00 

s=n 

n+ 

< 

s=n 

1 
+ 

''+p  /n+p        \ 

(=M  M-    \s=n         ' 

SO  that 


„     Z'^  +  P  X  1 

f-   \s=n 


for  all  integers  n  ^  m,  and  for  all  positive  integers  j9.     Hence  (Corollary,  §  29) 


n+p         I 


"SIJ. 


where  \2\=  r  ;  because  k,  being  greater  than  the  upper  limit  of  the  modulus 
of  the  above  series  for  all  the  values  of  z  considered,  is  greater  than  the  upper 
limit  of  its  modulus  for  values  of  z  such  that  \z  \=r.  It  therefore  follows 
that,  because  kr~'^  is  an  arbitrary  quantity  assigned  as  small  as  we  please, 
and  because  an  integer  m  can  be  chosen  such  that  the  above  inequality  holds 

00 

for  all  integers  n^m  and  for  all  positive  integers  p,  the  series  2  agf^  converges 
to  a  unique  finite  limit.     Denote  this  by  ^^.  • 

n  —  \  oc 

Let  %  as^=A^',     2  a,^  =  A^"; 

s=l  s=n 

then  regarding  kr{~i^  and  kr2~'^  as  two  assigned  quantities,  as  small  as  we 
please  (because  k  can  be  assigned  as  small  as  we  please,  and  Vi,  r^  are  finite 
non-vanishing  magnitudes),  the  convergence  of  the  series  whose  sum  is  A^ 
enables  us  to  choose  an  integer  n  such  that  A^"  is  smaller  than  each  of  the 
quantities  h\~i^  and  krfi^;  thus 

A''  <  h\-i^,     A^"  <  kr-^-f^. 


82.]  A   SERIES   OF    FUNCTIONS  155 

Now  consider  the  series  S^^^*^  for  a  value  of  z  such  that  r^<  \z\—T<r^. 
We  have 

S    !^/>.i<    2    k(-T<k      '^' 


!   -^-^^ 


t=  -QO 


H=  —03 


7' J  r  —  r. 


and  therefore  S    j  ^/V  |  <  k  -^^  +k    ^^ 


/y*  ^__  n™  /v«     ,    ,    /y* 


Hence  the  series   X-l/'^'^  converges.     Moreover,  each  of  the  power-series 
fi (z),  ...,  fn^i (z)  converges  uniformly ;    therefore 

"i'fs(z)  =  tA,'z>^, 

s  =  l 

and  the  latter  series  converges  uniformly.     The  two  series  "^A^'zi^,  '2,A^"z'^, 
can  therefore  be  combined  into  the  series 

which  accordingly  is  a  converging  series. 
Finally,  we  have 

i  /,  (z)  -  ^A^z'^  =1  fs  {z)  -  s^;^'^  -  s^/> 


and  therefore 


=  S/,(^)-2^/>, 


t  Mz)  -tA^z<^  \=    t  fs{z)-X  a;'z^ 

s=l  1^  1  s=n 


<\  S/.(^) 


+  2  1^/> 


^\k  +  k ^  +k '—. 


As  the  assigned  quantity  k  is  at  our  disposal,  we  can  choose  it  so  that  the 
quantity  on  the  right-hand  side  is  smaller  than  any  assignable  magnitude  : 
consequently,  for  the  values  of  z  under  consideration,  we  have 

00 

^f,(z)  =  lA^z^. 

s=l  f" 

83.  In  the  second  place,  consider  the  series  of  functions  f,  (z),  /g  (z), 
fs{z),  ...  more  generally.  The  region  of  continuity  may  be  supposed  to 
"consist  of  one  part  or  of  more  than  one  part :  let  such  a  part  be  denoted  by 


156  REGION   OF    CONTINUITY   OF  [83. 

A,  and  let  F{z)  denote  the  function  represented  by  the  series  within  A,  so 
that 

00 

F{z)=XMz), 

s  =  \ 

and  assume  that  within  A  (though  not  necessarily  at  points  on  its  boundary) 
the  function-series  converges  uniformly.  Let  a  denote  any  arbitrarily 
assumed  position  within  A ;  each  of  the  functions  fg  {z)  is  regular  in  the 
vicinity  of  a  and  is  expressible  in  the  form  of  a  power-series  Pg  (z  —  a) 
containing  only  positive  powers  of  z—  a.  By  the  preceding  investigation,  the 
function-series  can  be  represented  as  a  power-series,  and  we  have 

F{z)=P{z-a). 

00 

In  P{z—  a),  the  coefficient  of  (z  —  ay  is  J.yx,  which  is  2  ag^,  where  a^^  is  the 
coefficient  of  (z  —  aY  in  fg  {z) ;  accordingly 


di^P  (z  -  a) 
dzi^ 


d>^fg{z) 


s=i     dzi^ 


for  all  values  of  /x.  Since  a  is  any  arbitrarily  chosen  point  in  A,  it  follows 
that,  for  all  points  within  A,  we  have 

d^F{z)_  ^  di^fs(z) 
dz>^     ~  s^i     dzi^ 

As  the  function-series  ^  fg  (z)  converges  uniformly,  and  as  fg  (z)  is  regular  in 

s=l 

the  vicinity  of  a,  it  is  easy  to  see  that  the  series 

s=i     dzi- 

also  converges  uniformly ;  and  therefore  the  derivatives  of  the  function-series 
within  the  region  of  continuity  are  the  derivatives  of  the  function  the  series 
represents. 

The  expression  P {z—  a)  is  an  Element  of  the  function  F {z) :  and  within 
the  domain  of  a,  contained  in  the  region  A,  it  represents  the  function.  It  can 
be  used  for  the  continuation  of  F  (z)  so  long  as  the  domains  of  successive 
points  lie  within  A  ;  but  this  restriction  is  necessary,  and  the  full  continuation 
of  P  (z  —  a)  as  an  element  of  a  power-series  is  not  necessarily  limited  by 
the  region  A.  It  is  solely  in  that  part  of  its  region  of  continuity  which  is 
included  within  A  that  it  represents  the  function  F(z);  the  boundary  of  the 
region  A  must  not  be  crossed  in  forming  the  continuations  of  P  (^  —  a). 

It  therefore  appears  that  a  converging  series  of  functions  of  a  variable 
can  be  expressed  in  the  form  of  series  of  powers  of  the  variable,  which 
converge  within  the  parts  of  the  plane  where  the  series  of  functions 
converges  uniformly ;  but  the  equivalence  of  the  two  expressions  is  limited 


83.]  A   SEEIES   OF   FUNCTIONS  157 

to  such  parts  of  the  plane,  and  cannot  be  extended  beyond  the  boundary  of 
the  region  of  continuity  of  the  series  of  functions. 

If  the  region  of  continuity  of  a  series  of  functions  consist  of  several  parts 
of  the  plane,  then  the  series  of  functions  can  in  each  part  be  expressed  in 
the  form  of  a  set  of  converging  series  of  powers  :  but  the  sets  of  series  of 
powers  are  not  necessarily  the  same  for  the  different  parts,  and  they  are  not 
necessarily  continuations  of  one  another,  regarded  as  power-series. 

Suppose,  then,  that  the  region  of  continuity  of  a  series  of  functions 

F{z)^lf,{z) 

consists  of  several  parts  A^,  Ao_,  ....  Within  the  part  A^  let  F {z)  be 
represented,  as  above,  by  a  set  of  power-series.  At  every  point  within  A-^, 
the  values  of  F  {z)  and  of  its  derivatives  are  each  definite  and  unique  ;  so 
that,  at  every  point  which  lies  in  the  regions  of  convergence  of  two  of  the 
power-series,  the  values  which  the  two  power-series,  as  the  equivalents  of  F{z) 
in  their  respective  regions,  furnish  for  F  {z)  and  for  its  derivatives  must  be 
the  same.  Hence  the  various  power-series,  which  are  the  equivalents  of  ^(^) 
in  the  region  A-^,  are  continuations  of  one  another:  and  they  are  sufficient  to 
determine  a  uniform  monogenic  analytic  function,  say  F^  {z).  The  functions 
F{z)  and  F-^  {z)  are  equivalent  in  the  region  A-^ ;  and  therefore,  by  §  81,  the 
series  of  functions  represents  one  and  the  same  function  for  all  points  within 
one  continuous  'part  of  its  7'egion  of  continuity.  It  may  (and  frequently  does) 
happen  that  the  region  of  continuity  of  the  analytical  function  F^  (z)  extends 
beyond  A^;  and  then  Fi{z)  can  be  continued  beyond  the  boundary  of  ^j  by 
a  succession  of  elements.  Or  it  may  happen  that  the  region  of  continuity 
of  F-^{z)  is  completely  bounded  by  the  boundary  of  A^ ;  and  then  that  function 
cannot  be  continued  across  that  boundary.     In  either  case,  the  equivalence 

00 

of  Fj{z)  and  2  fs(z)  does  not  extend  beyond  the  boundary  of  A-^,  one 

s  =  l 

complete   and  distinct   part   of  the  region   of  continuity  of  2  fg  (z) ;    and 

therefore,  by  using  the  theorem  proved  in  §  81,  it  follows  that: — 

A  series  of  functions  of  a  variable,  tuhich  converges  within  a  continuous 
part  of  the  plane  of  the  variable  z,  is  either  a  partial  or  a  complete 
representation  of  a  single  uniform  analytic  function  of  the  variable  in  that 
part  of  the  plane. 

Further,  it  has  just  been  proved  that  the  converging  series  of  functions 
can,  in  any  of  the  regions  A,  be  changed  into  an  equivalent  uniform  analytic 
function,  the  equivalence  being  valid  for  all  points  in  that  region,  say 

lf{z)  =  F,{z). 

s=l 


^ 


158  A    CONVERGING   SERIES   OF   FUNCTIONS  [83. 

We  have  seen  that  every  derivative  of  J'^  {z)  at  any  point  within  A  is  the 
sum  of  the  corresponding  derivatives  oi  fg{z),  this  sum  converging  uniformly 
within  A.  The  equivalence  of  the  analytic  function  and  th^  series  of 
functions  has  not  been  proved  for  points  on  the  boundary ;  even  if  they  are 
equivalent  there,  the  function  i'^  {z)  cannot  be  proved  to  have  a  uniform 
finite  derivative  at  every  point  on  the  boundary  of  A,  and  therefore  it  cannot 

he  affirmed  that  X  fs  (z)  has,  of  necessity,  a  u7iiform  finite  derivative  at  points 

s=l 

00 

on  the  boundary  of  A ,  even  though  the  value  of  S  f  (z)  be  uniform  and  finite 

s  =  l 

at  every  point  on  the  boundary*. 

Ex.  In  illustration  of  the  last  inference,  regarding  the  derivative  of  a  function  at 
a  point  on  the  boundary  of  its  region  of  continuity,  consider  the  series 

g{z)=   2  6"2«'', 
«=o 

where  6  is  a  positive  quantity  less  than  unity,  and  a  is  a  positive  quantity  which  will  be 
taken  to  be  an  odd  integer. 

For  points  within  and  on  the  circumference  of  the  circle  |  2  |  =  1,  the  series  converges 
uniformly  and  unconditionally ;  and  for  all  points  without  the  circle  the  series  diverges. 
It  thus  defines  a  function  for  points  within  the  circle  and  on  the  circumference,  but  not 
for  points  without  the  circle. 

Moreover,  for  points  actually  within  the  circle,  the  function  has  a  first  derivative  and 
consequently  has  any  number  of  derivatives.  But  it  cannot  be  declared  to  have  a 
derivative  for  points  on  the  circle  :  and  it  will  in  fact  now  be  proved  that,  if  a  certain 
condition  be  satisfied,  the  derivative  for  variations  at  any  point  on  the  circle  is  not  merely 
infinite  but  that  the  sign  of  the  infinite  value  depends  upon  the  direction  of  the  variation, 
so  that  the  function  is  not  monogenic  for  the  circumference  t. 

Let  z=e^^ :  then,  as  the  function  converges  unconditionally  for  all  points  along  the 
circle,  we  take 

f{6)=   i   6™e«"^'-, 

■n.=0 

where  ^  is  a  real  variable.     Hence 

f{6  +  4>)-f{6)     ;  &\x(«+*)^-Xe^' 

<p  n=0  9 

m - 1  Ce*" (e+i>)i  _  ga"fl J) 

71=0  I  «"<^  i 

+     2    6'»  +  'M 

)l  =  0  I  (j) 

*  It  should  be  remarked  here,  as  at  the  end  of  §  21,  that  the  result  in  itself  does  not 

contravene  Eiemann's  definition  of  a  function,  according  to  which  (§  8)  -r-  must  have  the  same 

value  whatever  be  the  direction  of  the  vanishing  quantity  dz ;  at  a  point  on  the  boundary  of 
the  region  there  are  outward  directions  for  which  dw  is  not  defined. 

t  The  following  investigation  is  due  to  Weierstrass,  who  communicated  it  to  Du  Bois- 
Reymond:    see  Crelle,  t,  Ixxix,  (1875),  pp.  29 — 31;   Weierstrass,  Ges.  Werke,  t.  ii,  pp.  71—74. 


83.]  NOT   POSSESSING   A   DERIVATIVE  159 

assuming  m,  in  the  first  place,  to  be  any  positive  integer.     To  transform  the  first  sum  on 
the  right-hand  side,  we  take 

and  therefore  2    (ahY -I 

sin  (ia''(j))  I 


)B-1 

<    2    (ab)» 

H=0 


m-l  (ah')™- 


^     '    '  a.«(^  J      ^  ah-V 


'\i  ab>  \.     Hence,  on  this  hypothesis,  we  have 
2 

?l=0 

where  -y  is  a  complex  quantity  with  modulus  <  1. 

n 

To  transform  the  second  sum  on  the  right-hand  side,  let  the  integer  nearest  to  «™- 

be  a„j,  so  that 

6 

2  ^  **  —  a„j  ^  —  9- 

TT 

for  any  value  of  m  :  then  taking  « 

we  have    '  \tt^x>  -\it , 

and  cos  x  is  not  negative.     We  choose  the  quantity  0  so  that 

and  therefore  <±  = , 

which,  by  taking  m  sufiiciently  large  {a  is  >1),  can  be  made  as  small  as  we  please.     We 
now  have 

ga."'+"  i0-t-(}>)i_  ^a^'iri  (1  +  «„,)__/■_■[  N  a™ 

if  a  be  an  odd  integer,  and 

Hence  ^ — —  =  -  ( - 1  )'^»  ^^ a™, 

^  TT  — .r 

and  therefore         2    6™  +  M^ ^ l  =  _(-l)«.. 2  6"(l-fe«^*). 

J!=0  I  9  J  TT—  X  „  =  o 

The  real  part  of  the  series  on  the  right-hand  side  is 

2  6"  {1  -1-  cos  a'^x} ; 

71  =  0 

every  term  of  this  is  positive  and  therefore,  as  the  first  term  is  1  -t-cos  x,  the  real  part 

>  1  -1-  cos  X 
>1, 


160  A    CONVERGING   SERIES   OF    FUNCTIONS  [83, 

for  cos^  is  not  negative.     Also  it  is  finite,  for  it  is 


<  2  2   6" 

«=o 

2 

Moreover  ^n  <  tt  —  x  <  §77, 

TT  .  2 

so  tliat  is  positive  and  >  ;r .     Hence 

TT-X  3 

io'™i ^ }=-^-l)"''^3^' 

where  j;  is  a  finite  complex  quantity,  the  real  part  of  which  is  positive  and  greater  than 
vmity.     We  thus  have 

/(5  +  <^)-/(^) 


=  -(-l)«m  («&)'» 


where  i  y'  |  <  1,  and  the  real  part  of  rj  is  positive  and  >  1. 
Proceeding  in  the  same  way  and  taking 

TT           a'"    ' 
SO  that  Y  = , 

where  |  7/  j  <  1  and  the  real  part  of  t^j,  a  finite  complex  quantity,  is  positive  and  greater 
than  unity. 

If  now  we  take  ab  —  l>  fir, 

the  real  parts  of  -  -+y'  ~j: — ;  ,  say  of  ^, 

3  TT  (CO  —  i 

and  Of  |^+^/_L_,sayofCi,       _ 

are  both  positive  and  different  from  zero.     Then,  since 

and  /(^-x)-/W^(-i)°>.(a&y»^i, 

m  being  at  present  any  positive  integer,  we  have  the  right-hand  sides  essentially  different 
quantities,  because  the  real  part  of  the  first  is  of  sign  opposite^'to  the  real  part  of  the 
second. 

Now  let  m  be  indefinitely  increased ;  then  0  and  x  are  infinitesimal  quantities  which 

ultimately  vanish;  and  the  Hmit  of  -[f(6  +  cj))-f{d)]  for  ^  =  0  is  a  complex  infinite 

quantity  with  its  real  part  opposite  in  sign  to  the  real  part  of  the  complex  infinite 

quantity  which  is  the  limit  of  ^[/(^-;^)-/(^)]  for  x  =  0.     If /(^)  had  a  differential 

coefficient,  these  two  limits  would  be  equal :  hence  f{6)  has  not,  for  any  value  of  6, 
a  determinate  differential  coefUcient. 


83.]  NOT   POSSESSING   A   DERIVATIVE  161 

From  this  result,  a  remarkable  inference  relating  to  real  functions  may  be  at  ouce 
derived.     The  real  part  of  f{6)  is 

2   6»cos(a»(9), 

which  is  a  series  converging  uniformly  and  unconditionally.      The  real  parts  of 

-i-ir"' {abrC 
and  of   +(-l)''™(a&)™^i 

are  the  corresponding  magnitudes  for  the  series  of  real  quantities  :  and  they  are  of  opposite 
signs.     Hence  for  no  value  of  0  has  the  series 

2  6"cos(a"^) 

a  determinate  difFereutial  coefficient,  that  is,  we  can  choose  an  increase  (p  and  a  decrease  x 
of  d,  both  being  made  as  small  as  we  please  and  ultimately  zero,  such  that  the  limits  of 
the  expressions 

f{e  +  cl>)-f(d)^    f{6-x)-f{6) 

4>         '  ~x 

are  diflFerent  from  one  another,  provided  a  be  an  odd  integer  and  ab>  l+§ir. 

The  chief  interest  of  the  above  investigation  lies  in  its  application  to  functions  of  real 
variables,  continuity  in  the  value  of  which  is  thus  shewn  not  necessarily  to  imply  the 
existence  of  a  determinate  differential  coefficient  defined  in  the  ordinary  way.  The 
application  is  due  to  Weierstrass,  as  has  already  been  stated.  Further  discussions  will 
be  found  in  a  paper  by  Wiener,  Crelle,  t.  xc,  (1881),  pp.  221 — 252,  in  a  remark  by 
Weierstrass,  Ges.  Werke,  t.  ii,  p.  229,  and  in  a  paper  by  Lerch,  Crelle,  t.  ciii,  (1888), 
pp.  126 — 138,  who  constructs  other  examples  of  continuous  functions  of  real  variables; 
and  an  example  of  a  continuous  function  without  a  derivative  is  given  by  Schwarz, 
Ges.    Werke,  t.  ii,  pp.  269—274. 

The  simplest  classes  of  ordinary  functions  are  characterised  by  the  properties : — 
(i)    Within  some  region  of  the  plane  of  the  variable  they  are  uniform,  finite,  and 

continuous  : 
(ii)    At  all  points  within  that  region  (but  not  necessarily  on  its  boundary)  they  have 

a  differ ential  coefficient : 
(iii)    When  the  variable  is  real,  the  number  of  maximum  values  and  the  number  of 
minimum  values  within  any  given  range  is  finite. 

The  function  2  6"cos(a«^),  suggested  by  Weierstrass,  possesses  the  first  but  not  the 

71=0 

second  of  these  properties.  Kopcke  (Math.  Ann.,  t.  xxix,  pp.  123 — 140)  gives  an  example 
of  a  function  which  possesses  the  first  and  the  second  but  not  the  third  of  these 
properties. 

84.  In  each  of  the  distinct  portions  A^,  A.,,  ...  of  the  complete  region 
of  continuity  of  a  series  of  functions,  the  series  can  be  represented  by  a 
monogenic  analytic  function,  the  elements  of  which  are  converging  power- 
series.  But  the  equivalence  of  the  function-series  and  the  monogenic 
analytic  function  for  any  portion  A^  is  limited  to  that  region.  When  the 
monogenic  analytic  function  can  be  continued  from  J.i  into  A. 2,  the  continua- 
tion is  not  necessarily  the  same  as  the  monogenic  analytic  function  Avhich  is 

F.   F.  11 


162  ANALYTICAL  EXPRESSION  [84. 

00 

the  equivalent  of  the  series  2  fs{z)  in  A.^-    Hence,  if  the  monogenic  analytic 

s  =  l 

functions  for  the  two  portions  J-j  and  A.2  be  different,  the  function-series 
represents  different  functions  in  the  distinct  parts  of  its  region  of  continuity. 

A  simple  example  will  be  an  effective  indication  of  the  actual  existence 
of  such  variety  of  representation  in  particular  cases ;  that,  which  follows,  is 
due  to  Tannery*. 

Let  a,  b,  c  be  any  three  constants ;  then  the  fraction 

a  +  hcz''^ 

when  m  is  infinite,  is  equal  to  a  if  |  2^  j  <  1,  and  is  equal  to  c  if  |  2^  j  >  1. 

Let  nio,  nil,  m.2,  ...  be  any  set  of  positive  integers  arranged  in  ascending- 
order  and  be  such  that  the  limit  of  lUn,  when   n  =  co ,  is  infinite.     Then, 

since 

a  +  hcz'"^''     a  +  6c^*"«       ^    [a  ■\-  bcz'^i     a  +  hcz'^t-^ 


1  +  hz'^'^       1  +  bz'""      i=i  [l  +  bz'^i       1  +  bz'^i-^ 

a  +  bcz"""      ,  ,  .  ^   !   (2»^-'»i-i  -  1)  £r™i-i 


+  b{c-a)  _-.^  |(l  +  5^m,)(l  +  ^^m,_,^| 


1  +  bz""" 
the  function  <f)  (z),  defined  by  the  equation 

converges  uniformly  to  a  value  a  if  \z\^p<\,  and  converges  uniformly  to  a 
value  c  if  \z\^  p  >1.  But  if  [^|  =  1,  the  value  to  which  the  series  tends 
depends  upon  the  argument  of  z :  the  series  cannot  be  said  to  converge  for 
values  of  z  such  that  1 2^  j  =  1. 

The  simplest  case  occurs  when  6  =  —  1  and  nii  =  2^ ;  then,  denoting  the 
function  by  4>(^)>  ^^  ^^"^^ 


a  —  cz 


r2» 


<^  (^)  =  -. — -  +  (a  -  c)  2  --H — T 

i  —Z  i  =  {)Z^       —  1 

a—  cz       ,  ^  {  '  ^         ,        ^"        ,       ^       ,         ] 


that  is,  the  function  (f>  (z)  is  equal  to  a  if  j  ^  |  <  1,  and  it  is  equal  to  c  if 

kl  >!• 

*  It  is  contained  in  a  letter  of  Tannery's  to  Weierstrass,  who  communicated  it  to  the 
Berlin  Academy  in  1881,  Ges.  Werke,  t.  ii,  pp.  231— 23B.  A  similar  series,  which  indeed  is 
equivalent  to  the  special  form  of  (j>{z),  was  given  by  Schroder,  Schldm.  Zeitschrift,  t.  xxii,  (1876), 
p.  184;  and  Pringsheim,  3Iath.  Ann.,  t.  xxii,  (1883),  p.  110,  remarks  that  it  can  be  deduced, 
without  material  modifications,  from  an  expression  given  by  Seidel,  Crelle,  t.  Ixxiii,  (1871), 
pp.  297—299. 


84.]  EEPRESENTING   DIFFERENT   FUNCTIONS  163 

When  \2\  =  1,  the  function  can  have  any  value  whatever.  Hence  a  circle 
of  radius  unity  is  a  line  of  singularities,  that  is,  it  is  a  line  of  discontinuity 
for  the  series.  The  circle  evidently  has  the  property  of  dividing  the  plane 
into  two  parts  such  that  the  analytical  expression  represents  different 
functions  in  the  tiuo  pa,rts. 

If  we  introduce  a  new  variable  f  connected  with  2  by  the  relation* 

^^1  +^ 
^      1-z' 

then,  if  ^=  f  +  irj  and  z  =  x  +  iy,  we  have 

J,  _   1  —  ic^  —  2/2 

so  that  ^  is  positive  when  |  ^  j  <  1,  and  ^  is  negative  when  [  ^  |  >  1.     If  then 

the  function  %  (^)  is  equal  to  a  or  to  c  according  as  the  real  part  of  ^  is 
positive  or  negative. 

And,  generally,  if  we  take  ^  a  rational  function  of  z  and  denote  the 
modified  form  of  ^{^),  which  will  be  a  sum  of  rational  functions  of  z,  by 
(/>!  {z),  then  </>!  {z)  will  be  equal  to  a  in  some  parts  of  the  plane  and  to  c 
in  other  parts  of  the  plane.  The  boundaries  between  these  parts  are  lines 
of  singular  points  :  and  they  are  constituted  by  the  ^-curves  which  correspond 

to!  ^1  =  1. 

85.  Now  let  F  {z)  and  G{z)  be  two  functions  of  z  with  any  number  of 
singularities  in  the  plane :  it  is  possible  to  construct  a  function  which  shall 
be  equal  to  F{z)  within  a  circle  centre  the  origin  and  to  G{z)  without  the 
circle,  the  circumference  being  a  line  of  singularities.  For,  when  we  make 
a  =  \  and  c  =  0  in  j>  {z)  of  §  84,  the  function 

'\  z  z"  z^ 

^-^      1—z     z^  —  1      z*—l      z^  —  1 
is  unity  for  all  points  within  the  circle  and  is  zero  for  all  points  without  it : 

and  therefore 

G(z)+{F{z)-G{z)]d{z) 

is  a  function  which  has  the  required  property. 

Similarly  F^  (z)  +  {F^  (z)  -  F^  iz))  d  (z)  +  {F,  (z)  -  F^  (z)]  6  {^ 

is  a  function  which  has  the  value  F^  {z)  within  a  circle  of  radius  unity,  the  value  F^  (2) 
between  a  circle  of  radius  unity  and  a  concentric  circle  of  radius  r  greater  than  unity,  and 
the  value  F^  (z)  without  the  latter  circle.  All  the  singularities  of  the  functions  Fi,  F^,  F^ 
are  singularities  of  the  function  thus  represented ;  and  it  has,  in  addition  to  these,  the 
two  lines  of  singularities  given  by  the  circles. 

*  The  significance  of  a  relation  of  this  form  will  be  discussed  in  Chapter  XIX. 

11—2 


164  MONOGENIC    FUNCTIONALITY  [85. 


Again,  G{z)  +  {F{z)-0{z)}e(^^^ 


is  a  function  of  z,  which  is  equal  to  F{z)  on  the  positive  side'of  the  axis  of  y,  and  is  equal 
to  O  (z)  on  the  negative  side  of  that  axis. 

1+2 

Also,  if  we  take  ^e-?",  —pi  =  - , 

1  —  2 

where  oj  and  pi  are  real  constants,  as  an  equation,  defining  a  new  variable  ^+^v,  we  have 
I  cos  a^-TT)  sin  ai -joi  =  ^y_^Y+y^- ' 

so  that  the  two  regions  of  the  2-plane  determined  by  |  2  |  <  1  and  |  2  |  >  1  correspond  to  the 
two  regions  of  the  {"-plane  into  which  the  line  |  cos  a^+ri  sin  ai—pi  =  0  divides  it.     Let 

so  that  on  the  positive  side  of  the  line  |  cosai  +  j?sin  ai -J0i=0  the  function  6^  is  unitj- 
and  on  the  negative  side  of  that  line  it  is  zero.  Take  any  three  lines  defined  by 
ai5  i^i;  02?  ^"2;   035  pz  respectively;  then 

is  a  function  which  has  the  value  F  within 
the  triangle,  the  value  —F  in  three  of  the 
spaces  without  it,  and  the  value  zero  in  the 
remaining  three  spaces  without  it,  as  indi- 
cated in  the  figure  (fig.  13). 

And  for  every  division  of  the  plane  by    /g^ J- ^ \- /^\ 

lines,  into  which  a  circle  can  be  transformed  ~  ^  /  ^  ^T  ^ 

by  rational  equations,  as  will  be  explained  (1)/  \(2) 

when  conformal  representation  is  discussed  pj     y^ 

hereafter,  there  is  a  possibility  of  represent- 
ing discontinuous  functions,  by  expressions  similar  to  those  just  given. 

These  examples  are  sufficient  to  lead  to  the  following  result*,  which  is 
complementary  to  the  theorem  of  §  82 : — 

When  the  region  of  continuity  of  an  infimte  series  of  functions  consists 
of  several  distinct  parts,  the  series  represents  a  single  function  in  each  part 
hut  it  does  not  necessarily  represent  the  same  function  in  different  parts. 

It  thus  appears  that  an  analytical  expression  of  given  form,  which  con- 
verges uniformly  and  unconditionally  in  different  parts  of  the  plane  separated 
from  one  another,  can  represent  ditferent  functions  of  the  variable  in  those 
different  parts;  and  hence  the  idea  of  monogenic  functionality  of  a  complex 
variable  is  not  coextensive  with  the  idea  of  functional  dependence  expressible 
through  arithmetical  operations,  a  distinction  first  established  by  Weierstrass. 

86.  We  have  seen  that  an  analytic  function  has  not  a  definite  value  at 
an  essential  singularity  and  that,  therefore,  every  essential  singularity  is 
excluded  from  the  region  of  definition  of  the  function. 

*  Weierstrass,  Ges.  Werke,  t.  ii,  p.  221. 


86.]  LINE   OF   SINGULARITIES  165 

Again,  it  has  appeared  that  not  merely  must  single  points  be  on  occasion 
excluded  from  the  region  of  definition  but  also  that  functions  exist  with 
continuous  lines  of  essential  singularities  which  must  therefore  be  excluded. 
One  method  for  the  construction  of  such  functions  has  just  been  indicated : 
but  it  is  possible  to  obtain  other  analytical  expressions  for  functions  which 
possess  what  may  be  called  a  singular  line.  Thus  let  a  function  have  a 
circle  of  radius  c  as  a  line  of  essential  singularity*;  let  it  have  no  other 
singularities  in  the  plane  and  let  its  zeros  be  a^,  a._,  a.,  ...,  supposed  arranged 
in  such  order  that,  if  p„e^'^«  =  «„,  then 

\pn-c\  ^\pn+i-c\, 

so  that  the  limit  of  pn,  when  n  is  infinite,  is  c. 

Let  Cn  =  ce*^«,  a  point  on  the  singular  circle,  corresponding  to  an  which  is 
assumed  not  to  lie  on  it.  Then,  proceeding  as  in  Weierstrass's  theory  in  §  51 
if 

G{z)=  n  ]l_^e.«.(^) 
where       ^.  (.)  =  «!^^  +  1  f  ^^^t^V  +  . . .  +      1       "^'^-'=' 


Z  —  Cr, 


Z  -Cn         2\  Z  -Cn    '         '"        nin  -  1 

G  (z)  is  a  uniform  function,  continuous  everywhere  in  the  plane  except  along 
the  circumference  of  the  circle  which  may  be  a  line  of  essential  singularities. 
Special  simpler  forms  can  be  derived  according  to  the  character  of  the 
series  of  quantities  constituted  by  |  a„  -  c„  | .     If  there  be  a  finite  integer  m, 

such  that   2    \an  —  Cn\^  is  a  converging  series,  then  in  gni^)  only  the  first 
«.=i 

m  —  1  terms  need  be  retained. 
JEx.     Construct  the  function  when 

a„  =  (  I — - 


m  being  a  given  positive  integer  and  r  a  positive  quantity. 

Again,  the  point  Cn  was  associated  with  a„  so  that  they  have  the  same 
argument :  but  this  distribution  of  points  on  the  circle  is  not  necessary,  and 
it  can  be  made  in  any  manner  which  satisfies  the  condition  that  in  the  limited 

case  just  quoted  the  series  2  j  «„  —  c^  j*^  is  a  converging  series. 

11=1 

SingTilar  lines  of  other  classes,  for  example,  sections'^  in  connection  with  functions 
defined  by  integrals,  arise  in  connection  with  analytical  functions.  They  are  discussed 
by  Painleve,  S^w  les  lignes  singuUeres  des  fonctions  analytiques,  (Th^se,  Gauthier-Villars, 
Paris,  1887). 

Ex.  1.     Shew  that,  if  the  zeros  of  a  function  be  the  points 

h  +  c  —  {a  —  d)i 


A  = 


a  +  d-^{h-c)i'' 


*  This  investigation  is  due  to  Picard,  Comptes  Rendus,  t.  xci,  (1881),  pp.  690—692. 
t  Called  coupures  by  Hermite ;    see  §  103. 


166  LACUNARY  [86. 

where  a,  b,  c,  d  are  integers  satisfying  the  condition  ad  —  hc=\,  so  that  the  function 
has  a  circle  of  radius  unity  for  an  essential  singular  line,  then  if 

h-\-di 


B: 


'  d  +  W 


(Z-A       TT^l 

the  function  H  <——^e        Y, 


where  the  product  extends  to  all  positive  integers  subject  to  the  foregoing  condition 
ad-bc  =  l,  is  a  uniform  function  finite  for  all  points  in  the  plane  not  lying  on  the 
circle  of  radius  unity.  (Picard.) 

Ex.  2.     Examine  the  character  of  the  distribution  of  points  %  in  the  plane  of   z 
which  are  given  by 

2,„=('l+l^e^/2'»'rS  (7i=l,  2,  3,   ...). 

Consider  especially  the  neighbourhood  of  the  circle  whose  centre  is  the  origin  and  whose 
radius  is  1. 

Shew  that 

-   _  1 

represents  a  monogenic  function  of  z  at  all  points  within  the  circle ;  and  investigate  the 
possibility  of  an  analytical  continuation  of  this  function  beyond  the  circle. 

(Math.  Trip.,  Part  II.,  1896.) 

87.  In  the  earlier  examples,  instances  were  given  of  functions  which 
have  only  isolated  points  for  their  essential  singularities:  and,  in  the  latter 
examples,  instances  have  been  given  of  functions  which  have  lines  of 
essential  singularities,  that  is,  there  are  continuous  lines  for  which  the 
functions  do  not  exist.  We  now  proceed  to  shew  how  functions  can  be 
constructed  which  do  not  exist  in  assigned  continuous  spaces  in  the  plane. 
Weierstrass  was  the  first  to  draw  attention  to  lacunary  functions,  as  they 
may  be  called;  the  following  investigation  in  illustration  of  Weierstrass's 
theorem  is  due  to  Poincare*. 

Take  any  convex  curve  in  the  plane,  say  G :  and  consider  a  function- 
series  of  the  form 

CO  A 

where  the  constants  An  and  6«  are  subject  to  the  conditions 

00 

(i)      The  series  %  ^  .^  converges  unconditionally : 

(ii)      Each  of  the  points  hn  is  either  within  or  upon  the  curve  C : 

(iii)     When  any  arc  whatever  of  G  is  taken,  as  small  as  we  please,  that 
arc  contains  an  unlimited  number  of  the  points  hn. 

*  Acta  Soc.  Fenn.,  t.  xii,  (1883),  pp.  341—350 ;  Amer.  Journ.  Math.,  t.  xiv,  (1892), 
pp.   201—221. 


87.] 


FUNCTIONS 


167 


It  will  be  seen  that,  for  values  of  z  outside  C,  i^{z)  is  represented  by  a 
power-series,  which  cannot  be  continued  across  the  curve  C  into  the  interior, 
and  which  therefore  has  the  area  of  G  for  a  lacunary  space. 

00 

Let  8  denote  the  sum  of  the  converging  series  S  \An\:  then  denoting  by 

K  any  assigned  quantity,  as  small  as  we  please,  an  integer  p  can  always  be 
determined  so  that 


0«  —    '-I    \  -^1 


<K. 


Consider  the  function-series  in  the  vicinity  of  any  point  c  outside  C.  Let 
R  denote  the  distance  of  c  from  the  nearest  point  of  the  boundary*  of  C,  so 
that  72  is  a  finite  non-vanishing  quantity ;  and  draw  a  circle  of  radius  R  and 
centre  c,  which  thus  touches  C  externally.  Thus  for  all  the  points  h  except 
at  the  point  of  contact,  we  have 

\hn  —  c\>  R. 
Let  z  be  any  point  within  the  circle,  so  that 

\z  —  c\<R 

=  eR, 

say,  where  ^  is  a  positive  quantity  less  than  1.     Then 

Vz  —  hn\^\hn—  c\  —  \z—  C\ 


and  therefore 
Consequently 


^R{l-d); 


\Z  —  br 


R{l-6) 


M  =  0 


Z-hr. 


■^n 


s 


<,t,R{\-6)<R{\-dy 


so  that  the  function- series  converges  unconditionally.     Also 


oo  A 

n=m  Z  —  0% 


■A.n  I 


< 


-hr 


^R{\-e)' 

and  therefore  the  function-series  converges  uniformly :  that  is, 

cx>  A 

n=0  2  —  On 

*  This  will  be  either  the  shortest  normal  from  c  to  the  boundary,  or  the  distance  of  c  from 
some  point  of  abrupt  change  of  direction,  as  for  instance  at  the  angular  point  of  a  polygon  ;  for 
brevity  of  description  we  shall  assume  the  former  to  be  the  case. 


168 


LACUNARY 


[87. 


converges  uniformly  and  unconditionally  within  any  circle  concentric  with 
the  circle  of  radius  R  and  lying  within  it.  Accordingly,  by  Weierstrass  s 
investigation  (§§  82,  83),  this  is  expressible  in  the  form  of  a  converging  series 
P  {z  —  c);  manifestly 

00  oc  A 

(z  -  c)"\ 

}«=0  n=0  Vn        ^} 

We  have 

A 

(z  -  cY' 


00  oc  J 

=0  n=0  {O-ii  —  Cf' 

I  A,,  1  d'^R' 


{hn  -  cy^+' 


A..  I  e^ 


R 


and  therefore 


2    t 

0)1=0  n=0 


A  I  "1  00  00 


R{i-e)' 

that   is,   the    series   P  (z  —  c)   converges   unconditionally.     Let    C,„    denote 
2  An(bn-cy'-^;  then 

71  =  0 

P{z-c)  =  -  5  C,„(5-c)'". 

The  point  c  is  any  arbitrarily  chosen  point  outside  the  curve  C ;  and  therefore 
the  function  represented  by  <f)  (z)  for  points  z  outside  the  curve  C  is  a  uniform 
analytic  function. 

Any  power-series  representing  this  function  can  be  used  as  an  element 
for  continuation  outside  C  and  away  from  C :  we  proceed  to  prove  that  it 
cannot  be  continued  across  the  boundary  of  C.  If  this  were  possible,  it  would 
arise  through  the  construction  of  the  domain  of  some  point  Zq,  where  Zq  is  a 
point  outside  G  (say  within  such  a  circle  as  the  above,  centre  c),  and  where 
the  circle  bounding  the  domain  of  Zq  would  cut  oft'  some  arc  from  the  boundary 
of  G.  The  preceding  analysis  shews  that,  in  the  domain  of  z^,  the  function  is 
represented  by  a  power-series 

Q(2-z,)  =  -  ^  B„,  (z  -  5o)'^ 


where 


Rm  =    ^    An  {bn  —  Zq) 


it  must  be  shewn  that  the  series  diverges  for  points  z  within  G. 

In  the  first  place,  consider  the  series  P  {z  —  c);  in  order  that  it  may 
converge,  only  such  values  of  z  are  admissible  as  make  the  limit  of 
Gra  (^  —  c)"*  zero,  when  m  is  infinite.  Let  a  point  be  taken  on  the  circum- 
ference of  the  circle  G  of  radius  R ;  then  the  above  limit  can  only  be  zero  if 

Lt  a,.,i^»»=o, 


87.] 


FUNCTIONS 


169 


a  condition  that  is  not  satisfied,  as  will  now  be  proved.  This  circle  touches 
C  externally ;  let  the  point  of  contact  be  a  point  bk  (such  a  circle  can  always 
be  constructed,  by  drawing  the  outward  normal  at  a  point  b  and  choosing 
some  point  c  upon  it).  Let  any  arbitrary  quantity  e  be  assigned,  as  small 
as  we  please ;  and  let  an  integer  p  be  chosen  large  enough  to  secure  that 

Sjj  <  -^  eR, 

this  being  possible  because  ^S'^^,  the  remainder  of  the  converging  series  S  |-4„|, 
can  (by  choice  of  p)  be  made  less  than  any  assigned  quantity.  Either  the 
chosen  number  p  is  greater  than  k:  or  if  it  is  less  than  k,  then  some  other 
number  (>  p)  can  be  chosen  so  that  it  is  greater  than  k :  we  may  therefore 
assume  p  >  k. 

Draw  a  circle,  centre  c  and  radius  R'  greater  than  R,  so  as  to  include  the 
point  b/c,  and  exclude  the  points  bg,  ...,bp  with  the  exception  of  b^.  This  can 
be  done  :  for  if 

I  bk  -  bk-i  I  >  XE,         i  bk  -  bk+i  1  >  XE, 

where  A.  is  some  positive  quantity  as  small  as  we  please  (but  not  absolutely 

zero),  we  can  take 

R''  =R'  +  \^R' ; 
and  then 

\bn  —  c\<  R',     for     n  =  0,l,  ...,k—l,k  +1,  .. 


p. 


Let  q  denote  a  number  sufficiently  large  to  secure  that 


Then  as 


we  have 


8  fRy    , 
R'VR'J  ^^'■ 


G,-Ak{bk-c)-^-'=  S 


p-i 
+    S 


A. 


+  X 


A. 


=0  {bn  -  Cy+'  ^  n=k+l  (hn  "  0^+'       n=p  Q>n  "  c)5+^  ' 


and  therefore 


k-\ 


\R'i{Cq-Ak{bk-c)-<i-']\<  2 

n=0 


A^m 


p-\ 
+   % 


{bn-Cy+'\        n=?c+l 


AnR'i 


(6,.-c)3+i 


+    2 
n—p 


AnR^ 


{bn-cy+'\ 


RV  .   ^  \An\ 


.  =  k+l     R       \R  '         n=p 


R 


e  *-i  e    2'~l  8-,. 


^^^{S-\Ak\}+ie 


<  e. 


170  LACUNAEY   FUNCTIONS  [87. 

a  quantity  arbitrarily  assigned  as  small  as  we  please.     Accordingly  we  have 

Limit !  Ri  [Cq  -  Aj,  (h  -  c)-^-^}  |  =  0, 
that  is, 

Limit  i  GqR'i  j  =  Limit  |  A^Ri  {bk  -  c)-i-^  \  =  L^ , 

so  that  GqR'i  does  not  tend  to  zero  when  q  is  infinitely  large,  as  it  should  if 
P{z  —  c)  converges.     Thus  P(z  —  c)  does  not  converge  for  points  given  by 

\z  —  c\  =  R. 

Consider  now  the  domain  of  Zg,  assumed  to  include  points  within  G  and 
therefore  some  arc  of  G;  the  function  is  represented  throughout  that  domain 
by  Q{z  —  ^o).  On  the  included  arc  of  G  take  any  one  (say  hk)  of  the  un- 
limited number  of  points  h;  a,t  b^  draw  an  outward  normal  to  G  and  choose  a 
point  Zi  on  it  such  that  the  circle 

\z-z-,\  =  \bk-  z^l 

lies  wholly  within  the  domain  of  Zq.  The  function  is  represented  by  a  power- 
series  in  z  —  Z2_  throughout  this  circle ;  and  as  the  circle  lies  wholly  within 
the  domain  of  Zq,  the  representation  is  included  in  Q{z  —  z^).  But,  by  the 
preceding  investigation,  the  power-series  does  not  converge  on  the  circum- 
ference of  the  circle  \  z  —  z-^\  =  \b]c  —  z^\:  contradicting  the  supposition  that 
Q{z  —  Zq)  converges  in  a  domain  of  Zq  enclosing  this  circle.  Hence  the  power- 
series  P  (z  —  c)  cannot  be  continued  across  the  boundary  of  (7 ;  in  other  words, 
the  function  represented  by  P  (z  —  c)  and  its  continuations  has  the  area  of  G 
for  a  lacunary  space. 

The  discussion  of  the  significance  (if  any)  of  (f)  (z)  for  points  z  within 
G  depends  on  the  distribution  of  the  points  b^  within  G,  as  to  which  no 
hypothesis  has  been  made. 

As  an  example,  take  a  convex  polygon  having  Ui,  ,  ap  for  its  angular  points ; 

then  any  point 

mi«i  -t- +  nipttj, 

mi  + +  'mp     ' 

where  m^, ,mp  are  positive  integers  or  zero  (simultaneous  zeros  being  excluded),  is 

either  within  the  polygon  or  on  its  boundary :  and  any  rational  point  within  the  polygon 
or  on  its  boundary  can  be  represented  by . 

p 

2    lUrar 


p 
,.=1 


by  proper  choice  of  m^,  ,  nip,  a  choice  which  can  be  made  in  an  infinite  nvimber 

of  ways. 

Let  Ml, ,  zip  he  given  quantities,  the  modulus  of  each  of  which  is  less  than  unity : 

then  the  series 

2  ?«]'»! V**" 


87.]  LACUNARY   SPACES  iTl 

converges  unconditionally.     Then  all  the  assigned  conditions  are  satisfied  for  the  function 

^l{"^l u/ih 


m-iai  + +  mpap 

TOi  + +mp 

and  therefore  it  is  a  function  which  converges  uniformly  and  unconditionally  everywhere 
outside  the  polygon  and  which  has  the  polygonal  space  (including  the  boundary)  for 
a  lacunary  space. 

If,  in  particular,  ^  =  2,  we  obtain  a  function  which  has  the  straight  line 
joining  Oj  and  a,  as  a  line  of  essential  singularity.  When  we  take  (Xi  =  0, 
a2=l,  and  slightly  modify  the  summation,  we  obtain  the  function 

00        n     -).  ni„  n—m 


n=l  m=0      „       ^'*' 

Z 

n 
which,  when  \uj\  <1  and  \u2\<l,  converges  uniformly  and  unconditionally 
everywhere  in  the  plane  except  at  points  between  0  and  1  on  the  axis  of  real 
quantities,  this  part  of  the  axis  being  a  line  of  essential  singularity. 

For  the  general  case,  the  following  remarks  may  be  made  : — 

(i)     The  quantities  itj,  Wg.  •••  need  not  be  the  same  for  every  term;  a 
numerator,  quite  different  in  form,  might  be  chosen,  such  as 
{mi^  +  . . .  +  m/)~'^  where  2yu,  >  p;  all  that  is  requisite  is  that  the 
series,  made  up  of  the  numerators,  should  converge  uncondition- 
ally, 
(ii)    The  preceding  is  only  a  particular  illustration,  and  is  not  necessarily 
the  most  general  form  of  function  having  the  assigned  lacunary 
space. 
It  is  evident  that  one  mode  of  constructing  a  function,  which  shall  have 
any  assigned  lacunary  space,  would  begin  by  the  formation  of  some  expression 
which,  by  the  variation  of  the  constants  it  contains,  can  be  made  to  represent 
indefinitely  nearly  any  point  within  or  on  the  contour  of  the  space.     Thus 
for  the  space  between  two  concentric  circles,  of  radii  a  and  c  and  centre  the 
origin,  we  could  take 

TTiia  +  {n  —  mi)  b  ~^^i 
n 
which,  by  giving  m^  all  values  from  0  to  n,  m^  all  values  from  0  to  n-1,  and 
n  all  values  from  1  to  infinity,  will  represent  all  rational  points  in  the  space : 
and  a  function,  having  the  space  between  the  circles  as  lacunary,  would  be 
given  by 

CO        n        n-1    r  U^Hj'^^U2^ 

2^      2Li        2^ 


n=l  mi=0  m2=0 


m^a  +  {n  —  mi)  b  — '27ri[ 
n  J 


provided  j  i^  |  <  1,  |  Wj  j  <  1,  |  Wg  ]  <  1. 


172  EXAMPLES  [87. 

In  particular,  if  a  =  6,  then  the  common  circumference  is  a  line  of  essential  singularity 
for  the  corresponding  function.     It  is  easy  to  see  that  the  function 


2 

n=0 

2?l— 1 
^        in,  n     m,  n 

m=0                 —  TTl 

z  —  aen 

00 

2ji— 1      m          n 

s 

2      W           V 

m=l 

m=0     m,  n     m,  n 

provided  the  series 

converges  unconditionally,  is  a  function  having  the  circle  \z\  =  a  as  a  line  of  essential 
singularity.  It  can  be  expressed  as  an  analytic  function  within  the  circle,  and  as  another 
analytic  function  without  the  circle. 

Other  examples  will  be  found  in  memoirs  by  Goursat*    Poincaret,  and  Homenl. 

Ex.  1.     Shew  that  the  fimction 

2  2      (m  +  ?is)-2-'-, 


?K=o 


5!=-c 


where  r  is  a  real  positive  quantity  and  the  summation  is  for  all  integers  m  and  n  between 
the  positive  and  the  negative  infinities,  is  a  uniform  function  in  all  parts  of  the  plane 
except  the  axis  of  real  quantities  which  is  a  line  of  essential  singularity. 

Ex.  2.     Discuss  the  region  in  which  the  function 

CO         cc         CO      ^^-2^--Zp-2 

2       2       2 — 


n=\  m=l   ii=\   ^_(^    1    !_-,• 

\?i      n 
is  definite.  (Homen.) 

Ex.  3.     Prove  that  the  function 

2   ^-'^x^'" 

exists  only  within  a  circle  of  radius  unity  and  centre  the  origin.  (Poincare.) 

Ex.  4.     Prove  that  the  series 

CO  4 

2   -^ 

represents  a  uniform  meromorphic  function,  if  the  quantities  |  a^  \  increase  without  limit 
as  n  increases  and  if  the  series  |  An  Ian  I  converges. 

Ex.  5.     An  infinite  number  of  points  a^,  ao,  as, are  taken  on  the  circumference  of 

a  given  circle,  centre  the  origin,  so  that  they  form  the  aggregate  of  rational  points  on  the 
circumference.     Shew  that  the  series 

^     1       z 

2    s^^l 
n=l  **■    ^n      2 

can  be  expanded  in  a  series  of  ascending  powers  of  z  which  converges  for  points  within  the 
circle,  but  that  the  function  cannot  be  continued  across  the  circumference  of  the  circle. 

(Stieltjes.) 

*  Comptes  Rendus,  t.  xciv,  (1882),  pp.  715—718;  Bulletin  de  Darboux,  2'"«  Ser.,  t.  xi,  (1887), 
pp.  109—114. 

t  In  the  memoirs,  quoted  p.  166,  aud  Comptes  Rendus,  t.  xcvi,  (1883),  pp.  1134 — 1136. 
+  Acta  Soc.  Fenn.,  t.  xii,  (1883),  pp.  445—464. 


87.]  EXAMPLES  173 


Ex.  6.     Prove  that  the  infinite  continued  fraction 

1111 


converges  for  all  values  of  z,  provided  the  series 


2   a„ 

H  =  l 


diverges,  the  quantities  a  being  real.     Discuss,  in  particular,  the  cases,  (i)  when  z  has  real 
positive  values,  (ii)  when  z  has  real  negative  values. 

(Stieltjes.) 

Ex.  7.     Denoting  by  «„  a  positive  quantity  less  than  1,  prove  that  the  infinite  product 


n  ^    1--    e»     1 I  e^+^n 

«=i  iV       nj        \       n  +  e, 

converges  ;  and  that  the  series 

i  l(^ L 

»=i  fn  \s-n-en      z-n 
converges. 

Shew  that,  if  a  new  series  be  constructed  by  separating  the  two  fractions  in  the  single 
term  so  as  to  provide  two  terms,  this  new  series  does  not  converge  when  e^  =  n~^.  Does 
the  same  consequence  follow  when  e^^=n~'^1 


(Borel.) 


Ex.  8.     Prove  that  the  series 


'2  ,  „     2    *     °°    f  z  1 

TT  TT  _^  _„  \{l  —  2m—2nzi)  {2m  +  2nzi)'^) 


2        <=c        00        ^ 

+  -    2     2    i- 


[{I  -27n-2nz-H)  {2m  +  2nz-H)^j  ' 

where  the  summation  extends  over  all  positive  and  negative  integral  values  of  m  and  of  n 
except  simultaneous  zeros,  converges  uniformly  and  unconditionally  for  all  points  in 
the  finite  part  of  the  plane  which  do  not  lie  on  the  axis  of  i/ ;  and  that  it  has  the 
value  +1  or  —1,  according  as  the  real  part  of  z  is  positive  or  negative. 

.  (Weierstrass.) 

Ex.  9.     Prove  that  the  region  of  continuity  of  the  series 

consists  of  two  parts,  separated  by  the  circle  1^1  =  1  which  is  a  line  of  infinities  for 
the  series:  and  that,  in  these  two  parts  of  the  plane,  it  represents  two  different 
functions. 

bi'n 

If  two  complex  quantities  a  and  a  be  taken,  such  that  z=e  "'^  and  the  real  part  of 
— .  is  positive,  and  if  they  be  associated  with  the  elliptic  function  g>  (u)  as  its  half-periods, 
then  for  values  of  s,  which  lie  within  the  circle  |  2  |  =  1, 

"1  ft)   0-3  (<a)  ,  1 

in  the  usual  notation  of  Weierstrass's  theory  of  elliptic  functions. 

Find  the  function  which  the  series  represents  for  values  of  z  without  the  circle  |  2  |  =  1. 

(Weierstrass.) 


174  EXAMPLES  [87. 

Ex.  10.      Discuss    the    descriptive    properties    of  the   functions  represented  by  the 
expressions :    . 

for  all  values  of  the  complex  argument  z.  (Math.  Trip.,  Part  II.,  1893.) 

E:v.  11.     Four  circles  are  drawn  each  of  radius  -j-  having  their  centres  at  the  points 

1,  i,  —1,  —i  respectively ;  the  two  parts  of  the  plane,  excluded  by  the  four  circumferences, 
are  denoted  the  interior  and  the  exterior  parts.     Shew  that  the  function 


sm  J?^7^ 


2      ^ 


is  equal  to  tt  in  the  interior  part  and  is  zero  in  the  exterior  part.  (Appell.) 

Ux.  12.     Obtain  the  values  of  the  function 

^(*^)"-(^TT?"(^^i)^} 

in  the  two  parts  of  the  area  within  a  circle  centre  the  origin  and  radius  2  which  lie 
without  two  circles  of  radius  unity,  having  their  centres  at  the  points  1  and  —  1 
respectively.  (Appell.) 

Ex.13.       If  f(s)=Ul+U2  + +  Un, 

and  £.,„=F,„(.)-^-l^^  +  (.-«,„-l)|^^-^^  }, 

where  the  regions  of  continuity  of  the  functions  F  extend  over  the  whole  plane,  then  f{z) 
is  a  function  existing  everywhere  except  within  the  circles  of  radius  unity  described  round 
the  points  «! ,  a2 , ,  «« .  (Teixeira.) 

Ex.  14.      Let  there  be  n  circles  having  the   origin   for  a  common    centre,  and    let 

C^,  C2, ,  On,  Cn  +  ihe7i  +  l  arbitrary  constants  ;  also  let  a^,  ag, ,  ««  be  any  71  points 

lying  respectively  on  the  circumferences  of  the  first,  the  second, ,  the  wth  circles. 

Shew  that  the  expression 


1   (^"-(^ 
27r  J  0     \ze^^ 


has  the  value  C^  for  points  z  lying  between  the  (wi  — l)th  and  the  ■>7ith  circles,  and  the 
value  C'n  +  i  for  points  lying  without  the  nth  circle. 

Construct  a  function  which  shall  have  any  assigned  values  in  the  various  bands  into 
which  the  plane  is  divided  by  the  circles.  (Pincherle.) 

Ex.  15.     Examine  the  nature  of  the  functions  defined  by  the  series 

(02-a2)» 


(i) 
(ii)      2 


„=i  2  (s  -  a)2"  -  5  (^2  -  a2)»  +  2{z  +  af^ ' 

(22-a2)» 


„=i(3-a)2«  +  2(3+a)2'" 
where  a  is  a  real  positive  constant.  (Math.  Trip.,  Part  II.,  1897.) 


88.]  CLASSIFICATION   OF   SINGULARITIES  175 

88.  In  §  32  it  was  remarked  that  the  discrimination  of  the  various 
species  of  essential  singularities  could  be  effected  by  means  of  the  properties 
of  the  function  in  the  immediate  vicinity  of  the  point. 

Now  it  was  proved,  in  §  63,  that  in  the  vicinity  of  an  isolated  essential 
singularity  h  the  function  could  be  represented  by  an  expression  of  the  form 


«C4j)+^(-'') 


for  all  points  in  the  space  without  a  circle  centre  b  of  small  radius  and  within 
a  concentric  circle  of  radius  not  large  enough  to  include  singularities  at 
a  finite  distance  from  b.  Because  the  essential  singularity  at  b  is  isolated, 
the  radius  of  the  inner  circle  can  be  diminished  to  be  all  but  infinitesimal : 

the   series  P{z-b)  is   then   unimportant  compared  with   GI y],  which 

can  be  regarded  as  characteristic  for  the  singularity  of  the  function. 

Another  method  of  obtaining  a  function,  which  is  characteristic  of  the 
singularity,  is  provided  by  §  68.  It  was  there  proved  that,  in  the  vicinity  of 
an  essential  singularity  a,  the  function  could  be  represented  by  an  expression 
of  the  form 


i^-TB[^^Qi^-o), 


where,  within  a  circle  of  centre  a  and  radius  not  sufficiently  large  to  include 
the  nearest  singularity  at  a  finite  distance  from  a,  the  function  Q  {z  —  a)  is 
finite  and  has  no  zeros :  all  the  zeros  of  the  given  function  within  this  circle 
(except  such  as  are  absorbed  into  the  essential  singularity  at  a)  are  zeros  of 

the    factor  ^f j ,  and  the  integer-index  n  is  affected  by  the  number  of 

these  zeros.     When  the  circle  is  made  small,  the  function 


{z-aYH{-^) 
'        \z  —  a) 


can  be  regarded  as  characteristic  of  the  immediate  vicinity  of  a  or,  more 
briefly,  as  characteristic  of  a. 

It  is  easily  seen  that  the  two  characteristic  functions  are  distinct.  For 
if  F  and  F^  be  two  functions,  which  have  essential  singularities  at  a  of  the 
same  kind  as  determined  by  the  first  characteristic,  then 

Fiyz)  -  F^  (z)  =  P(z-a)-P,(z-a) 

=  P,{z-a), 

while  if  their  singularities  at  a  be  of  the  same  kind  as  determined  by  the 
second  characteristic,  then 

F(z)  _Qiz-a)  _^ 


176  CLASSIFICATION  [88. 

in  the  immediate  vicinity  of  a,  since  Qi  has  no  zeros.     Two  such  equations 
cannot  subsist  simultaneously,  except  in  one  instance. 

Without  entering  into  detailed  discussion,  the  results  obtained  in  the 
preceding  chapters  are  sufficient  to  lead  to  an  indication  of  the  classification 
of  singularities  *. 

Singularities  are  said  to  be  of  the  first  class  when  they  are  accidental ; 
and  a  function  is  said  to  be  of  the  first  class  when  all  its  singularities  are  of 
the  first  class.  It  can,  by  §  48,  have  only  a  finite  number  of  such  singularities, 
each  singularity  being  isolated. 

It  is  for  this  case  alone  that  the  two  characteristic  functions  are  in 
accord. 

When  a  function,  otherwise  of  the  first  class,  fails  to  satisfy  the  last 
condition,  solely  owing  to  failure  of  finiteness  of  multiplicity  at  some  point, 
say  at  z  —  oo  ,  then  that  point  ceases  to  be  an  accidental  singularity.  It  has 
been  called  (§  32)  an  essential  singularity ;  it  belongs  to  the  simplest  kind  of 
essential  singularity ;  and  it  is  called  a  singularity  of  the  second  class. 

A  function  is  said  to  be  of  the  second  class  when  it  has  some  singularities 
of  the  second  class ;  it  may  possess  singularities  of  the  first  class.  By  an 
argument  similar  to  that  adopted  in  |  48,  a  function  of  the  second  class 
can  have  only  a  limited  number  of  singularities  of  the  second  class,  each 
singularity  being  isolated. 

When  a  function,  otherwise  of  the  second  class,  fails  to  satisfy  the  last 
condition  solely  owing  to  unlimited  condensation  at  some  point,  say  at  £;  =  oo  , 
of  singularities  of  the  second  class,  that  point  ceases  to  be  a  singularity 
of  the  second  class :  it  is  called  a  singularity  (necessarily  essential)  of  the 
third  class. 

A  function  is  said  to  be  of  the  third  class  when  it  has  some  singularities 
of  the  third  class  ;  it  may  possess  singularities  of  the  first  and  the  second 
classes.  But  it  can  have  only  a  limited  number  of  singularities  of  the  third 
class,  each  singularity  being  isolated. 

Proceeding  in  this  gradual  sequence,  we  obtain  an  unlimited  number  of 
classes  of  singularities  :  and  functions  of  the  various  classes  can  be  constructed 
by  means  of  the  theorems  which  have  been  proved.  A  function  of  class  7i 
has  a  limited  number  of  singularities  of  class  n,  each  singularity  being 
isolated,  and  any  number  of  singularities  of  lower  classes  which,  except  in  so 
far  as  they  are  absorbed  in  the  singularities  of  class  n,  are  isolated  points. 

*  For  a  detailed  discussion,  reference  should  be  made  to  Guichard,  Theorie  des  points 
singuliers  essentiels  (These,  Gauthier-Villars,  Paris,  1883),  who  gives  adequate  references  to  the 
investigations  of  Mittag-Letfler  in  the  introduction  of  the  classification  and  to  the  researches  of 
Cantor.  See  also  Mittag-Leffier,  Acta  Math.,  t.  iv,  (1884),  pp.  1 — 79  ;  Cantor,  Crelle,  t.  Ixxxiv, 
(1878),  pp.  242—258,  Acta  Math.,  t.  ii,  (1883),  pp.  311—328. 


88.]  ■  OF   SINGULARITIES  177 

The  effective  limit  of  this  sequence  of  classes  is  attained  when  the 
number  of  the  class  increases  beyond  any  integer,  however  large.  When 
once  such  a  limit  is  attained,  we  have  functions  with  essential  singularities  of 
unlimited  class,  each  singularity  being  isolated  ;  when  we  pass  to  functions 
which  have  their  essential  singularities  no  longer  isolated  but,  as  in  previous 
class-developments,  of  infinite  condensation,  it  is  necessary  to  add  to  the 
arrangement  in  classes  an  arrangement  in  a  wider  group,  say,  in  species*. 

Calling,  then,  all  the  preceding  classes  of  functions  functions  of  the  first 
species,  we  may,  after  Guichard  (I.e.),  construct,  by  the  theorems  already 
proved,  a  function  which  has  at  the  points  a^,  an, ...  singularities  of  classes 
1,  2,  ..  ,  both  series  being  continued  to  infinity.  Such  a  function  is  called 
a  function  of  the  second  species. 

By  a  combination  of  classes  in  species,  this  arrangement  can  be  continued 
indefinitely  ;  each  species  will  contain  an  infinitely  increasing  number  of 
classes ;  and  when  an  unlimited  number  of  species  is  ultimately  obtained, 
another  wider  group  must  be  introduced. 

This  gradual  construction,  relative  to  essential  singularities,  can  be  carried 
out  without  limit ;  the  singularities  are  the  characteristics  of  the  functions. 

*  Guichard  (I.e.)  uses  the  term  genre. 


F.   F.  12 


CHAPTER   YIII. 

Multiform  Functions. 

89.  Having  now  discussed  some  of  the  more  important  general  properties 
of  uniform  functions,  we  proceed  to  discuss  some  of  the  properties  of  multiform 
functions. 

Deviations  from  uniformity  in  character  may  arise  through  various  causes  : 
the  most  common  is  the  existence  of  those  points  in  the  2^-plane,  which  have 
already  (§  12)  been  defined  as  branch-points. 

As  an  example,  consider  the  two  power-series 

u  =  l  -i^'  — l'^'"—  •••,  v  =  — (1  -^/  -i/'—  ...), 

which,  for  points  in  the  plane  such  that  \z'  \  is  less  than  unity,  are  the  two 
values  of  (1  —  z')^ ;  they  may  be  regarded  as  representing  the  two  branches 
of  the  function  w,  say  w^  and  Wg,  defined  by  the  equation 

w^  =  1  —  z'  =  z. 

Let  z'  describe  a  small  curve  (say  a  circle  of  radius  r)  round  the  point 
z'  =  l,  beginning  on  the  axis  of  x ;  the  point  1  is  the  origin  for  z.  Then  z 
is  r  initially,  and  at  the  end  of  the  first  description  of  the  circle  z  is  ?'e^''i 
The  branch  of  the  function,  which  initially  is  equal  to  u,  changes  continuously 
during  the  description  of  the  circle.  The  series  for  u,  and  the  continuations 
of  that  series,  give  rise  to  the  complete  variation  of  the  branch  of  the  function 
which  originally  is  u.  Its  initial  value  is  ?'*,  and  its  final  value  is  r^e'^',  that 
is^  —r~^  ;  so  that  the  final  value  of  the  branch  is  v.  Similarly  for  the  branch 
of  the  function,  which  initially  is  equal  to  v;  it  is  continuously  changed 
during  the  description  of  the  circle ;  the  series  for  v,  and  the  continuations 
of  that  series,  give  rise  to  the  complete  variation  of  the  branch  of  the  function 
which  originally  is  v ;  and  the  branch  acquires  u  as  its  final  value.  Thus  the 
effect  of  the  single  circuit  is  to  change  Wi  into  Wg  ^md  Wg  into  w^ ,  that  is,  the 
effect  of  a  circuit  round  the  point,  at  which  w^  and  lUo  coincide  in  value,  is  to 
interchange  the  values  of  the  two  branches. 

If,  however,  z  describe  a  circuit  which  does  not  include  the  branch-point, 
Wi  and  W2  return  each  to  its  initial  value. 


89.]  CONTINUATION   OF   MULTIFORM   FUNCTION  179 

Instances  have  already  occurred,  e.g.  integrals  of  uniform  functions,  in 
which  a  variation  in  the  path  of  the  variable  has  made  a  difference  in  the 
result;  but  this  interchange  of  value  is  distinct  from  any  of  the  effects 
produced  by  points  belonging  to  the  families  of  critical  points  which  have 
been  considered.  The  critical  point  is  of  a  new  nature;  it  is,  in  fact,  a 
characteristic  of  multiform  functions  at  certain  associated  points. 

We  now  proceed  to  indicate  more  generally  the  character  of  the  relation 
of  such  points  to  functions  affected  by  them. 

The  method  of  constructing  a  monogenic  analytic  function,  described  in 
§  34,  by  forming  all  the  continuations  of  a  power-series,  regarded  as  a  given 
initial  element  of  the  function,  leads  to  the  aggregate  of  the  elements  of  the 
function  and  determines  its  region  of  continuity.  When  the  process  of  con- 
tinuation has  been  completely  carried  out,  two  distinct  cases  may  occur. 

In  the  first  case,  the  function  is  such  that  any  and  every  path,  leading 
from  one  point  a  to  another  point  z  by  the  construction  of  a  series  of 
successive  domains  of  points  along  the  path,  gives  a  single  value  at  z  as  the 
continuation  of  one  initial  value  at  a.  When,  therefore,  there  is  only  a 
single  value  of  the  function  at  a,  the  process  of  continuation  leads  to  only  a 
single  value  of  the  function  at  any  other  point  in  the  plane.  The  function  is 
uniform  throughout  its  region  of  continuity.  The  detailed  properties  of  such 
functions  have  been  considered  in  the  preceding  chapters. 

In  the  second  case,  the,  function  is  such  that  different  paths,  leading  from 
a  to  z,  do  not  give  a  single  value  at  z  as  the  continuation  of  one  and  the 
same  initial  value  at  a.  There  are  different  sets  of  elements  of  the  function, 
associated  with  different  sets  of  consecutive  domains  of  points  on  paths  from 
a  to  z,  which  lead  to  different  values  of  the  function  at  z ;  but  any  change 
in  a  path  from  a  to  ^^  does  not  necessarily  cause  a  change  in  the  value  of  the 
function  at  z.  The  function  is  multiform  in  its  region  of  continuity.  The 
detailed  properties  of  such  functions  will  now  be  considered. 

90.  In  order  that  the  process  of  continuation  may  be  completely  carried 
out,  continuations  must  be  effected,  beginning  at  the  domain  of  any  point  a 
and  proceeding  to  the  domain  of  any  other  point  h  by  all  possible  paths  in 
the  region  of  continuity,  and  they  must  be  effected  for  all  points  a  and  b. 
Continuations  must  be  effected,  beginning  in  the  domain  of  every  point  a 
and  returning  to  that  domain  by  all  possible  closed  paths  in  the  region  of 
continuity.  When  they  are  effected  from  the  domain  of  one  point  a  to  that 
of  another  point  6,  all  the  values  at  any  point  z  in  the  domain  of  a  (and  not 
merely  a  single  value  at  such  points)  must  be  continued :  and  similarly  when 
they  are  effected,  beginning  in  the  domain  of  a  and  returning  to  that  domain. 
The  complete  region  of  the  plane  will  then  be  obtained  in  which  the  function 
can  be  represented  by  a  series  of  positive  integral  powers  :  and  the  boundary 
of  that  region  will  be  indicated. 

12—2 


180 


BRANCHES   OF 


[90. 


Fig.  14. 


In  the  first  instance,  let  the  boundary  of  the  region  be  consti'tuted  by  a 
number,    either    finite    or   infinite,    of 

isolated    points,    say    Xj,    Xo,    L-^,    

Take   any  point   A   in   the  region,  so 

that    its    distance    from    any    of    the 

points    L    is    not    infinitesimal ;    and 

in    the    region    draw    a    closed    path 

ABG...EFA    so    as    to    enclose    one 

point,  say   Xj,   but  only  one  point,  of 

the    boundary  and    to    have   no    point 

of  the  curve  at  a  merely  infinitesimal  distance  from  L^.     Let  such  curves  be 

drawn,  beginning  and  ending  at  A,  so  that  each  of  them  encloses  one  and 

only  one  of  the  points  of  the  boundary :   and  let  Kr  be  the  curve  which 

encloses  the  point  Lr. 

Let  Wi  be  one  of  the  power-series  defining  the  function  in  a  domain  with 
its  centre  at  A  :  let  this  series  be  continued  along  each  of  the  curves  Kg  by 
successive  domains  of  points  along  the  curve  returning  to  A.  The  result 
of  the  description  of  all  the  curves  will  be  that  the  series  Wj  cannot  be 
reproduced  at  A  for  all  the  curves,  though  it  may  be  reproduced  for  some 
of  them ;  otherwise,  tUi  would  be  a  uniform  function.  Suppose  that  w^,  w^,  ..-., 
each  in  the  form  of  a  power-series,  are  the  aggregate  of  new  distinct  values 
thus  obtained  at  A]  let  the  same  process  be  effected  on  w^,  lu..,  ...  as  has 
been  effected  on  w-^,  and  let  it  further  be  effected  on  any  new  distinct  values 
obtained  at  A  through  w^,  w^,  ...,  and  so  on.  When  the  process  has 
been  carried  out  so  far  that  all  values  obtained  at  A,  by  continuing  any 
series  round  any  of  the  curves  K  back  to  A ,  are  included  in  values  already 
obtained,  the  aggregate  of  the  values  of  the  function  at  A  is  complete :  they 
are  the  values  at  A  of  the  branches  of  the  function. 

We  shall  now  assume  that  the  number  of  values  thus  obtained  is  finite, 
say  n,  so  that  the  function  has  n  branches  at  ^  :  if  their  values  be  denoted 
by  Wi,  W2,  ...,  Wn,  these  n  quantities  are  all  the  values  of  the  function  at  A. 
Moreover,  n  is  the  same  for  all  points  in  the  plane,  as  may  be  seen  by  con- 
tinuing the  series  at  A  to  any  other  point  and  taking  account  of  the  corollaries 
at  the  end  of  the  present  section. 

The  boundary-points  L  may  be  of  two  kinds.  It  may  (and  not  infre- 
quently does)  happen  that  a  point  Lg  is  such  that,  whatever  branch  is  taken 
at  A  as  the  initial  value  for  the  description  of  the  circuit  A'^,  that  branch  is 
reproduced  at  the  end  of  the  circuit.  Let  the  aggregate  of  such  points  be 
/i,  I2,  ....  Then  each  of  the  remaining  points  L  is  such  that  a  description 
of  the  circuit  round  it  effects  a  change  on  at  least  one  of  the  branches,  taken 
as  an  initial  value  for  the  description ;  let  the  aggregate  of  these  points  be 
5i,  jBg,  ....  They  are  the  branch -points ;  their  association  with  the  definition 
in  02  will  be  made  later. 


90.] 


MULTIFORM    FUNCTIONS 


181 


Fig.  15. 


When  account  is  taken  of  the  continuations  of  the  function  from  a  point 
A  to  another  point  B,  we  have  n  values  at  B  as  the  continuations  of  n  values 
at  A.  The  selection  of  the  individual  branch  at  B,  which  is  the  continuation 
of  a  particular  branch  at  A,  depends  upon  the  path  of  z  between  A  and  B; 
it  is  governed  by  the  following  fundamental  proposition : — 

■  The  final'  value  of  a  branch  of  a  function  for  two  paths  of  variation  of  the 
independent  variable  from  one  point  to  another  will  he  the  same,  if  one  path 
can  he  deforvied  into  the  other  without  passing  over  a  branch-point. 

Let  the  initial  and  the  final  points  be  a  and  b,  and  let  one  path  of 
variation  be  acb.     Let  another  path  of   variation  be  aeb,  . 

both  paths  lying  in  the  region  in  which  the  function  can 
be  expressed  by  series  of  positive  integral  powers  :  the  two 
paths  are  assumed  to  have  no  point  within  an  infinitesimal 
distance  of  any  of  the  boundary-points  L  and  to  be  taken 
so  close  together,  that  the  circles  of  convergence  of  pairs  of 
points  (such  as  Ci  and  e^,  Cg  and  e^,  and  so  on)  along  the  two 
paths  have  common  areas.  When  we  begin  at  a  with  a 
branch  of  the  function,  values  at  Ci  and  at  e^  are  obtained, 
depending  upon  the  values  of  the  branch  and  its  derivatives  at  a  and  upon 
the  positions  of  Cj  and  e^ ;  hence,  at  any  point  in  the  area  common  to  the 
circles  of  convergence  of  these  two  points,  only  a  single  value  arises  as 
derived  through  the  initial  value  at  a.  Proceeding  in  this  way,  only  a  single 
value  is  obtained  at  any  point  in  an  area  common  to  the  circles  of  con- 
vergence of  points  in  the  two  paths.  Hence  ultimately  one  and  the  same 
value  will  be  obtained  at  b  as  the  continuation  of  the  value  of  the  one  branch 
at  a  by  the  two  different  paths  of  variation  which  have  been  taken  so  that 
no  boundary-point  L  lies  between  them  or  infinitesimally  near  to  them. 

Now  consider  any  two  paths  from  a  to  b,  say  acb  and  adb,  such  that 
neither  of  them  is  near  a  boundar3'--point  and  that  the 
contour  they  constitute  does  not  enclose  a  boundary-point. 
Then  by  a  series  of  successive  infinitesimal  deformations  we 
can  change  the  path  acb  to  adb  ;  and  as  at  h  the  same  value 
of  w  is  obtained  for  variations  of  z  from  a  to  6  along  the 
successive  deformations,  it  follows  that  the  same  value  of  w 
is  obtained  at  b  for  variations  of  z  along  acb  as  for  varia- 
tions along  adb. 

Next,  let  there  be  two  paths  acb,  adb  constituting  a  closed  contour, 
enclosing  one  (but  not  more  than  one)  of  the  points  /  and  none  of  the  points 
B.  When  the  original  curve  K  which  contains  the  point  /  is  described,  the 
initial  value  is  restored :  and  hence  the  branches  of  the  function  obtained  at 
any  point  of  K  by  the  two  paths  from  any  point,  taken  as  initial  point,  are 
the  same.     By  what  precedes,  the  parts  of  this  curve  K  can  be  deformed 


182  EFFECT   OF   DEFORMATION    OF  [90. 

into  the  parts  of  achda  without  affecting  the  branches  of  the  function  :  hence  . 
the  value  obtained  at  b,  by  continuation  along  acb,  is  the  same  as  the  value 
there  obtained  by  continuation  along  adb.  It  therefore  follows  that  a  path 
between  two  points  a  and  b  can  be  deformed  over  any  point  I  without 
affecting  the  value  of  the  function  at  6 ;  so  that,  when  the  preceding 
results  are  combined,  the  proposition  enunciated  is  proved. 

By  the  continued  application  of  the  theorem,  we  are  led  to  the  following 
results : — 

Corollary  I.  Whatever  be  the  effect  of  the  description  of  a  circuit  on  the 
initial  value  of  a  function,  a  reversal  of  the  circuit  restores  the  original  value 
of  the  function. 

For  the  circuit,  when   described  positively  and  negatively,  may  be   re- 
garded as  the  contour  of  an  area  of  infinitesimal  breadth,  which  encloses  no 
branch-point    within    itself  and    the    description    of  the    contour  of  which  ' 
therefore  restores  the  initial  value  of  the  function. 

Corollary  II.  ^1  circuit  can  be  deformed  into  any  other  circuit  ivithout 
aff^ecting  the  final  value  of  the  function,  provided  that  no  branch-point  be  crossed 
in  the  process  of  deformation. 

It  is  thus  justifiable,  and  it  is  often  convenient,  to  deform  a  path  con- 
taining a  single  branch-point  into  a  loop  round  the 

point.     A  loop*  consists  of  a  line  nearly  to  the  point,        O  Iv V'^ 

nearly  the  whole  of  a  very  small  circle  round  the  point,  p-     -^rj 

and  a  line  back  to  the  initial  point ;  see  figure  17. 

Corollary  III.  The  value  of  a  function  is  unchanged  when  the  variable 
describes  a  closed  circuit  containing  no  branch-point ;  it  is  likeivise  unchanged 
ivhen  the  variable  describes  a  closed  circuit  containing  all  the  branch-points. 

The  first  part  is  at  once  proved  by  remarking  that,  without  altering  the 
value  of  the  function,  the  circuit  can  be  deformed  into  a  point. 

For  the  second  part,  the  simplest  plan  is  to  represent  the  variable  on 
Neumann's  sphere.  The  circuit  is  then  a  curve  on  the  sphere  enclosing  all 
the  branch -points :  the  effect  on  the  value  of  the  function  is  unaltered  by 
any  deformation  of  this  curve  which  does  not  make  it  cross  a  branch-point. 
The  curve  can,  without  crossing  a  branch-point,  be  deformed  into  a  point 
in  that  other  part  of  the  area  of  the  sphere  which  contains  none  of  the 
branch-points;  and  the  point,  which  is  the  limit  of  the  curve,  is  not  a 
branch-point.  At  such  a  point,  the  value  of  the  function  is  unaltered ;  and 
therefore  the  description  of  a  circuit,  which  encloses  all  the  branch-points, 
restores .  the  initial  value  of  the  function. 

Corollary  IV.     If  the  values  of  lu  at  b  for  variations  along  tivo  paths 
*  French  writers  use  the  word  lacet,  German  writers  the  word  Schleife. 


90.] 


PATH   OF   THE   VAEIABLE 


183 


acb,  adb  be  not  the  same,  then  a  description  of  acbda  will  not  restore  the  initial 
value  of  w  at  a. 

In  particular,  let  the  path  be  the  loop  OeceO  (fig.  17),  and  let  it  change  iv 
at  0  into  w'.  Since  the  values  of  lu  at  0  are  different  and  because  there  is 
no  branch-point  in  Oe  (or  in  the  evanescent  circuit  OeO),  the  values  of  w  at 
€  cannot  be  the  same :  that  is,  the  value  with  which  the  infinitesimal  circle 
round  a  begins  to  be  described  is  changed  by  the  description  of  that  circle. 
Hence  the  'part  of  the  loop  that  is  effective  for  the  change  in  the  value  of  w  is 
the  small  circle  round  the  point;  and  it  is  because  the  description  of  a  small 
circle  changes  the  value  of  w  that  the  value  of  tu  is  changed  at  0  after  the 
description  of  a  loop. 

itf{z)  be  the  value  of  ^u  which  is  changed  into  /i  (z)  by  the  description  of 
the  loop,  so  that  /  (z)  and  f  (z)  are  the  values  at  0,  then  the  foregoing 
explanation  shews  that /"(e)  and  /i  (e)  are  the  values  at  e,  the  branch /(e) 
being  changed  by  the  description  of  the  circle  into  the  branch /i  (e). 

From  this  result  the  inference  can  be  derived  that  the  points  B^,  B.2,... 
are  branch-points  as  defined  in  §  12.  Let  a  be  any  one  of  the  points,  and 
let/(^)  be  the  value  of  w  which  is  changed  into/i(^)  by  the  description  of 
a  very  small  circle  round  a.  Then  as  the  branch  of  w  is  monogenic,  the 
difference  between  /  (z)  and  f  (z)  is  an  infinitesimal  quantity  of  the  same 
order  as  the  length  of  the  circumference  of  the  circle :  so  that,  as  the  circle 
is  infinitesimal  and  ultimately  evanescent,  \f(z)  —f  (z)  j  can  be  made  as  small 
as  we  please  with  decrease  of  \z  —  a\  or,  in  the  limit,  the  values  of  /  (a)  and 
f  (a)  at  the  branch-point  are  equal.  Hence  each  of  the  points  B  is  such 
that  two  or  more  branches  of  the  f  miction  have  the  same  value  at  the  point, 
and  there  is  interchange  among  these  branches  ivhen  the  variable  describes  a 
small  circuit  round  the  point:  which  affords  a  definition  of  a  branch-point, 
more  complete  than  that  given  in  |  12. 

CoEOLLARY  V.  If  a  closed  circuit  contain  several  branch-points,  the  effect 
luhich  it  produces  can  be  obtained  by  a  combination  of  the  effects  produced  in 
succession  by  a  set  of  loops  each  going  round  only  one  of  the  branch-points. 

If  the  circuit  contain  several  branch-points,  say  three  as  at  a,  b,  c,  then 
a  path  such  as  AEFD,  in  fig.  18,  can  without 
crossing  any  branch-point,  be  deformed  into  the 
loops  A  aB,  BbC,  CcD ;  and  therefore  the  complete 
circuit  AEFD  A  can  be  deformed  validly  into 
AaBbCcDA,  and  the  same  effect  will  be  produced 
by  the  two  forms  of  circuit.     When  D  is  made  D^ 

practically  to  coincide  with  A.  the  whole  of  the  Fig.  18. 

second  circuit  is  composed  of  the  three  loops.     Hence  the  corollary. 

This  corollary  is  of  especial  importance  in  the  consideration  of  integrals 
of  multiform  functions. 


184  BRANCHES   OF  [90. 

Corollary  VI.  In  a  continuous  'part  of  the  plane  where  there  are  no 
branch-points,  each  branch  of  a  multiform  function  is  uniform. 

Each  branch  is  monogenic  and,  except  at  isolated  points,  continuous; 
hence,  in  such  regions  of  the  plane,  all  the  propositions  which  have  been 
proved  for  monogenic  analytic  functions  can  be  applied  to  each  of  the 
branches  of  a  multiform  function. 

91.  If  there  be  a  branch-point  within  the  circuit,  then  the  value  of  the 
function  at  b  consequent  on  variations  along  acb  may,  but  will  not  necessarily, 
differ  from  its  value  at  the  same  point  consequent  on  variations  along  adb. 
Should  the  values  be  different,  then  the  description  of  the  whole  curve  a^bda 
will  lead  at  a  not  to  the  initial  value  of  w,  but  to  a  different  value. 
The  test  as  to  whether  such  a  change  is  effected  by  the  description  is 
immediately  derivable  from  the  foregoing  proposition;  and  as  in  Corollary 
IV.,  §  90,  it  is  proved  that  the  value  is  or  is  not  changed  by  the  loop, 
according  as  the  value  of  w  for  a  point  near  the  circle  of  the  loop  is  or 
is  not  changed  by  the  description  of  that  circle.  Hence  it  follows  that,  if 
there  be  a  branch-point  which  affects  the  branch  of  the  fanction,  a  path  of 
variation  of  the  independent  variable  cannot  be  deformed  across  the  branch- 
point without  a  change  in  the  value  of  w  at  the  extremity  of  the  path. 

And  it  is  evident  that  a  point  can  be  regarded  as  a  branch-point  for  a 
function  only  if  a  circuit  round  the  point  interchange  some  {or  all)  of  the 
branches  of  the  function  which  are  equal  at  the  point.  It  is  not  necessary  that 
all  the  branches  of  the  function  should  be  thus  affected  by  the  point :  it  is 
sufficient  that  some  should  be  interchanged*. 

Further,  the  change  in  the  value  of  w  for  a  single  description  of  a  circuit 
enclosing  a  branch-point  is  unique. 

For,  if  a  circuit  could  change  w  into  w  or  w" ,  then,  beginning  with  w" 
and  describing  it  in  the  negative  sense  we  should  return  to  iv  and  afterwards 
describing  it  in  the  positive  sense  with  w  as  the  initial  value  we  should 
obtain  w'.  Hence  the  circuit,  described  and  then  reversed,  does  not  restore 
the  original  value  w"  but  gives  a  different  branch  w' ;  and  no  point  on 
the  circuit  is  a  branch-point.  This  result  is  in  opposition  to  Corollary  I., 
of  §  90;  and  therefore  the  hypothesis  of  alternative  values  at  the  end  of 
the  circuit  is  not  valid,  that  is,  the  change  for  a  single  description  is 
unique. 

But  repetitions  of  the  circuit  may,  of  course,  give  different  values  at  the 
end  of  successive  descriptions. 

*  In  what  precedes,  certain  points  were  considered  which  were  regular  singularities  (see 
p.  192,  note)  and  certain  which  were  branch-points.  Frequently  points  will  occur  which  are 
at  once  branch-points  and  infinities ;   proper  account  must  of  course  be  taken  of  them. 


92.]  MULTIFORM   FUNCTIONS  185 

92.     Let  0  be   any  ordinary  point  of  the  function ;    join  it  to   all  the 
branch-points    (generally    assumed    finite    in 
number)  in  succession  by  lines  which  do  not  ;!;    * 

meet  each  other :  then  each  branch  is  uniform  % 

for  each  path  of  variation  of  the  variable  which  % 

meets  none  of  these  lines.     The  effects  pro-  'm  ::P^^ 

duced  by  the  various  branch-points  and  their         %       ^.p^         -yi^ 
relations  on  the  various  branches  can  be  indi-         %     -;#'    -><^^ 
cated   by   describing   curves,  each    of  which        %'f^l;^^^^ 
begins    at    a   point    indefinitely  near    0  and       ^^----- ------ zz::z:"~"::::::"":"i  Bj 

returns  to  another  point  indefinitely  near  it 

after  passing  round  one  of  the  branch-points, 

and  by  noting  the  value  of  each  branch  of  the  function  after  each  of  these 

curves  has  been  described. 

The  law  of  interchange  of  branches  of  a  function  after  description  of  a 
circuit  round  a  branch-point  is  as  follows  \— 

All  the  branches  of  a  function,  which  are  affected  hy  a  hranch-point  as  such, 
can  either  he  arranged  so  that  the  order  of  interchange  (for  description  of  a 
path  round  the  point)  is  cyclical,  or  he  divided  into  sets  in  each  of  which  the 
order  of  interchange  is  cyclical. 

Let  Wi,  W2,  Ws,...  be  the  branches  of  a  function  for  values  of  z  near  a 
branch-point  a  which  are  affected  by  the  description  of  a  small  closed  curve 
C  round  a  :  they  are  not  necessarily  all  the  branches  of  the  function,  but  only 
those  affected  by  the  branch-point. 

The  branch  w^  is  changed  after  a  description  of  C ;  let  w.2  be  the  branch 
into  which  it  is  changed.  Then  Wa  cannot  be  unchanged  by  C ;  for  a  reversed 
description  of  C,  which  ought  to  restore  Wj,  would  otherwise  leave  w^  un- 
changed. Hence  lu.^  is  changed  after  a  description  of  C;  it  may  be  changed 
either  into  tv^  or  into  a  new  branch,  say  Wg.  If  into  m/i,  then  Wj  and  Wg  form 
a  cyclical  set. 

If  the  change  be  into  w^,  then  tUs  cannot  remain  unchanged  after  a 
description  of  G,  for  reasons  similar  to  those  that  before  applied  to  the 
change  of  lUo ;  and  it  cannot  be  changed  into  Wg,  for  then  a  reversed  de- 
scription of  C  would  change  w^  into  Wg,  and  it  ought  to  change  w^  into  w^. 
Hence,  after  a  description  of  G,  w^  is  changed  either  into  w^  or  into  a  new 
branch,  say  w^.     If  into  lUj,  then  w^,  w..,  tu.^  form  a  cyclical  set. 

If  the  change  be  into  w^,  then  w^  cannot  remain  unchanged  after  a 
description  of  G ;  and  it  cannot  be  changed  into  VJ2  or  w^ ,  for  by  a  reversal 
of  the  circuit  that  earlier  branch  would  be  changed  into  lu^  whereas  it  ought 
to  be  changed  into  the  branch,  which  gave  rise  to  it  by  the  forward  descrip- 
tion— a  branch  which  is  not  tv^.  Hence,  after  a  description  of  G,  w^  is 
changed  either  into  w^  or  into  a  new  branch.  If  into  w^,  then  Wi,  iv^.,  w^,  w^ 
form  a  cyclical  set. 


186  INTERCHANGE   OF   BRANCHES  *       [92. 

If  Wi  he  changed  into  a  new  branch,  we  proceed  as  before  with  that  new 
branch  and  either  complete  a  cyclical  set  or  add  one  more  to  the  set.  By 
repetition  of  the  process,  we  complete  a  cyclical  set  sooner  or  later. 

If  all  the  branches  be  included,  then  evidently  their  complete  system 
taken  in  the  order  in  which  they  come  in  the  foregoing  investigation  is  a 
system  in  which  the  interchange  is  cyclical. 

If  only  some  of  the  branches  be  included,  the  remark  applies  to  the  set 
constituted  by  them.  We  then  begin  with  one  of  the  branches  not  included 
in  that  set  and  evidently  not  inclusible  in  it,  and  proceed  as  at  first,  until 
we  complete  another  set  which  may  include  all  the  remaining  branches  or 
only  some  of  them.  In  the  latter  case,  we  begin  again  with  a  new  branch 
and  repeat  the  process;  and  so  on,  until  ultimately  all  the  branches  are 
included.  The  whole  system  is  then  arranged  in  sets,  in  each  of  which  the 
order  of  interchange  is  cyclical. 

93.  The  analytical  test  of  a  branch-point  is  easily  obtained  by  con- 
structing the  general  expression  for  the  branches  of  a  function  which  are 
interchanged  there. 

Let  z  =  ahe  Si  branch-point  where  n  branches  Wj,  Wg,  ...,  w„  are  cyclically 
interchanged.  Since  by  a  first  description  of  a  small  curve  round  a,  the 
branch  Wj  changes  into  lu^,  the  branch  Wo  into  Ws,  and  so  on,  it  follows  that 
by  r  descriptions  w^  is  changed  into  Wr+i  and  by  n  descriptions  w^  reverts  to 
its  initial  value.  Similarly  for  each  of  the  branches.  Hence  each  branch 
returns  to  its  initial  value  after  n  descriptions  of  a  circuit  round  a  branch- 
point where  n  branches  of  the  function  are  interchangeable. 

Now  let  z-a=^Z'^; 

then,  when  z  describes  circles  round  a,  Z  moves  in  a  circular  arc  round  its 
origin.     For   each    circumference  described  by  z,  the   variable   Z  describes 

-th  part  of  its  circumference;   and  the  complete  circle  is  described  by  Z 

n       ^ 

round  its  origin  when  n  complete  circles  are  described  by  z  round  a.     Now 

the  substitution  changes  tUr  as  a  function  of  z  into  a  function  of  Z,  say  into 
Wr',  and,  after  n  complete  descriptions  of  the  ^^-circle  round  a,  w,.  returns 
to  its  initial  value.  Hence,  after  the  description  of  a  .^'-circle  round  its 
origin,  Wr  returns  to  its  initial  value,  that  is,  Z  =  0  ceases  to  be  a  branch- 
point for  Wr.     Similarly  for  all  the  branches  W. 

But  no  other  condition  has  been  associated  with  a  as  a  point  for  the 
function  w;  and  therefore  Z  =  0  maybe  any  point  for  the  function  W,  that 
is,  it  may  be  an  ordinary  point,  or  a  singularity.  In  every  case,  we  have  W 
a  uniform  function  of  Z  in  the  immediate  vicinity  of  the  origin  ;  and  therefore 
in  that  vicinity  it  can  be  expressed  in  the  form 


e(^)+P(2). 


93.]  ANALYTICAL   TEST  187 

with  the  significations  of  P  and  G  ah^eady  adopted.  When  Z=0  is  an 
ordinary  point,  (r  is  a  constant  or  zero ;  when  it  is  an  accidental  singularity, 
(r  is  a  polynomial  function ;  and,  when  it  is .  an  essential  singularity,  G  is 
a  transcendental  function. 

The  simpler  cases  are,  of  course,  those  in  which  the  form  of  G  is  poly- 
nomial or  constant  or  zero  ;  and  then  W  can  be  put  into  the  form 

Z'^P{Z), 

where  P  is  an  infinite  series  of  positive  powers  and  m  is  an  integer.  As  this 
is  the  form  of  W  in  the  vicinity  of  ^  =  0,  it  follows  that  the  form  of  w  in  the 
vicinity  of  ^^  =  a  is 

m  1 

(z  -  afP  [{z  -  af]  ; 
and  the  various  n  branches  of  the  function  are  easily  seen  to  be  given  by 
substituting  in  the  above  for  {z  —  a)'^  the  values 

27rsi  1 

e'^(z-aY, 

where  s  =  0,  1, . . . ,  »  —  1.  We  therefore  infer  that  the  general  expression  for 
the  n  branches  of  a  function,  which  are  interchanged  by  circuits  round  a 
branch-point  z  =  a,  assumed  not  to  be  an  essential  singularity,  is 

m  _  1 

1 
whei^e  in  is  an  integer,  and  luhere  to  (z  —  a)"  its  n  values  are  in  tarn  assigned 
to  obtain  the  different  branches  of  the  function. 

There  may  be,  however,  more  than  one  cyclical  set  of  branches.  If  there 
be  another  set  of  r  branches,  then  it  may  similarly  be  proved  that  their 
general  expression  is 

{z-afQ{{z-af], 
where  ?«!  is  an  integer,  and  Q  is  an  integral  function ;  the  various  branches 

are  obtained  by  assigning  to  {z  —  af  its  r  values  in  turn. 

And  so  on,  for  each  of  the  sets,  the  members  of  which  are  cyclically 
interchangeable  at  the  branch-point. 

When  the  branch-point  is  at  infinity,  a  different  form  is  obtained.  Thus 
in  the  case  of  a  set  of  n  cyclically  interchangeable  branches  we  take 

z  =  %~**, 

so  that  n  negative  descriptions  of  a  closed  ^-curve,  excluding  infinity  and  no 
other  branch-point,  require  a  single  positive  description  of  a  closed  curve 
round  the  ^t-origin.  These  n  descriptions  restore  the  value  of  w  as  a  function 
of  z  to  its  initial  value ;  and  therefore  the  single  description  of  the  it-curve 
round  the  origin  restores  the  value  of   U — the  equivalent  of  w  after  the 


188  BRANCHES   OF  [93. 

change  of  the  independent  variable — as  a  function  of  u.  Thus  u  =  0  ceases 
to  be  a  branch -point  for  the  function  U\  and  therefore  the  form  of  U  is 

(?(l)+p(«). 

where  the  symbols  have  the  same  general  signification  as  before. 

If,  in  particular,  ^r  =  oo  be  a  branch-point  but  not  an  essential  singularity, 
then  G  is  either  a  constant  or  a  polynomial  function ;    and  then  U  can  be 

expressed  in  the  form  _ 

w-'«  P  (u), 

where  m  is  an  integer.  When  the  variable  is  changed  from  u  to  z,  then  the 
general  expression  for  the  n  branches  of  a  function  which  are  interchangeable 
at  z  =  ^ ,  assumed  not  to  be  an  essential  singularity,  is 

in  1 

^P{z~% 

1 

where  m  is  an  integer  and  where  to  s"  its  n  values  are  assigned  to  obtain  the 
different  branches  of  the  function. 

If,  however,  the  branch-point  z  =  a  in  the  former  case  or  z=  cc  in  the 
latter  be  an  essential  singularity,  the  forms  of  the  expressions  in  the  vicinity 
of  the  point  are 

G{(z-a)  ^]+P{(z-ay], 

1  _T_ 

and  G(z'')-\-P{z  «), 

respectively. 

Note.  When  a  multiform  function  is  defined,  either  explicitly  or  im- 
plicitly, it  is  practically  always  necessary  to  consider  the  relations  of  the 
branches  of  the  function  for  z  =  co  as  well  as  their  relations  for  points  that 
are  infinities  of  the  function.  The  former  can  be  determined  by  either 
of  the  processes  suggested  in  §  4  for  dealing  with  z—co;  the  latter  can  be 
determined  as  in  the  present  section. 

Moreover,  the  total  number  of  branches  of  the  function  has  been  assumed 
to  be  finite.  The  cases,  in  which  the  number  of  branches  is  unlimited,  need 
not  be  discussed  in  general :  it  will  be  sufficient  to  consider  them  when  they 
arise,  as  they  do  arise,  e.g.,  when  the  function  is  of  the  form  of  an  algebraical 
irrational  with  an  irrational  index  such  as  z^'^ — hardly  a  function  in  the 
ordinary  sense — ,  or  when  the  function  is  the  logarithm  of  a  function  of  z, 
or  is  the  inverse  of  a  periodic  function.  In  the  nature  of  their  multiplicity 
of  branching  and  of  their  sequence  of  interchange,  they  are  for  the  most  part 
distinct  from  the  multiform  functions  with  only  a  finite  number  of  branches. 

Ex.  The  simplest  illustrations  of  multiform  functions  are  furnished  by  functions 
defined  by  algebraical  equations,  in  particular,  by  algebraic  irrationals. 


98.] 


MULTIFORM   FUNCTIONS 


189 


The  general  type  of  the  algebraical  irrational  is  the  product  of  a  nimiber  of  functions 
of  the  form  iv  =  {A  (z  —  a^)  (z  -  a2) (s-«n)}™)  ™  and  n  being  integers. 

This  particular  function  has  ?h  branches  ;  the  points  a^,  a^, ,  a„  are  branch-points. 

To  find  the  law  of  interchange,  we  take  z - a,.^ pe^"^ ;  then  when  a  small  circle  of  radius  p 
is  described  round  ar,  so  that  z  returns  to  its  initial  position,  the  value  of  d  increases  by 

2rr  and  the  new  value  of  w  is  aiv,  where  a  is  the  mth.  root  of  unity  defined  by  em  "^  Taking 
then  the  various  branches  as  given  by  w,^aiv,  ahv, ,  a™-%,  we  have  the  law  of  inter- 
change for  description  of  a  small  curve  round  any  one  branch-point  as  given  by  this 
succession  in  cyclical  order.  The  law  of  succession  for  a  circuit  enclosing  more  than 
one  of  the  branch-points  is  derivable  by  means  of  Corollary  V.,  §  90. 

To  find  the  relation  of  z=cc  to  w,  we  take  zz'  =  l  and  consider  the  new  function  W  in 

the  vicinity  of  the  s'-origin.     We  have 

1    -'1 
^W={A  (l—aiz')(l-a2z') {l-anz')}mz'  m. 

If  the  variable  z'  describe  a  very  small  circle  round  the  origin  in  the  negative  sense,  then 


-2771 


z'  is  multiplied  by  e~  "^^  and  so  W  acquires  a  factor  e""'™,  that  is,  W  is  changed  unless 
this  acquired  factor  is  unity.  It  can  be  unity  only  when  n/m  is  an  integer ;  and  therefore 
except  when  nj/n  is  an  integer,  z  =  oc  is  a  branch-point  of  the  function.      The  law  of 

succession  is  the  same  as  that  for  negative  description  of  the  /-circle,  viz.,  w,  aP-iv,  a^Hv, ; 

the  m  vakies  form  a  single  cycle  only  if  n  be  prime  to  m,  and  a  set  of  cycles  if  n  be  not 
prime  to  m. 

Thus  s=oo  is  a  branch-point  for  w  =  {4z^  —  ff2^-ff3)~'^;  it  is  not  a  branch-point  for 
w  =  {(I  - z^)  {I  —  kh^)}~ '^ ;  and  z  =  b  is  a  branch-point  for  the  function  defined  by 

{z  —b)iv'^  =  z  —  a, 
but  z  =  b  is  not  a  branch-point  for  the  function  defined  by  {z—})fvfl  =  z  —  a. 

Again,  if  p  denote  a  particular  value  of  2^,  when  z  has  a  given  value,  and  q  similarly 

lz-\\k  2  J 

denote  a  particular  value  of  ( r-j  ,  then  w=p-\-q  is  a  six- valued  function,  the  values 

being 

Wx=     p  +  q,  w^=     p+aq,  tv^=     p  +  a^q, 

W2=  -p  +  q,         «'4=  -p  +  ciq,         "2^6=  -p  +  a^q, 
where  a  is  a  primitive  cube  root  of  unity.     The  branch-points  are  —  1,  0,  1,  oo  ;  and  the 
orders  of  change  for  small  circuits  round  one   (and  only  one)  of  these  points  are  as 
follows : — 


For  a  small  circuit  round 

-1 

0 

1 

oc 

Wi  changes  to 

Wo 

W2 

W3 

W2 

1^2              „ 

We 

Wi 

W4 

Wi 

^3              „ 

Wi 

W4 

Wr, 

Wi 

-4                „ 

W.2 

Ws 

We 

W3 

W5                „ 

w^ 

We 

Wi 

lOg 

■"'6                „ 

Wi 

Wr, 

W2 

Ws 

190  ALGEBRAIC  [93. 

Combinations  can  at  once  be  effected ;  thus,  for  a  positive  circuit  enclosing  both  1  and  oc 
but*  not  - 1  or  0,  the  succession  is 

Wi,  W4,  w^,  rv2,  W3,  tVQ 
in  cyclical  order. 

94.  It  has  already  been  remarked  that  algebraic  irrationals  are  a  special 
class  of  functions  defined  by  algebraical  equations.  Functions  thus  generally 
defined  by  equations,  which  are  polynomial  so  far  as  concerns  the  dependent 
variable  but  need  not  be  so  in  reference  to  the  independent  variable,  are 
often  called  algebraical.  The  term,  in  one  sense,  cannot  be  strictly  applied 
to  the  roots  of  an  equation  of  every  degree,  seeing  that  the  solution 
of  equations  of  the  fifth  and  higher  degrees  can  be  effected  only  by 
transcendental  functions;  but  what  is  implied  is  that  a  finite  number  of 
determinations  of  the  dependent  variable  is  given  by  the  equation f. 

The  equation  is  polynomial  in  relation  to  the  dependent  variable  w,  that 
is,  it  will  be  taken  to  be  of  finite  degree  n  in  w.  The  coefficients  of  the 
different  powers  will  be  supposed  to  be  uniform  functions  of  z :  were  they 
multiform  (with  a  limited  number  of  values  for  each  value  of  z)  in  any  given 
equation,  the  equation  could  be  transformed  into  another,  the  coefficients  of 
which  are  uniform  functions.  And  the  equation  is  supposed  to  be  irreducible, 
that  is,  if  the  equation  be  taken  in  the  form 

f{w,z)  =  0, 
the  left-hand  member  f{w,  z)  cannot  be  resolved  into  factors  of  a  form  and 
character  as  regards  lu  and  z  similar  to  /  itself. 

The  existence  of  equal  roots  of  the  equation  for  general  values  of  z 
requires  that 


f{w,z)    and  .-^;— 


dw 

shall  have  a  common  factor,  which  will  be  uniform  owing  to  the  form  of 
f{w,  z).  This  form  of  factor  is  excluded  by  the  irreducibility  of  the  equation ; 
so  that  /=  0,  as  an  equation  in  w,  has  not  equal  roots  for  general  values 
of  z.  But  though  the  two  equations  are  not  both  satisfied  in  virtue  of  a 
simpler  equation,  they  are  two  equations  determining  values  of  w  and  z; 
and  their  form  is  such  that  they  will  give  equal  values  of  w  for  special 
values  of  5. 

Since  the  equation  is  of  degree  n,  it  may  be  taken  to  be 

yjn  j^  ^n-ip^  (^2)  +  W'^-'F^  (z)+...+  wFn-i  (z)  +  Fn  {z)  =  0, 

where  the  functions  F-^,  F2,...  are  uniform.  If  all  their  singularities  be 
accidental,  they  are  rational  meromorphic  functions  of  z  (unless  z=  cc  is  the 

*  Such  a  circuit,  if  drawn  on  the  Neumann's  sphere,  may  be  regarded  as  excluding  - 1  and  0, 
or  taking  account  of  the  other  portion  of  the  surface  of  the  sphere,  it  may  be  regarded  as  a 
negative  circuit  including  - 1  and  0,  the  cyclical  interchange  for  which  is  easily  proved  to  be 
if?!,  M'4,  iv^,  W2,  W3,  1*^6  as  in  the  text. 

t  Such  a  function  is  called  Men  defini  by  Liouville. 


94]  FUNCTIONS  191 

only  singularity,  in  which  case  they  are  holomorphic) ;  and  the  equation  can 
then  be  replaced  by  one  which  is  equivalent  and  has  all  its  coefficients 
holomorphic,  the  coefficient  of  w'^  being  the  least  common  multiple  of  all  the 
denominators  of  the  meromorphic  functions  in  the  first  form.  This  form 
cannot  however  be  deduced,  if  any  of  the  singularities  be  essential. 

The  equation,  as  an  equation  in  lu,  has  n  roots,  all  functions  of  z;  let 
these  be  denoted  by  Wj,  Wo, . . .,  w„,  which  are  the  n  branches  of  the  function  lu. 
When  the  geometrical  interpretation  is  associated  with  the  analytical  relation, 
there  are  n  points  in  the  w'-plane,  say  ttj,..., ct^, which  correspond  with  a  point 
in  the  4?-plane,  say  with  a-^\  and  in  general  these  n  points  are  distinct. 
Further,  as  will  appear  from  the  investigations  in  §  97  (p.  207),  the  n  roots  w 
are  continuous  functions  of  z;  that  is  to  say,  any  small  change  in  the  value 
of  z  entails  corresponding  small  changes  in  the  value  of  each  of  the  n  roots  lu. 
Hence,  when  z  varies  so  as  to  move  in  its  own  pfene,  each  of  the  i^-points 
inoves  in  their  common  plane ;  and  thus  there  are  n  w-paths  corresponding 
to  a  given  z-path.     These  n  curves  may  or  may  not  meet  one  another. 

If  they  do  not,  there  are  n  distinct  w-paths,  leading  from  ai,  ...,a„  to 
/3i, ...,  yS„,  respectively  corresponding  to  the  single  ^-path  leading  from  a  to  h. 

If  two  or  more  of  the  w-paths  do  meet  one  another,  and  if  the  describing 
■w-points  coincide  at  their  point  of  intersection,  then  at  such  a  point  of 
intersection  in  the  ?(;-plane,  the  associated  branches  w  are  equal ;  and 
therefore  the  point  in  the  2^-plane  is  a  point  that  gives  equal  values  for  w. 
It  is  one  of  the  roots  of  the  equation  obtained  by  the  elimination  of  w 
between 

/('->=«■  '^-'■' 

the  analytical  test  as  to  whether  the  point  is  a  branch-point  will  be 
considered  later.  The  march  of  the  concurrent  'ly-branches  from  such  a 
point  of  intersection  of  two  iy-paths  depends  upon  their  relations  in  its 
immediate  vicinity. 

When  no  such  point  lies  on  a  ^^-path  from  a  to  h,  no  two  of  the  w-points 
coincide  during  the  description  of  their  paths.  By  |  90,  the  ^-path  can  be 
deformed  (provided  that,  in  the  deformation,  it  does  not  cross  a  branch-point) 
without  causing  any  two  of  the  w-points  to  coincide.  Further,  if  z  describe 
a  closed  curve  which  includes  none  of  the  branch-points,  then  each  of  the 
w-branches  describes  a  closed  curve  and  no  two  of  the  tracing  points  ever 
coincide. 

Note.  The  limitation  for  a  branch-point,  that  the  tracing  w-points 
coincide  at  the  point  of  intersection  of  the  w-curves,  is  of  essential .  im- 
portance. 

What  is  required  to  establish  a  point  in  the  i;-plane  as  a  branch-point, 
is  not   a   mere  geometrical  intersection   of  a  couple  of  completed  zy-paths 


192  ALGEBRAIC    FUNCTIONS  [94. 

but  the  coincidence  of  the  w-points  as  those  paths  are  traced,  together  with 
interchange  of  the  branches  for  a  small  circuit  round  the  point.  Thus  let  there 
be  such  a  geometrical  intersection  of  two  w-curves,  without  coincidence  of  the 
tracing  points.  There  are  two  points  in  the  ^•-plane  corresponding  to  the 
geometrical  intersection ;  one  belongs  to  the  intersection  as  a  point  of  the 
w-path  which  first  passed  through  it,  and  the  other  to  the  intersection  as  a 
point  of  the  w-path  which  was  the  second  to  pass  through  it.  The  two 
branches  of  w  for  the  respective  values  of  z  are  undoubtedly  equal ;  but  the 
equality  would  not  be  for  the  same  value  of  z.  And  unless  the  equality  of 
branches  subsists  for  the  same  value  of  z,  the  point  is  not  a  branch-point. 

A  simple  example  will  serve  to  illustrate  these  remarks.      Let  w  be  defined  hj  the 
equation 

so  that  the  branches  ivi  and  Wg  are  given  by 

CWi  =  CS  4-2  (S2  +  C2)4,  CIV2  =  CZ  -  Z  {z'^  +  G^f'  ] 

it  is  easy  to  prove  that  the  equation  resulting  from  the  elimination  of  w  between  /=0  and 

1^=0  is 

aw 

s2(22  +  c2)=0, 

and  that  only  the  two  points  z=  ±ic  are  branch-points. 

The  values  of  z  which  make  lOi  equal  to  the  vakie  of  Wo  for  z  =  a  (supposed  not  equal  to 
either  0,  ci  or  -  ci)  are  given  by 

cz  +  z  (0^+ c^)^  =  ca  —  a  {o?  +  c^)  ^ , 

which  evidently  has  not  z=a  for  a  root.  Rationalising  the  equation  so  far  as  concerns  z 
and  removing  the  factor  z-a,  as  it  has  just  been  seen  not  to  furnish  a  root,  we  find  that  z 
is  determined  by 

s3  +  z'^a  +  sa2  -1-  cfS  +  2ac^  -  2ac  {a^  -f  c^)^  =  0, 

the  three  roots  of  which  are  distinct  from  a,  the  assumed  point,  and  from  +  ci,  the  branch- 
point. Each  of  these  three  values  of  z  will  make  W]  equal  to  the  value  of  lo^  for  z  =  a:  we 
have  geometrical  intersection  without  coincidence  of  the  tracing  points. 

95.  When  the  characteristics  of  a  function  are  required,  the  most  im- 
portant class  are  its  infinities :  these  must  therefore  now  be  investigated. 
It  is  preferable  to  obtain  the  infinities  of  the  function  rather  than  the 
singularities  alone,  in  the  vicinity  of  which  each  branch  of  the  function 
is  uniform*  :  for  the  former  will  include  these  singularities  as  well  as  those 
branch-points  which,  giving  infinite  values,  lead  to  regular  singularities  when 
the  variables  are  transformed  as  in  §  93.  The  theorem  which  determines 
them  is : — 

The  infinities  of  a  function  determined  by  mi  algebraical  equation  are  the 
singularities  of  the  coefficients  of  the  equation. 

Let  the  equation  be 

^^Jn  +  ^o''-'F,  (z)  +  tu''-'F,  {z)+...+  wFn-i  (z)  +  Fn  {z)  =  0, 

*  These  singularities  will,  for  the  sake  of  brevity,  be  called  regular. 


95. J  THEIR    INFINITIES  193 

and  let  w'  be  any  branch  of  the  function;    then,  if  the  equation  which 
determines  the  remaining  branches  be 

we  have  F^  {z)  =  —  w'Gn-i  (z), 

i^,^_l  (Z)  =  -  w'Gn-,  (Z)  +  Gn-,  (z), 
Fn-2  {z)  =  -  w'Gn-3  (Z)  +  Gn-2  (z), 


F,{z)  =  -w'  +  G,(z). 

Now  suppose  that  a  is  an  infinity  of  w' ;  then,  unless  it  be  a  zero  of  order 
at  least  equal  to  that  of  Gn-i  (z),  a  is  an  infinity  of  Fn  (z).  If,  however,  it  be 
a  zero  of  Gn-i  (z)  of  sufficient  order,  then  from  the  second  equation  it  is  an 
infinity  of  Fn^-^  (z)  unless  it  is  a  zero  of  order  at  least  equal  to  that  of 
Gn-i{z);  and  so  on.  The  infinity  must  be  an  infinity  of  some  coefficient  not 
earlier  than  Fi  {z)  in  the  equation,  or  it  must  be  a  zero  of  all  the  functions 
G  which  are  later  than  Gi--^{z).  If  it  be  a  zero  of  all  the  functions  Gr,  so 
that  we  may  not,  without  knowing  the  order,  assert  that  it  is  of  rank  at 
least  equal  to  its  order  as  an  infinity  of  lu,  still  from  the  last  equation  it 
follows  that  a  must  be  an  infinity  of  F-^^  (z).  Hence  any  infinity  of  w  is  an 
infinity  of  at  least  one  of  the  coefficients  of  the  equation. 

Conversely,  from  the  same  equations  it  follows  that  a  singularity  of  one 
of  the  coefficients  is  an  infinity  either  of  w'  or  of  at  least  one  of  the  co- 
efficients G.  Similarly  the  latter  alternative  leads  to  an  inference  that  the 
infinity  is  either  an  infinity  of  another  branch  w"  or  of  the  coefficients  of  the 
(theoretical)  equation  which  survives  when  the  two  branches  have  been 
removed.  •  Proceeding  in  this  way,  we  ultimately  find  that  the  infinity  either 
is  an  infinity  of  one  of  the  branches  or  is  an  infinity  of  the  coefficient  in  the 
last  equation,  that  is,  of  the  last  of  the  branches.  Hence  any  singularity 
of  a  coefficient  is  an  infinity  of  at  least  one  of  the  branches  of  the  function. 

It  thus  appears  that  all  the  infinities  of  the  function  are  included  among, 
and  include,  all  the  singularities  of  the  coefficients ;  but  the  order  of  the 
infinity  for  a  branch  does  not  necessarily  make  that  point  a  regular 
singularity  nor,  if  it  be  a  regular  singularity,  is  the  order  necessarily  the 
same  as  for  the  coefficient. 

The  following  method  is  effective  for  the  determination  of  the  order  of 
the  infinity  of  the  branch. 

Let  a  be  an  accidental  singularity  of  one  or  more  of  the  F  functions, 
say  of  order  m^  for  the  function  Fi ;  and  assume  that,  in  the  vicinity  of  a, 
we  have 

Fi  {z)  =  {z-  a)-'^i  [d  +  di(z  -  a) +  ei(z-ay +...]. 

F.  F.  13 


/Ao 

A 

/ 

/A«-t 

An 

^^^.A 

.A 

•A 

/ 

An-. 

A„- 

y 

194  INFINITIES   OF  [95. 

Then  the  equation  which  determines  the  first  term  of  the  expansion  of  w  in 
a  series  in  the  vicinity  of  a  is 

w"  +  Ci  (^  -  ay-'^^w^-^  -\-c^{z-  aY'^^'up-^  4- ... 

Mark  in  a   plane,  referred   to    two    rectangular   axes,  points   n,   0;   n—1, 

—  m-^;  w  —  2,  —  ma ;  . . . ,  0,  —  m„ ;  let  these  

be  Aq,  J.1,  ...,  An  respectively.     Any  line 
through  Ai  has  its  equation  of  the  form 

y  -\-mi  =  X[x  —  {n  —  i)], 
that  is, 

y  —  \x  =  —  \{n  —  i)  —  vii. 

If  then  w  =  {z  —  a)~^f{z),  where  f{z)  is 

finite  when  z  =  a,  the  intercept  of  the  Fig-  20. 

foregoing  line  on  the  negative  side  of  the  axis  of  y  is  equal  to  the  order  of 

the  infinity  in  the  term 

This  being  so,  we  take  a  line  through  An  coinciding  in  direction  with  the 
negative  part  of  the  axis  of  y,  and  we  turn  it  about  4^  in  a  trigonometrically 
positive  direction  until  it  first  meets  one  of  the  other  points,  say  An-r',  then 
we  turn  it  about  An-r  until  it  meets  one  of  the  other  points,  say  Ans',  and 
so  on  until  it  passes  through  Aq.  There  will  thus  be  a  line  from  An  to 
Aq,  generally  consisting  of  a  number  of  parts;  and  none  of  the  points  A 
will  be  outside  the  figure  bounded  by  this  line  and  the  axes. 

The  perpendicular  from  the  origin  on  the  line  through  An-r  and  Ans  is 
evidently  greater  than  the  perpendicular  on  any  parallel  line  through  a 
point  A,  that  is,  on  any  line  through  a  point  A  with  the  same  value 
of  A.;    and,  as  this  perpendicular  is 

it  follows  that  the  order  of  the  infinite  terms  in  the  equation,  when  the  par- 
ticular substitution  is  made  for  w,  is  greater  for  terms  corresponding  to  points 
lying  on  the  line  than  it  is  for  any  other  terms. 

If  f{z)  =  6  when  z  =  a,  then  the  terms  of  lowest  order  after  the  substitu- 
tion of  (z  —  a)~^f{z)  for  w  are 

{Z  -  ay'^n-r->^r  \cn-rd''  +  . . .  "F  Cn-sd% 

as  many  terms  occurring  in  the  bracket  as  there  are  points  A  on  the  line 
joining  An-r  to  Ans-  Since  the  equation  determining  w  must  be  satisfied, 
terms  of  all  orders  must  disappear,  and  therefore 

Cn—s  t/         +  . .  •  +  Cn—r  ^^  ^j 

an  equation  determining  s  —  r  values  of  6,  that  is,  the  first  terms  in  the 
expansions  of  s  —  r  branches  w. 


95.]  ALGEBRAIC   FUNCTIONS  195 

Similarly  for  each  part  of  the  line :  for  the  first  part,  there  are  r  branches 
with  an  associated  value  of  \ ;  for  the  second,  s  —  r  branches  with  another 
associated  value ;  for  the  third,  t  —  s  branches  with  a  third  associated  value ; 
and  so  on. 

The  order  of  the  infinity  for  the  branches  is  measured  by  the  tangent 
of  the  angle  which  the  corresponding  part  of  the  broken  line  makes  with  the 
axis  of  x;  thus  for  the  line  joining  A^-r  to  A^s  the  order  of  the  infinity 
for  the  s  —  r  branches  is 

where  nin-r  and  mns  are  the  orders  of  the  accidental  singularities  of  Fn_r  C-^) 
and  Fn-s  (2). 

If  any  part  of  the  broken  line  should  have  its  inclination  to  the  axis  of 
X  greater  than  I^tt  so  that  the  tangent  is  negative  and  equal  to  —  //,,  then  the 
form  of  the  corresponding  set  of  branches  w  is  (z  —  a)i^g{z)  for  all  of  them, 
that  is,  the  point  is  not  an  infinity  for  those  branches.  But  when  the 
inclination  of  a  part  of  the  line  to  the  axis  is  <  -^tt,  so  that  the  tangent  is 
positive  and  equal  to  X,  then  the  form  of  the  corresponding  set  of  branches 
w  is  (z  —  a)~''f{z)  for  all  of  them,  that  is,  the  point  is  an  infinity  of  order  \ 
for  those  branches. 

In  passing  from  An  to  A^,  there  may  be  parts  of  the  broken  line  which 
have  the  tangential  coordinate  negative,  implying  therefore  that  a  is  not  an 
infinity  of  the  corresponding  set  or  sets  of  branches  w.  But  as  the  revolving 
line  has  to  change  its  direction  from  A^y  to  some  direction  through  Ao, 
there  must  evidently  be  some  part  or  parts  of  the  broken  line  which  have 
their  tangential  coordinate  positive,  implying  therefore  that  a  is  an  infinity 
of  the  corresponding  set  or  sets  of  branches. 

Moreover,  the  point  a  is,  by  hypothesis,  an  accidental  singularity  of  at 
least  one  of  the  coefficients,  and  it  has  been  supposed  to  be  an  essential 
singularity  of  none  of  them;  hence  the  points  Aq,  A^,  ...,  An  are  all  in  the 
finite  part  of  the  plane.  And  as  no  two  of  their  abscissae  are  equal,  no  line 
joining  two  of  them  can  be  parallel  to  the  axis  of  y,  that  is,  the  inclination 
of  the  broken  line  is  never  ^-tt  and  therefore  the  tangential  coordinate  is 
finite,  that  is,  the  order  of  the  infinity  for  the  branches  is  finite  for  any 
accidental  singularity  of  the  coefficients. 

If  the  singularity  at  a  be  essential  for  some  of  the  coefficients,  the 
corresponding  result  can  be  inferred  by  passing  to  the  limit  which  is 
obtained  by  making  the  corresponding  value  or  values  of  m  infinite.  In 
that  case  the  corresponding  points  A  move  to  infinity  and  then  parts  of  the 
broken  line  pass  through  A^  (which  is  always  on  the  axis  of  x)  parallel  to 
the  axis  of  y,  that  is,  the  tangential  coordinate  is  infinite  and  the  order  of 

13—2 


196  INFINITIES  [95. 

the  infinity  at  a  for  the  corresponding  branches  is  also  infinite.  The  point  is 
then  an  essential  singularity  (and  it  may  be  also  a  branch-point). 

It  has  been  assumed  implicitly  that  the  singularity  is  at  a  finite  point  in 
the  ^^-plane ;  if,  however,  it  be  at  oo ,  we  can,  by  using  the  transformation 
zz'  =  1  and  discussing  as  above  the  function  in  the  vicinity  of  the  origin, 
obtain  the  relation  of  the  singularity  to  the  various  branches.  We  thus 
have  the  further  proposition : — 

The  order  of  the  infinity  of  a  branch  of  an  algebraical  function  at  a 
singulanty  of  a  coefficient  of  the  equation,  which  determines  the  function,  is 
finite  or  infinite  according  as  the  singularity  is  accidental  or  essential. 

If  the  coefficients  Fi  of  the  equation  be  holomorphic  functions,  then 
^  =  00  is  their  only  singularity  and  it  is  consequently  the  only  infinity  for 
branches  of  the  function.  If  some  of  or  all  the  coefficients  Fi  be  mero- 
morphic  functions,  the  singularities  of  the  coefficients  are  the  zeros  of 
the  denominators  and,  possibly,  z—  ao;  and,  if  the  functions  be  rational, 
all  such  singularities  are  accidental.  In  that  case,  the  equation  can  be 
modified  to 

h,  {z)  w'^  +  h^  (z)  w'^-i  +  h^  (z)  io«-2  +  . . .  =  0, 

where  Ao(^)  is  the  least  common  multiple  of  all  the  denominators  of  the 
functions  F^.  The  preceding  results  therefore  lead  to  the  more  limited 
theorem : — 

When  a  function  w  is  determined  by  an  algebraical  equation  the  coefficients 
of  which  are  holomorphic  functions  of  z,  then  each  of  the  zeros  of  the  coefficient 
of  the  highest  power  of  w  is  an  infinity  of  some  of  {and  it  may  be  of  all)  the 
branches  of  the  function  w,  each  such  infinity  being  of  finite  order.  The  point 
z=  CO  may  also  be  an  infinity  of  the  function  w ;  the  order  of  that  infinity  is 
finite  or  infinite  according  as  z=  <X)  is  an  accidental  or  an  essential  singularity 
of  any  of  the  coefficients. 

It  will  be  noticed  that  no  precise  determination  of  the  forms  of  the 
branches  w  at  an  infinity  has  been  made.  The  determination  has,  however, 
only  been  deferred :  the  infinities  of  the  branches  for  a  singularity  of  the 
coefficients  are  usually  associated  with  a  branch-point  of  the  function,  and 
therefore  the  relations  of  the  branches  at  such  a  point  will  be  of  a  general 
character  independent  of  the  fact  that  the  point  is  an  infinity. 

If,  however,  in  any  case  a  singularity  of  a  coefficient  should  prove  to  be, 
not  a  branch-point  of  w  but  only  a  regular  singularity^  then  in  the  vicinity  of 
that  point  the  branch  of  w  is  a  uniform  function.  A  necessary  (but  not  suffi- 
cient) condition  for  uniformity  is  that  {mn-r  —  nins)  -^{s  —  r)  be  an  integer. 

Note.  The  preceding  method  can  be  applied  to  determine  the  leading 
terms  of  the  branches  in  the  vicinity  of  a  point  a  which  is  an  ordinary  point 
for  each  of  the  coefficients  F. 


96.]  BRANCH-POINTS  197 

96.  There  remains  therefore  the  consideration  of  the  branch-points  of  a 
function  determined  by  an  algebraical  equation. 

The  characteristic  property  of  a  branch-point  is  the  equality  of  branches 
of  the  function  for  the  associated  value  of  the  variable,  coupled  with  the 
interchange  of  some  of  (or  all)  the  equal  branches  after  description  by  the 
variable  of  a  small  contour  enclosing  the  point. 

So  far  as  concerns  the  first  part,  the  general  indication  of  the  form  of  the 
value  has  already  (§  93)  been  given.  The  points,  for  which  values  of  tu 
determined  as  a  function  of  z  by  the  equation 

are  equal,  are  determined  by  the  solution  of  this  equation  treated  simul- 
taneously with 

df(w,  z)_ 

.    aw      ~     ' 

and  when  a  point  z  is  thus  determined,  the  corresponding  values  of  w,  which 
are  equal  there,  are  obtained  by  substituting  that  value  of  z  and  taking  M, 

the  greatest  common  measure  of  /  and  —- .     The  factors  of  M  then  lead  to 

the  value  or  the  values  of  w  at  the  point ;  the  index  m  of  a  linear  factor 
gives  at  the  point  the  multiplicity  of  the  value  which  it  determines,  and 
shews  that  m+1  values  of  w  have  a  common  value  there,  though  they  are 
distinct  at  infinitesimal  distances  from  the  point.  Values  of  w,  determined 
by/=0  but  not  occurring  in  a  factor  oi M,  are  isolated  values;  each  of  them 
determines  a  branch  that  is  uniform  at  the  point. 

Let  z  =  a,  w  =  a.  be  a  value  of  z  and  a  value  of  w  thus  obtained ;  and 
suppose  that  m  is  the  number  of  values  of  w  that  are  equal  to  one  another. 
The  point  ^  =  a  is  not  a  branch-point  unless  some  interchange  among  the 
m  values  of  w  is  effected  by  a  small  circuit  round  a;  and  it  is  therefore 
necessary  to  investigate  the  values  of  the  branches*  in  the  vicinity  of  z  =  a. 

Let  iu  =  a.-\-w',  z  =  a  +  z' ;  then  we  have 

/(a-t-w,  a-l-/)  =  0, 

that  is,  on  the  supposition  that/(w,  z)  has  been  freed  firom  fractions, 

/(a,  a)  +  X^ArsZ''w'  =  0, 

r,s 

so  that,  since  a  is  a  value  of  w  corresponding  to  the  value  a  of  z,  we  have 
w'  and  z'  connected  by  the  relation 

r   s 

*  The  following  investigations  are  founded  on  the  researches  of  Puiseux  on  algebraic 
functions;  they  are  contained  in  two  memoirs,  Liouville,  1"  Ser.,  t.  xv,  (1850),  pp.  365 — 480, 
ih.,  t.  xvi,  (1851),  pp.  228—240.  See  also  the  chapters  on  algebraic  functions,  pp.  19—76, 
in  the  second  edition  of  Briot  and  Bouquet's  Theorie  des  fonctions  elliptiques. 


198  BRANCH-POINTS  [96. 

When  z'  is  0,  the  zero  value  of  w'  must  occur  m  times,  since  a  is  a  root 
m  times  repeated;  hence  there  are  terms  in  the  foregoing  equation  inde- 
pendent of  /,  and  the  term  of  lowest  index  among  them  is  w'"^.  Also  when 
w  =^,  z'=0  is  a  possible  root ;  hence  there  must  be  a  term  or  terms 
independent  of  w'  in  the  equation. 

First,  suppose  that  the  lowest  power  of  /  among  the  terms  independent 
of  w'  is  the  first.     The  equation  has  the  form 

Az'  +  higher  powers  of  z' 
+  Bw'"^  -H  higher  powers  of  w' 
-\-  terms  involving  z  and  w  =  0, 

where  A  is  the  value  of       \         for  w  =  a,  z  =  a.     Let  /  =  K"^,  w'  =  vt',  the 

oz 

last  form  changes  to 

{A  +  Bv'^)  ^™  -1-  terms  with  ^"^+1  as  a  factor  =  0  ; 

and  therefore  A  4-  Bv^  +  terms  involving  ^  =  0. 

Hence  in  the  immediate  vicinity  of  ^r  =  a,  that  is,  of  ^  =  0,  we  have 

A  +  Bv"^  =  0. 

Neither  A  nor  B  is  zero,  so  that  all  the  m  values  of  v  are  finite.  Let  them 
be  Vi,...,  Vm,  so  arranged  that  their  arguments  increase  by  27r/m  through 
the  succession.     The  corresponding  values  of  w'  are 

^VrZ'"^, 

for  r=l, ...,  m.  Now  a  ^^-circuit  round  a,  that  is,  a  /-circuit  round  its 
origin,  increases  the  argument  of  z'  by  27r;  hence  after  such  a  circuit,  we 

1^     27n  1 

have  the  new  value  of  w,/  as  Vrz''^  e'^,  that  is,  it  is  Vr+iz'"^  which  is  the  value 
of  wV+i-  Hence  the  set  of  values  w\,  w\,...,  w'^  form  a  complete  set  of 
interchangeable  values  in  their  cyclical  succession;  all  the  m  values,  which 
are  equal  at  a,  form  a  single  cycle  and  the  point  is  a  branch-point. 

Next,  suppose  that  the  lowest  poM^er  of  z'  among  the  terms  independent 
of  w'  is  z'^,  where  I  >1.     The  equation  now  has  the  form 
0  =  Az'^  +  higher  powers  of  / 
+  Bw  ^  +  higher  powers  of  w 

l-l  m-\ 
+    S       S    ArsZ'^'w'+l.^GrsZ'^w'', 

where  in  the  last  summation  r  and  s  are  not  zero  and  in  every  term  either 
(i),  r  is  equal  to  or  greater  than  I  or  (ii),  s  is  equal  to  or  greater  than  m 
or  (iii),  both  (i)  and  (ii)  are  satisfied.     As  only  terms  of  the  lowest  orders 


96.] 


BRANCH-POINTS 


199 


Fig.  21. 


need  be  retained  for  the  present  purpose,  which  is  the  derivation  of  the  first 
term  of  w'  in  its  expansion  in  powers  of  z ,  we  may  use  the  foregoing  equation 
in  the  form 

r=l  s=l 

To  obtain  this  first  term  we  proceed  in  a  manner  similar  to  that  in  §  95  *. 
Points  Aq,  ...,  Arn  are  taken  in  a  plane 
referred  to  rectangular  axes  having  as  co- 
ordinates 0,1; ... ;  s,r ;...',  m,-0  respectively. 
A  line  is  taken  through  Am  and  is  made  to 
turn  round  A^  from  the  position  AmO  until 
it  first  meets  one  of  the  other  points ;  then 
round  the  last  point  which  lies  in  this 
direction,  say  round  Aj,  until  it  first  meets 
another ;  and  so  on. 

Any  line  through  Ai  (the  point  Si,  r^)  is 

of  the  form 

y-ri=  -\(x-  Si). 

The  intercept  on  the  axis  of  ^'-indices  is  Xsi  +  Vi,  that  is,  the  order  of  the 
term  involving  Ar.g.  for  a  substitution  w'  =  vz'\  The  perpendicular  from  the 
origin  for  a  line  through  A^  and  Aj  is  less  than  for  any  parallel  line  through 
other  points  with  the  same  inclination ;  and,  as  this  perpendicular  is 

{\Si  +  n)  (1  +  x^y-', 

it  follows  that,  for  the  particular  substitution  w'  =  vz'^,  the  terms  correspond- 
ing to  the  points  lying  on  the  line  with  coordinate  \  are  the  terms  of  lowest 
order,  and  consequently  they  are  the  terms  which  give  the  initial  terms  for 
the  associated  set  of  quantities  w'. 

Evidently,  from  the  indices  retained  in  the  equation,  the  quantities  \ 
for  the  various  pieces  of  the  broken  line  from  A^,  to  A^^  are  positive  and 
finite. 

Consider  the  first  piece,  from  A^  to  A^  say;  then  taking  the  value  of  \  for 
that  piece  as  fx^,  so  that  we  write  v-^z*^^  as  the  first  term  of  w ,  we  have  as  the 
set  of  terms  involving  the  lowest  indices 

Bw'"^  +  ll^ArsZ^'w'  -f-  Ar.,z"iW''3, 

Sj  being  the  smallest  value  of  s  retained ;  and  then 


so  that 


r 


n- 


H-i  = 


m  —  s     m  —  Sj 


*  Eeference  in  this  connection  may  be  made  to  Chrystal's  Algebra,  cb.  xxx.,  -with  great 
advantage,  as  well  as  the  authorities  quoted  on  p.  197,  note. 


200  GROUPING   OF   BRANCHES  [96. 

Let  piq  be  the  equivalent  value  of  /^i  as  the  fraction  in  its  lowest  terms ;  and 

write  /=^?.     Then  iv'  =  v^z' ^  =  v-^^p  ;  all  the  terms  except  the  above  group 
are  of  order  >mp,  and -therefore  the  equation  leads  after  division  by  ^"^Pv^jto 

an  equation  which  determines  m  -  Sj  values  for  v-^,  and  therefore  the  initial 
terms  of  m  —  Sj  of  the  iw-branches. 

Consider  now  the  second  piece,  from  A^  to  Ai  say ;  then  taking  the  value 
of  \  for  that  piece  as  jx^,  so  that  we  write  Va^'"^  as  the  first  term  of  w',  we 
have  as  the  set  of  terms  involving  the  lowest  indices  for  this  value  of  yttg 

Ar  s  Z^'w'^i  +  SS^rs^'V^  +  Ar.,z"''W% 

J    3  *    ' 

where  s,-  is  the  smallest  value  of  s  retained.     Then 
Sjix^  +  Vj  =  SjX2  +  r  =  Sifi2  +  n. 
Proceeding  exactly  as  before,  we  find 

Ar.s.Vi-'i  +  t^ArsV^''  +  A  ...s.  =  0 

as  the  equation  determining  Sj  —  si  values  for  v^,  and  therefore  the  initial 
terms  of  Sj  —  Si  of  the  ^f;-branches. 

And  so  on,  until  all  the  pieces  of  the  line  are  used ;  the  initial  terms  of 
all  the  w-branches  are  thus  far  determined  in  groups  connected  with  the 
various  pieces  of  the  line  A,nAjAi...A^.  By  means  of  these  initial  terms, 
the  m  branches  can  be  arranged  for  their  interchanges,  by  the  description  of 
a  small  circuit  round  the  branch-point,  according  to  the  following  theorem  : — 

Each  group  can  he  resolved  into  systems,  the  'members  of  each  of  which  are 
cyclically  interchangeable. 

It  will  be  sufficient  to  prove  this  theorem  for  a  single  group,  say  the 
group  determined  by  the  first  piece  of  broken  line :  the  argument  is 
general. 

Since  ^  is  the  equivalent  of  and  of  — ^ —  and  since  Sj  <  s,  we  have 

q  ^  m  —  s  m  —Sj  ■' 

m  —  s  =  kq,         m  —  Sj  =  kjq,         kj  >  k ; 
and  then  the  equation  which  determines  v^  is 

Bv^hi  +  ^Arsvfi-^^  3  +  Ar.s.  =  0, 
that  is,  an  equation  of  degree  kj  in  Vj^  as  its  variable.     Let  U  be  any  root  of 
it ;  then  the  corresponding  values  of  t'l  are  the  values  of  C/"?.     Suppose  these 


.1 
possible,  because  p  is  prime  to  q.     Then  the  q  values  of  tv',  being  the  values 


q  values  to  be  arranged  so  that  the  arguments  increase  by  27r-,  which  is 

2 


of  ViZ'l^^ 

are 

p 

p 

p 

ViiZ'l, 

VizZ'l, 

VnZ'i, 

9  6. J  GROUPING   OF   BRANCHES  201 

where  v^^,  is  that  value  of  U^  which  has   ^^  for  its  argument.     A  circuit 

q  ° 

round  the  /-origin  evidently  increases  the  argument  of  any  one  of  these 

w'-values  by  ^Trp/g,  that  is,  it  changes  it  into  the  value  next  in  the  succession ; 

and  so  the  set  of  q  values  is  a  system  the  members  of  which  are  cyclically 

interchangeable. 

This  holds  for  each  value  of  U  derived  from  the  above  equation ;  so  that 
the  whole  set  of  m  —  Sj  branches  are  resolved  into  %  systems,  each  containing 
q  members  with  the  assigned  properties. 

It  is  assumed  that  the  above  equation  of  order  kj  in  v^^  has  its  roots  unequal. 
If,  however,  it  should  have  equal  roots,  it  must  be  discussed  ab  initio  by  a 
method  similar  to  that  for  the  general  equation;  as  the  order  kj  (being  a 
factor  of  m  -  Sj)  is  less  than  m,  the  discussion  will  be  shorter  and  simpler, 
and  will  ultimately  depend  on  equations  with  unequal  roots  as  in  the  case 
above  supposed. 

It  may  happen  that  some  of  the  quantities  /j,  are  integers,  so  that  the 
coiTesponding  integers  q  are  unity :  a  number  of  the  branches  would  then  be 
uniform  at  the  point. 

It  thus  appears  that  z  =  a  is  a  branch-point  and  that,  under  the  present 
circumstances,  the  m  branches  of  the  function  can  be  arranged  in  systems, 
the  members  of  each  one  of  which  are  cyclically  interchangeable. 

Lastly  it  has  been  tacitly  assumed  in  what  precedes  that  the  common 
value  of  w  for  the  branch-point  is  finite.  If  it  be  infinite,  this  infinite  value 
can,  by  §  95,  arise  only  out  of  singularities  of  the  coefficients  of  the  equation : 
and  there  is  therefore  a  reversion  to  the  discussion  of  §§  95,  96.  The  dis- 
tribution of  the  various  branches  into  cyclical  systems  can  be  carried  out 
exactly  as  above. 

Another  method  of  proceeding  for  these  infinities  would  be  to  take 
luw'  =  1,  z  =  c  +  z' ;  but  this  method  has  no  substantial  advantage  over  the 
earlier  one  and,  indeed,  it  is  easy  to  see  that  there  is  no  substantial 
difference  between  them. 

Note.  In  the  first  case  considered,  a  single  transformation  of  the  variables 
represented  by  /  =  ^^",  w'  =  v^,  was  sufficient  to  discriminate  among  the  m 
branches. 

In  the  second  case,  the  number  of  different  directions  in  the  broken  'line 
of  fig.  21  is  finite  (^  m) ;  to  each  such  direction  there  corresponds  a  trans- 
formation of  the  variables  which  leads  to  a  discrimination  among  one  of  the 
groups  out  of  the  m  branches,  and  therefore  the  whole  number  of  trans- 
formations needed  to  discriminate  among  the  m  branches  is  finite. 

If  the  m  branches  are  infinite  at  the  point,  the  corresponding  analysis 
shews  that  the  whole  number  of  transformations  needed  to  discriminate  among 
those  m  branches  is  finite. 


202  EXAMPLES  [96. 

Moreover  m  is  finite,  being  ^  n ;  hence  the  various  branches  of  the 
function  w  are  discriminated,  at  a  branch-point,  by  a  finite  number  of  trans- 
formations. 

Ux.  1.     As  an  example,  consider  the  function  determined  by  the  equation 

8ziv^+{l-z)(3w  +  l)  =  0. 
The  equation  determining  the  values  of  z  which  give  equ^l  roots  for  w  is 

8^(2-1)2  =  4(2-1)3, 

SO  that  the  values  are  2«=1  (repeated)  and  z=  —  l. 

When  z  —  1,  then  w=0,  occurring  thrice;  and  if  z=l+z',  then 

8w'^=^z\ 

that  IS,  w  =  ^2  3. 

The  three  values  are  branches  of  one  system  in  cyclical  order  for  a  circuit  round  2=1. 

When  2=  —  1,  the  equation  for  w  is 

4w3_3^_l=0, 

that  is,  (w-l)(2w  + 1)2  =  0, 

so  that  w=l,  or  w=  -\,  occurring  twice. 

For  the  former  of  these  we  easily  find  that,  for  2  =  —  1  +  2',  the  value  of  w  is 
14. 2  2'^ ^  an  isolated  branch  as  is  to  be  expected,  for  the  value  1  is  not  repeated. 

For  the  latter  we  take  w—  —^-\-w\  and  find 

^'^  =  2^^'  + » 

so  that  the  two  branches  are      ^ 

l_ 

2^6' 


W=-^  +  ^;-j^z'^  +  . 


w=-^- 


J-2'U 

2V6      +' 


,   and  they  are  cyclically  interchangeable  for  a  small  circuit  round  2=-l. 

These  are  the  finite  values  of  w  at  branch-points.  For  the  infinities  of  w,  which  may 
arise  in  connection  with  the  singularities  of  the  coefficients,  we  take  the  zeros  of  the 
coefficient  of  the  highest  power  of  w  in  the  integral  equation,  viz.,  2=0,  which  is  thus  the 

only  infinity  of  w.     To  find  its  order  we  take  w=z-"'f{z)=yz~'^-\- ,  where  y  is  a 

constant  and  f{z)  is  finite  for  2  =  0;  and  then  we  have 

-^ -y3  + =  3y2-'^-H -HI. 

1  —2 

Thus  l-3n=-n, 

provided  both  of  them  be  negative;  the  equality  gives  n  =  ^  and  satisfies  the  condition. 
And  8y3=  -3y.  Of  these  values  one  is  zero,  and  gives  a  branch  of  the  function  without 
an  infinity;  the  other  two  are  ±|>/^  and  they  give  the  initial  term  of  the  two 
branches  of  w,  which  have  an  infinity  of  order  --^  at  the  origin  and  are  cyclically 
interchangeable  for  a  small  circuit  round  it.  The  three  values  of  w  for  infinitesimal 
values  of  2  are 


/3._i      1       7         /3.1       4  275         /3     ^        4    ^ 

«'!=      Vs''      +6~T8   \/8''"'8T'-1944   V8'''~729^  + 

/3  .  _i      1       7         /3  .  i      4         275        /3     | 
^^=-\/8''       +6  +  T8   VS*^    -81^  +  1944   \/ s'^    " 


^22-. 

729 


18  8     , 

^^=-3  +  81^  +  729^  +■ 


96.]  ALGEBEAIC    FUNCTIONS  203 

The  first  two  of  these  form  the  system  for  the  branch-point  at  the  origin,  which  is  neither 
an  infinity  nor  a  critical  point  for  the  third  branch  of  the  function. 

Ex.  2.     Obtain  the  branch-points  of  the  functions  which  are  defined  by  the  following 
equations,  and  determine  the  cyclical  systems  at  the  branch-points : — 


(i) 

w^  —  tv+z=0; 

(ii) 

w3_3^2  +  26  =  0.: 

(iii) 

w3-3iv  +  2z^{2-z^)  =  0; 

(iv) 

iv^  —  Szw+z^  =  0; 

(V) 

iv^-{l-z^)wi-^^z^{l- 

-z^Y 

=0.  (Briot  and  Bouquet.) 

Also  discuss  the  branches,  in  the  vicinity  of  ^=0  and  of  3  =  oo ,  of  the  functions  defined 
by  the  following  equations : — 

( vi)     aw'  ■^hitT'z  +  mo*z*  +  dw^^  +  ewz''  +fz^ +fftv^  +  hiu^z^  +  kz^^ = 0 ; 
( vii)     '?<;™2™  +  w»  +  s'» = 0. 

97.     Having  shewn  how  to  discriminate  at  any  point  among  the  various 
branches  of  the  algebraic  function  defined  by  the  equation 

/(w,  z)  =  ho  (z)  w'^+ih  (z)  W^-^  +  h^  (z)  W'-^  -^  . . .  =  0, 
where  the  quantities  ho(z),  h-^{z),  h^iz),  ...   are   holomorphic  functions,   we 
proceed  to  indicate  the  character  of  the  various  branches  near  the  point.    After 
the  preceding  discussions,  it  will  be  sufficient  to  consider  only  finite  values 
of  z ;  the  consideration  of  infinite  values  can  be  obtained  through  the  zero 

values  of  z',  where  —  is  substituted  for  z.  It  is  only  for  zeros  of  ho  (z)  that 
an  infinite  value  (or  several  infinite  values)  of  w  can  arise  :  they  can  be 
discussed  through  the  zero  values  of  w ,  where  — ,  is  substituted  for  w. 

Accordingly,  let  a  denote  a  finite  value  of  z,  and  let  a  denote  a  finite 
value  of  w  for  z  =  a,  where  a  may  be  a  simple  root  or  multiple  root  of 
y  (a,  a)  =  0.  Take  w  =  a  +  y,  2^  =  a  -1-  ^,  so  as  to  consider  some  vicinity  of  the 
point  a  and  the  character  of  w  in  that  vicinity ;  and  let 
f{w,  z)  =/(«  +  y,a  +  w)  =  F(y,a;), 
where  jP  is  a  polynomial  in  y  of  degree  not  greater  than  n,  and  the  coefficients 
are  holomorphic  functions  of  oc  which  are  polynomials  when  all  the  coefficients 
ho,  hi,  ...  are  polynomials.  We  have  F (0,  0)=0,  so  that  there  is  no  term 
free  from  cc  and  y  in  F(y,  x).  Also  F  (y,  0)  does  not  vanish  for  all  values  of 
y ;  for  that  would  imply  that  some  integral  power  of  a;  is  a  factor  of  F(y,  x) 
and  therefore  that  some  integral  power  of  ^  -  a  is  a  factor  of  f{w,  z),  which 
is  not  the  case.  Hence  there  is  at  least  one  term  in  the  polynomial  F {y,  x), 
which  has  a  constant  for  its  coefficient,  and  there  may  be  more  than  one 
such  term ;  let  the  term  of  lowest  order  in  y  be  By"^,  and  let  the  aggregate 
of  such  terms  be  denoted  by  Fo{y).  Denoting  the  rest  by  Fi{y,  x),  where 
JPi  is  a  polynomial  in  y  that  has  holomorphic  functions  of  x  for  its  coefficients, 

we  have 

F{y,x)  =  F,{y)-F,{y,x)- 


204  THEOREM    OF  [97. 

clearly  F^{y,  x)  vanishes  when  x  =  Q  for  all  values  of  y,  in  any  vicinity  of 
2/ =  0.  Hence*  we  can  choose  a  region  in  the  vicinity  of  y  =  0,  x  =  0, 
such  that 

!^ol>|i^i|; 

but  as  ^0  vanishes  when  y  =  0,  there  may  be  some  limit  oi  \y\  other  than 
zero,  at  and  below  which  the  inequality  does  not  hold.  Accordingly,  assume 
as  the  range  for  the  inequality 

\po\<\y\<P>         \cc\<r. 
For  such  values  we  have,  on  taking  logarithmic  derivatives  of  the  equation 

the  relation 

Fd^~Fo  dy      dx^=,\Fo'- 
Since  Fo  is  a  polynomial  in  y  that  is  divisible  by  y"^,  we  have 

1   dFo     m      ^  ,  . 


where  G  is  a,  converging  series  of  integral  powers.     Similarly 

J?  K  cc 

—  =lGy      V-m^+l^ 

where  the  quantities  (ta,^^  are  converging  series  of  integral  powers  of  x,  each 

of  them  vanishing  with  x.     As  the  series  ^  r-  Wr  converges  uniformly,  we 

may  gather  together  the  various  terms  that  involve  the  same  power  of  y ; 
and  we  then  have 

00        1         Zp\  00 

where  each  of  the  coefficients  Op  is  a  converging  power-series  in  x  which 
vanishes  with  x.     Thus 

IdF  m  ^  ^,  .  d  Z  n  V 
-n^  =  —  +  Gr{y)-^  2  G^yP, 
Fdy      y        ^^'     a2/p=-oo   "^"^ 

where  the  only  term  on  the  right-hand  side  in  y-'^  is  —  . 

Now  let  %,  ...,  »7s  denote  the  zeros  of  F{y,  k),  for  values  of  y  such  that 
\y\<p  and  for  a  parametric  value  k  oi  x  such  that  |  /c  |  <  r :  it  might  be  that 
there  are  no  such  zeros  (though  this  will  be  seen  not  to  be  the  case) :  repeated 
zeros  are  given  by  repetition  in  the  quantities  rj.     Then 

1  dF{y,  k)      4       1 
F      dy         i=\  y-vi 

*  What  follows  is  a  special  case  of  an  important  theorem,  elite  to  Weierstrass,  Ges.  Werke, 
t.  ii,  p.  135. 


97.]  WEIERSTRASS  205 

is  finite  for  all  values  of  y  within  the  range,  and  therefore  it  can  be  expanded 
in  a  converging  series  of  positive  powers,  so  that 

Now  choose  values  of  y,  still  such  that  \y\<  p,  and  also  such  that  they  give 
moduli  greater  than  the  greatest  of  the  quantities ,  1 7/2 1 ;  the  fractions  on  the 
right-hand  side  can  be  expanded  in  descending  powers  of  y,  and  we  have 

1  dF(y,  k)      „  ,    ,       s       ^  „ 
F       dy  ^^'     y     ^=1  ^"^ 

where  S^  =  7)^1^ +  7}^'^+  ...  +  r^f. 

The  parametric  value  k  in  this  expansion  can  be  replaced  by  x ;  and  thus 
comparing  the  two  expansions  for  ^  -^  ,  we  have 

s  =  m,         S^  =  /j,G-fj.. 

The  first  of  these  results  shews  that  there  are  m  roots  of  F  within  the 
range.  The  second  of  them  expresses  the  sums  of  the  positive  powers  of 
7)j,  ...,  Tji  as  converging  series  of  positive  powers  of  w  which  vanish  with  as ; 
hence  the  symmetric  integral  functions  of  t^j,  ..,,  rn  are  regular  functions  of  x 
in  the  vicinity  of  iz;  =  0  and  vanish  with  x.     Let 

g(y,^)  =  (y-Vi)(y-V2)--.{y-vi) 
=  f''  +  9iy'"'~'  +  ---  +  91, 

where  gi  are  regular  functions  of  x  and  vanish  with  x. 
A  further  comparison  of  the  expansions  shews  that 

P(y)  =  G(y)-i(q+\)G,+,y^ 

q  =  0 

where  T  (y,  x)  is  a  regular  function  of  y  and  x,  given  by 


Hence 


and  therefore 


r(y,x)=      G(y)dy-tG,+,y'^+\ 

J  0  q  =  0 

1  dF     4        1  d  ^^-       . 

^^=2    +—T{y,x), 

F  dy      1=1  y-vi      oy 


where  D^  is  a  quantity  independent  of  y.     Now  when  x  is  zero,  U  is  B ;  hence 
generally  • 

11=  B(l  +  positive  powers  of  x) 

=  Be^, 


206  AN   ALGEBRAIC    FUNCTION  [97. 

where  ^  is  a  regular  function  of  x  vanishing  with  x.     Writing  Q{y,  x)  for 
r  {y,  x)  +  ^,  where  G  (0,  0)  =  0,  we  have 

F=Bg{y,  x)  e^^V'^^ 

with  the  defined  significance  of  g  (y,  x)  and  G(y,  x). 

Our  immediate  purpose  is  with  such  values  of  y,  being  functions  of  x,  as 
make  F  vanish  in  the  region  considered.  Clearly  the  exponential  term  does 
not  vanish  ;  and  therefore  we  have  the  values  of  y  given  by 

g(y,  x)  =  y'^  +  g^y'^-'  +  g.y'^-'-h...  +gn,  =  0, 

where  gi,  g<i,  ...,  gm  are  regular  functions  of  x  that  vanish  with  x. 

Case  1.  The  simplest  case  arises  when  m  =  1  ;  the  root  a  is  then  a  simple 
root  of /(a,  a)  =  0,  and  we  have 

that  is, 

to  -  a  =  y  =  -  g^=  Q  (z  -  a), 

or  in  the  vicinity  of  the  point  a,  the  branch  associated  with  the  simple  root 
of  /(a,  a)  =  0  is  a  regular  function  of  2  —  a. 

The  same  result  holds  for  each  simple  root  a  of  the  equation  /(a,  a)  =  0. 

Case  2.  Let  m  >  1,  so  that  the  root  a  is  a  multiple  root  of  /(a,  a)  =  0, 
and  z  =  a  may  be  (and  generally  is)  a  branch-point.     The  equation 

g(y,x)  =  y-  +  g^y-^-'  +  g,y^-^  +  ...+g^  =  0 

determines  in  branches.     By  |  96  these  branches  can  be  arranged  in  groups, 

p 
each  group  corresponding  to  a  particular  order  j  i/ 1  x  |  ^r  | «  for  sufficiently  small 

values  of\y\  and  \x\,  and  the  order  being  determined  by  a  portion  of  a  broken 

line  in  a  Puiseux  diagram. 

Thus  for  the  first  portion  of  the  line,  take  x  =^i,y=  v^p;  then  the  equation 
becomes  of  the  form 

V'"  +  iKrV''-'-  +  KsV^-'  +  ^P  {V,  0  =  0, 

where  P  {v,  ^)  is  a  regular  function  of  its  arguments.     When  ^=0,  we  have 

v^  +  2  a:,.  ■?;*■"'■  +  Kg  =  0, 

rejecting  the  zero  values  of  v.  If  v  =  Vi  be  a  simple  root  of  this  equation, 
then  in  the  earlier  equation  we  write  v  =  v^  +  u;  and  it  then  follows,  by 
Case  1  above,  that 

where  jR  is  a  regular  function  of  ^  that  vanishes  when  ^=0.  Accordingly 
for  every  simple  root  of  the  equation  in  v  when  ^  is  zero,  we  have 

z-a=^^,         ^v-a  =  ^P{v,  +  R(^)}, 


97.]  IS   ANALYTIC  207 

shewing  that  the  corresponding  branch  of  the  algebraic  function  is  a. uniform 

1 
function  of  (2  —  a)«.     When  q  is  1,  the  branch  is  a  regular  function  of  ^^  —  a. 
When  q>l,  there  is  a  system  of  roots  of  the  same  form. 

It  may  happen  that  Vi  is  a  multiple  root*  of 

V'  +  iKr-if-''  +  Ks  =  0. 

This  equation  is  of  degree  s,  being  less  than  771,  the  degree  of  the  original 
equation.  To  it  we  apply,  for  the  multiple  root,  the  preceding  process :  and 
so  gradually  reach  the  stage  in  which  each  of  the  branches  is  discriminated 
and  analytically  expressed. 

Similarly  for  the  remaining  portions  of  the  broken  line  in  the  Puiseux 
diagram  of  §  96. 

It  therefore  follows  that  all  the  branches  (if  the  branches  be  more  than 
one)  of  the  function,  defined  by  the  equation  f{w,  z)  =  0  and  acquiring  the 
value  a  when  z  =  a,  where  /(a,  a)  =  0,  can  be  represented  in  the  analytical 
form 

z-a  =  ^'i,         w-ci=^PS(^), 

where  S{^)  is  a  regular  function  of  its  argument  which  does  not  vanish  when 
^=0,  and  where  p,  q  are  positive  integers  not  necessarily  the  same  for  all 
the  branches.  (As  already  remarked,  we  have  assumed  a  and  a  to  be  finite. 
It  is  easy  to  see  that  for  an  infinite  value  of  w  when  z  =  a,  we  have  a  branch 
of  the  form 

where  p'  is  a  finite  integer;  and  similarly  for  infinite  values  of  z.)  Conse- 
quently the  function  defined  by  the  equation  f(w,  z)  =  0,  which  is  polynomial 
in  w  and  uniform  in  z,  has  tyi  branches  at  any  point  a,  each  of  the  branches 

being  expressible  as  a  uniform  analytic  function  of  (z  —  a)'^.  If  f(w,z)  is 
polynomial  in  5  as  well  as  in  w,  the  non-regular  points  of  the  branches  are 
poles  and  branch-points:  no  point  in  the  plane  is  an  essential  singularity  for 
any  branch. 

Corollary.  We  have  the  theorem,  originally  due  to  Cauchy,  as  an 
inference  from  the  whole  investigation: — 

The  roots  lu  of  an  equation  f{iv,z)  =  0,  which  is  polynomial  in  w  and 
uniform  in  z,  are  continuous  functions  of  z. 

It  follows  at  once  from  the  two'  relations 

*  Such  is  the  case  for  the  equation 

w^-15w*z-  2wh  +  15w%^  +  Qwz^  +  z^-z^  =  0. 


208  SIMPLE   BRANCH-POINTS  [97. 

Note.     If  Vi  be  a  multiple  root  of  its  equation,  the  form 

is  still  valid :  but  p  and  q  are  then  not  necessarily  prime  to  each  other.  (The 
equation  represented  by 

is  an  example.)  The  condition  is  that,  if  the  indices  in  the  expression  for 
w  —  a  have  a  common  factor  y,  then  y  is  not  a  factor  of  q. 

98.  There  is  one  case  of  considerable  importance  which,  though  limited 
in  character,  is  made  the  basis  of  Clebsch  and  Gordan's  investigations*  in  the 
theory  of  Abelian  functions — the  results  being,  of  course,  restricted  by  the 
initial  limitations.  It  is  assumed  that  all  the  branch-points  are  simple,  that 
is,  are  such  that  only  one  pair  of  branches  of  w  are  interchanged  by  a  circuit 
of  the  variable  round  the  point ;  and  it  is  assumed  that  the  equation  /=  0  is 
pol3momial  not  merely  in  w  but  also  in  z.  The  equation  /=  0  can  then  be 
regarded  as  the  generalised  form  of  the  equation  of  a  curve  of  the  nth.  order, 
the  generalisation  consisting  in  replacing  the  usual  coordinates  by  complex 
variables ;  and  it  is  further  assumed,  in  order  to  simplify  the  analysis,  that  all 
the  multiple  points  on  the  curve  are  (real  or  imaginary)  double-points.  But, 
even  with  the  limitations,  the  results  are  of  great  value  in  themselves ;  and 
the  theory  of  birational  transformation  (§§  245 — 252)  brings  them  within  the 
range  of  unrestricted  generality.  It  is  therefore  desirable  to  establish  the 
results  that  belong  to  the  present  section  of  the  subject. 

We  assume,  therefore,  that  the  branch-points  are  such  that  only  one 
pair  of  branches  of  w  are  interchanged  by  a  small  closed  circuit  round  any 
one  of  the  points.  The  branch-points  are  among  the  values  of  z  determined 
by  the  equations 

/./       N      f.  df{w,z) 

When  /"=  0  has  the  most  general  form  consistent  with  the  assigned 
limitations,  f(w,  z)  is  of  the  nth.  degree  in  z ;  the  values  of  z  are  determined 
by  the  eliminant  of  the  two  equations  which  is  of  degree  n  {n  —  1),  and  there 
are,  therefore,  n(n-  1)  values  of  z  which  must  be  examined. 

First,  suppose  that        „  ' — -    does   not   vanish    for   a   value   of  z,  thus 

obtained,  and  the  corresponding  value  of  lu;  then  we  have  the  first  case 
in  the  preceding  investigation.  And,  on  the  hypothesis  adopted  in  the 
present  instance,  ??z  =  2 ;    so  that  each  such  point  z  is  a  branch-point. 

*  Clebsch  und  Gordan,  Theorie  der  AheVschen  Fiinctionen,  (Leipzig,  Teubner,  1866).  It  will 
be  proved  hereafter  (§  252)  that  any  algebraical  equation  can  be  transformed  birationally  into  an 
equation  of  the  kind  indicated.  The  actual  transformations,  however,  tend  to  become  extremely 
complicated;  and,  in  particular  instances,  detailed  results  would  be  obtained  more  simply  by 
proceeding  directly  from  the  original  equation. 


98.]  SIMPLE   BRANCH-POINTS  209 

Next,  suppose  that       \         vanishes  for  some  of  the  n  {n  —  1)  values  of  z ; 

the  value  of  m  is  still  2,  owing  to  the  hypothesis.     The. case  will  now  be  still 

d^f(iu  z) 
further  limited  by  assuming  that     ^  )    '    '  does  not  vanish  for  the  value  of  z 

oz^ 

and  the  corresponding  value  of  w ;  and  thus  in  the  vicinity  of  ^^  =  a,  w  =  a  we 
have  an  equation 

0  =  Az'^  +  2Bz'w'  +  Cw'^  -t-  terms  of  the  third  degree  + , 

where  A,  B,  C  are  the  values  of  t—  ,  7~- ,  ~;  for  z  =  a,w  =  a. 

02^    ozotu    ovr 

\i  B"^  '^  A  C,  this  equation  leads  to  the  solution 

Ctu  +  Bz  oc  uniform  function  of  z'. 

The  point  z  =  a,  lu  =  a  is  not  a  branch-point ;  the  values  of  w,  equal  at  the  point, 
are  functionally  distinct.  Moreover,  such  a  point  z  occurs  doubly  in  the 
eliminant ;  so  that,  if  there  be  h  such  points,  they  account  for  28  in  the  eliminant 
of  degree  n{n  —  V);  and  therefore,  on  their  score,  the  number  n{n  —  l)  must 
be  diminished  by  2S.  The  case  is,  reverting  to  the  generalisation  of  the 
geometry,  that  of  a  double  point  where  the  tangents  are  not  coincident. 

If,  however,  5'-=  AG,  the  equation  leads  to  the  solution 

Ciu  +  Bz'  =  Lz''^  +  Mz'^  +  Nz'^  + 

The  point  z  =  a,  w  =  a.  is  a  point  where  the  two  values  of  z  interchange. 
Now  such  a  point  z  occurs  triply  in  the  eliminant ;  so  that,  if  there  be  k 
such  points,  they  account  for  3/c  of  the  degree  of  the  equation.  Each  of 
them  provides  only  one  branch-point,  and  the  aggregate  therefore  provides  k 
branch-points ;  hence,  in  counting  the  branch-points  of  this  type  as  derived 
through  the  degree  of  the  eliminant,  the  degree  must  be  diminished  by  2/c. 
The  case  is,  reverting  to  the  generalisation  of  the  geometry,  that  of  a  double 
point  (real  or  imaginary)  where  the  tangents  are  coincident. 

It  is  assumed  that  all  the  n{n  —  \)  points  z  are   accounted  for  under 

the  three  classes  considered.      Hence  the  number  of  branch-points  of  the 

equation  is 

0  =  7i(w-l)-28-2«, 

where  n  is  the  degree  of  the  equation,  S  is  the  number  of  double  points 
(in  the  generalised  geometrical  sense)  at  which  tangents  to  the  curve  do  not 
coincide,  and  k  is  the  number  of  double  points  at  which  tangents  to  the 
curve  do  coincide. 

And  at  each  of  these  branch-points,  O  in  number,  two  branches  of  the 
function  are  equal  and,  for  a  small  circuit  round  it,  interchange. 

F.  F.  14 


210  FUNCTIONS   POSSESSING  [99. 

99.  The  following  theorem  is  a  combined  converse  of  many  of  the 
theorems  which  have  been  proved : — 

A  function  w,  which  has  n  (and  only  n)  values  for  each  value  of  z,  and 
which  has  a  finite  number  of  infinities  and  of  branch-points  in  any  part  of  the 
plane,  is  a  root  of  an  equation  in  w  of  degree  n,  the  coefficients  of  which  are 
uniform  functions  of  z  in  that  part  of  the  plane. 

We  shall  first  prove  that  every  integral  symmetric  function  of  the  n 
values  is  a  uniform  function  in  the  part  of  the  plane  under  consideration. 

n 

Let  Sjc  denote  %  w/,  where  A;  is  a  positive  integer.     At  an  ordinary  point 

^  =  l 

of  the  plane,  S^  is  evidently  a  one-valued  function  and  that  value  is  finite  ; 
Sjc  is  continuous ;  and  therefore  the  function  8^  is  uniform  in  the  immediate 
vicinity  of  an  ordinary  point  of  the  plane. 

For  a  point  a,  which  is  a  branch-point  of  the  function  w,  we  know  that 
the  branches  can  be  arranged  in  cyclical  systems.  Let  w-^,  ...,  w^^  be  such  a 
system.  Then  these  branches  interchange  in  cyclical  order  for  a  description 
of  a  small  circuit  round  a ;  and,  \{  z  —  a  =  Z^,  it  is  known  (§  93)  that,  in  the 
vicinity  of  Z  =  0,  a  branch  w  is  a  uniform  function  of  Z,  say 


w 

-^V 

P{Z). 

Therefore 

w^ 

=«'&)+ 

Pu{Z), 

in  the 

vicinity 

of^ 

=  0; 

say 

w^  = 

=  ^k  + 

%    B]c,mZ~ 
m  =  l 

m  =  \ 

m 

Now  the  other  branches  of  the  function,  which  are  equal  at  a,  are  derivable 
from  any  one  of  them  by  taking  the  successive  values  which  that  one 
acquires  as  the  variable  describes  successive  circuits  round  a.  A  circuit 
of  w  round  a  changes  the  argument  oi  z  —  a  by  27r,  and  therefore  gives  Z 
reproduced  but  multiplied  by  a  factor  which  is  a  primitive  /^th  root  of  unity, 
say  by  a  factor  a;  a  second  circuit  will  reproduce  Z  with  a  factor  o? ;  and  so 
on.     Hence 

w.}  =  Aj,+  t  Bk,rna-^^Z-'^'+   2   Cfc,„,a'-Z^ 

m=l  m  =  l  ' 


w,+i^-=^^+  2  5jfc,„,a-™^-™+  2  G.^^oa-^Z^, 


99.]  A    FINITE    NUMBER   OF   BRANCHES  211 

and  therefore 

S  w/  =  fiAj,  +  S  Bj,,nZ-^  (1  +  «-'«  +  a-2"^  +  . . .  +  a-"''^+'«) 

»•  =  !  »j  =  l 

+  S  C;!;^^'*^  (1  +  a»^  +  a^'*^  +  . . .  +  a»"^-'"). 

in  =  l 

Now,  since  a  is  a  primitive  fxth  root  of  unity, 

1  +  a«  +  a^s-  +  ...  +  as<A^-i) 

is  zero  for  all  integral  values  of  s  which  are  not  integral  multiples  of  fi,  and  it 
is  fi  for  those  values  of  .<?  which  are  integral  values  of  /x ;  hence 

1     '^ 

-   twr^  =  Ak  +  Bj,^ ^Z-''  +  Bk, ,^Z--'-  4-  Bj,^ s^Z-'^  +  ... 

+  Gk,^Z>^  +  C,,,,Z^-^  +  Gk,,,Z'^'^  +  ... 

=Ak  +  B'k,,  {z  -  a)-i  +  B\,.  {z  -  a)-^  +  B\^^  {z  -  a)-'  +  ... 

+  C'k,  1  {z  -  a)  +  C"^,o  {z -  af  +  Cj,^,  {z-af+.... 

Hence  the  point  z  =  a  may  be  a  singularity,  of  S  w/  but  it  is  not  a  branch- 

r=l 

point  of  the  function ;  and  therefore  in  the  immediate  vicinity  of  ^^  =  a  the 

E,  .  . 

quantity  X  w,.*  is  a  uniform  function. 

The  point  a  is  an  essential  singularity  of  this  uniform  function,  if  the 
order  of  the  infinity  of  w  at  a  be  infinite  :  it  is  an  accidental  singularity,  if 
that  order  be  finite. 

This  result  is  evidently  valid  for  all  the  cyclical  systems  at  a,  as  well  as 
for  the  individual  branches  which  may  happen  to  be  one-valued  at  a.     Hence 

8]^,  being  the  sum  of  sums  of  the  form  2  wj'  each  of  which  is  a  uniform 

r  =  l 

function  of  z  in  the  vicinity  of  a,  is  itself  a  uniform  function  of  z  in  that 
vicinity.  Also  a  is  an  essential  singularity  of  8]c,  if  the  order  of  the  infinity 
Sit  z  =  a  for  any  one  of  the  branches  of  w  be  infinite ;  and  it  is  an  accidental 
singularity  of  ^S^^,  if  the  order  of  the  infinity  at  ^^  =  a  for  all  the  branches  of  w 
be  finite.  Lastly,  it  is  an  ordinary  point  of  Sjc,  if  there  be  no  branch  of  w  for 
which  it  is  an  infinity.     Similarly  for  each  of  the  branch-points. 

Again,  let  c  be  a  regular  singularity  of  any  one  (or  more)  of  the  branches 
of  w ;  then  c  is  a  regular  singularity  of  every  power  of  each  of  those  branches, 
the  singularities  being  simultaneously  accidental  or  simultaneously  essential. 
Hence  c  is  a  singularity  of  Sk'-  and  therefore  in  the  vicinity  of  c,  S^  is  a 
uniform  function,  having  c  for  an  accidental  singularity  if  it  be  so  for  each  of 
the  branches  w  affected  by  it,  and  having  c  for  an  essential  singularity  if  it 
be  so  for  any  one  of  the  branches  w. 

It  thus  appears  that  in  the  part  of  the  plane  under  consideration  the 
function  S^-is  one-valued ;  and  it  is  continuous  and  finite,  except  at  certain 

U—2 


212  FUNCTIONS   POSSESSING  [99. 

isolated  points  each  of  which  is  a  singularity.  It  is  therefore  a  uniform 
function  in  that  part  of  the  plane  ;  and  the  singularity  of  the  function  at  any 
point  is  essential,  if  the  order  of  the  infinity  for  any  one  of  the  branches  w 
at  that  point  be  infinite,  but  it  is  accidental,  if  the  order  of  the  infinity  for 
all  the  branches  w  there  be  finite.  And  the  number  of  these  singularities 
is  finite,  being  not  greater  than  the  combined  number  of  the  infinities  of  the 
function  w,  whether  regular  singularities  or  branch-points. 

Since  the  sums  of  the  Mh  powers  for  all  positive  values  of  the  integer  k 
are  uniform  functions,  and  since  any  integral  symmetric  function  of  the 
n  values  is  a  rational  integral  function  of  the  sums  of  the  powers,  it  follows 
that  any  integral  symmetric  function  of  the  n  values  is  a  uniform  function 
of  z  in  the  part  of  the  plane  under  consideration;  and  every  infinity  of  a 
branch  w  leads  to  a  singularity  of  the  symmetric  function,  which  is  essential 
or  accidental  according  as  the  orders  of  infinity  of  the  various  branches  are 
not  all  finite  or  are  all  finite. 

Since  w  has  n  (and  only  n)  values  Wy,  . . . ,  w^  for  each  value  oiz,  the  equation 
which  determines  w  is 

{W  —  Wi)  (W  —  W2)  ...{W  —  Wn)  =  0. 

The  coefficients  of  the  various  powers  of  w  are  symmetric  functions  of  the 
branches  Wi,  ...,Wn',  and  therefore  they  are  uniform  functions  of  z  in  the  part 
of  the  plane  under  consideration.  They  possess  a  finite  number  of  singularities, 
which  are  accidental  or  essential  according  to  the  character  of  the  infinities  of 
the  branches  at  the  same  points. 

Corollary.  If  all  the  infinities  of  the  branches  in  the  finite  part  of  the 
whole  plane  he  of  finite  order,  then  the  finite  singularities  of  all  the  coefficients 
of  the  powers  of  w  in  the  equation  satisfied  by  w  are  all  accidental ;  and  the 
coefiicients  themselves  then  take  the  form  of  a  quotient  of  an  integral  uniform 
function  (which  may  be  either  transcendental  or  merely  polynomial,  in  the  sense 
o/§  47)  by  another  function  of  a  similar  character. 

If  2r  =  00  be  an  essential  singularity  for  at  least  one  of  the  coefficients, 
through  being  an  infinity  of  unlimited  order  for  a  branch  of  w,  then  one 
or  both  of  the  functions  in  the  quotient-form  of  one  at  least  of  the  coefficients 
must  be  transcendental. 

If  2r  =  00  be  an  accidental  singularity  or  an  ordinary  point  for  all  the 
coefficients,  through  being  either  an  infinity  of  finite  order  or  an  ordinary 
point  for  the  branches  of  w,  then  all  the  functions  which  occur  in  all  the 
coefficients  are  rational  expressions.  When  the  equation  is  multiplied 
throughout  by  the  least  common  multiple  of  the  denominators  of  the 
coefficients,  it  takes  the  form 

w'^h,  {z)  +  w''-'h,  (z)+  ...+  whn-i  (z)  +  hn  (z)  =  0, 
where  the  functions  h^ (z),  h^ {z), ...,  hn (z)  are  poljmomials  in  z. 


99.]  A   FINITE   NUMBER   OF   BRANCHES  213 

A  knowledge  of  the  number  of  infinities  of  w  gives  an  upper  limit  of  the 
degree  of  the  equation*  in  z  in  the  last  form.  Thus,  let  a^  be  a  regular 
singularity  of  the  function;  and  let  oii,^i,<yi, ...  be  the  orders  of  the  infinities 
of  the  branches  at  a^- ;  then  ' 

where  X^  denotes  a^  +  yg^  +  7^  +  . . . ,  is  finite  (but  not  zero)  for  ^  =  a^ 

Let  Ci  be  a  branch-point,  which  is  an  infinity;  and  let  /*  branches  w  form  a 

system  for  Ci,  such  that  w{z-  Ci)f^  is  finite  (but  not  zero)  at  the  point ;  then 

W1W2    ...lUy,{Z    —    d)     * 

is  finite  (but  not  zero)  at  the  point,  and  therefore  also 

Wi...w,(^-cO^^  +  '^^  +  ^^  +  - 

is  finite,  where  6i,  (pi,  -^i, ...  are  numbers  belonging  to  the  various  systems; 
or,  if  ei  denote  ^^  +  ^j  +  -v/r^  +  . . . ,  then 

w^...Wn{z-Ciy' 

is  finite  for  z  =  d.     Similarly  for  other  symmetric  functions  of  w. 

Hence,  if  ai,  a.^, ...  be  the  regular  singularities  with  numbers  Xj,  Xg,  ••• 
defined  as  above,  and  if  Ci,  c^, ...  be  the  branch-points,  that  are  also  infinities, 
with  numbers  ej,  e^,  ...  defined  as  above,  then  the  product 

(w-wi) {w-iun)  n  {z-tti)^  n  {z-dY^ 

i=l  i=l 

is  finite  at  all  the  points  a^  and  at  all  the  points  d.  The  points  a  and  the 
points  c  are  the  only  points  in  the  finite  part  of  the  plane  that  can  make  the 
product  infinite :  hence  it  is  finite  everywhere  in  the  finite  part  of  the  plane, 
and  it  is  therefore  polynomial  in  z. 

Lastly,  let  p  be  the  number  for  z=  <x>  corresponding  to  X^  for  a^-  or  to  ei 
for  Ci,  so  that  for  the  coefficient  of  any  power  of  w  in  {w  —JWi)  ...{w-  Wn)  the 
greatest  difference  in  degree  between  the  numerator  and  the  denominator  is 
p  in  favour  of  the  excess  of  the  former. 

Then  the  preceding  product  is  of  order 

p  +  SXi  +  Sej, 

which  is  therefore  the  degree  of  the  equation  in  z  when  it  is  expressed  in  a 
holomorphic  form. 


CHAPTER   IX. 

Periods  of  Definite  Integrals,  and  Periodic  Functions  in  general, 

100.  Instances  have  already  occurred  in  which  the  value  of  a  function 
of  z  is  not  dependent  solely  upon  the  value  of  z  but  depends  also  on  the 
course  of  variation  by  which  z  obtains  that  value ;  for  example,  integrals  of 
uniform  functions,  and  multiform-  functions.  And  it  may  be  expected  that, 
a  fortiori,  the  value  of  an  integral  connected  with  a  multiform  function  will 
depend  upon  the  course  of  variation  of  the  variable  z.  Now  as  integrals 
which  arise  in  this  way  through  multiform  functions  and,  generally,  integrals 
connected  with  differential  equations  are  a  fruitful  source  of  new  functions, 
it  is  desirable  that  the  effects  on  the  value  of  an  integral  caused  by  variations 
of  a  £^-path  be  assigned  so  that,  within  the  limits  of  algebraic  possibility,  the 
expression  of  the  integral  may  be  made  completely  determinate. 

There  are  two  methods  which,  more  easily  than  others,  secure  this  result ; 
one  of  them  is  substantially  due  to  Cauchy,  the  other  to  Riemann. 

The  consideration  of  Riemann's  method,  both  for  multiform  functions  and 
for  integrals  of  such  functions,  will  be  undertaken  later,  in  Chapters  XV,, 
XVI,  Cauchy's  method  has  already  been  used  in  preceding  sections  relating 
to  uniform  functions,  and  it  can  be  extended  to  multiform  functions.  Its 
characteristic  feature  is  the  isolation  of  critical  points,  whether  regular 
singularities  or  branch-points,  by  means  of  small  curves  each  containing  one 
and  only  one  critical  point. 

Over  the  rest  of  the  plane  the  variable  z  ranges  freely  and,  under  certain 
conditions,  any  path  of  variation  of  z  from  one  point  to  another  can,  as  will 
be  proved  immediately,  be  deformed  without  causing  any  change  in  the 
value  of  the  integral,  provided  that  the  path  does  not  meet  any  of  the  small 
curves  in  the  course  of  the  deformation.  Further,  from  a  knowledge  of  the 
relation  of  any  point  thus  isolated  to  the  function,  it  is  possible  to  calculate 
the  change  caused  by  a  deformation  of  the  ^■-path  over  such  a  point ;  and 
thus,  for  defined  deformations,  the  value  of  the  integral  can  be  assigned 
precisely. 


100.]  INTEGKAL   OF   A   BRANCH  215 

The  properties  proved  in  Chapter  II.  are  useful  in  the  consideration  of 
the  integrals  of  uniform  functions ;  it  is  now  necessary  to  establish  the 
propositions  which  give  the  effects  of  deformation  of  path  on  the  integrals 
of  multiform  functions.  The  most  important  of  these  propositions  is  the 
following  : — 

If  w  he  a  multiform  function,  the  value  of  I    wdz,   taken    between   two 


ordinary  points,  is  unaltered  for  a  deformation  of  the  path,  provided  that  the 
initial  branch  of  w  be  the  same  and  that  no  branch-point  or  infinity  be  crossed 
in  the  deformation. 

Consider  two  paths  ach,  adb,  (fig.  16,  p.  181),  satisfying  the  conditions 
specified  in  the  proposition.  Then  in  the  area  between  them  the  branch  w 
has  no  infinity  and  no  point  of  discontinuity ;  and  there  is  no  branch-point 
in  that  area.  Hence,  by  §  90,  Corollary  VI.,  the  branch  w  is  a  uniform 
monogenic  function  for  that  area ;  it  is  continuous  and  finite  everywhere 
within  it  and,  by  the  same  Corollary,  we  may  treat  w  as  a  uniform,  mono- 
genic, finite  and  continuous  function.     Hence,  by  §  17,  we  have 

rb  ra 

(c)  I     ludz  +  (d)       wdz  =  0, 

J  a  b 

the  first  integral  being  taken  along  acb  and  the  second  along  bda;  and 
therefore 


rb  ra  rb 

(c)  1    ludz  =  —  (d)  I    wdz  =  (d)  I    wdz, 

J  a  •'  b  J  a 


shewing  that  the  values  of  the  integral  along  the  two  paths  are  the  same 
under  the  specified  conditions. 

It  is  evident  that,  if  some  critical  point  be  crossed  in  the  deformation, 
the  branch  w  cannot  be  declared  uniform  and  finite  in  the  area,  and  the 
theorem  of  §  17  cannot  then  be  applied. 

Corollary  I.  The  integral  round  a  closed  curve  containing  no  critical 
point  is  zero. 

.  Corollary  II.  A  cm-ve  round  a  branch-point,  containing  no  other 
critical  point  of  the  function,  can  be  deformed  into  a  loop 
without  altering  the  value  of  Jtudz ;  for  the  deformation 
satisfies  the  condition  of  the  proposition.  Hence,  when 
the  value  of  the  integral  for  the  loop  is  known,  the 
value  of  the  integral  is  known  for  the  curve. 

Corollary  III.  From  the  proposition  it  is  possible 
to  infer  conditions,  under  which  the  integral  Jwdz  round 
the  whole  of  any  curve  remains  unchanged,  when  the  whole 
curve  is  deformed,  without  leaving  an  infinitesimal  arc 
common  as  in  Corollary  II. 


216  INTEGRATION  [100. 

Let  GDC ,  ABA'  be  the  curves :  join  two  consecutive  points  AA'  to  two 

consecutive  points  GC.     Then  if  the  area  GABA'G'DG 

enclose  no  critical  point  of  the  function  w,  the  value  of 

jwdz  along  GDG'  is  by  the  proposition  the  same  as  its 

value  along  GABA'G'.     The  latter  is  made  up  of  the 

value  along  GA,  the  value  along  ABA',  and  the  value 

along  A'G',  say 

••A  r  rC 

wdz  +  1    ludz  +       w'dz, 

C  J  B  J  A' 

where  w'  is  the  changed  value  of  tu  consequent  on  the  description  of  a  simple 
curve  reducible  to  B  (§  90,  Cor.  II.). 

Now  since  w  is  finite  everywhere,  the  difference  between  the  values  of  w 
at  A  and  at  A'  consequent  on  the  description  of  ABA'  is  finite:  hence  as 
A'A  is  infinitesimal  the  value  of  ^wdz  necessary  to  complete  the  value  for 
the  whole  curve  B  is  infinitesimal  and  therefore  the  complete  value  can  be 

taken  as  the  foregoing  integral       wdz.     Similarly  for  the    complete   value 

J  B 

along  the  curve  D :  and  therefore  the  difference  of  the  integrals  round  B  and 

round  D  is 

rA  rC 


A  rv 

wdz  -\-  I    w'dz, 

C  J  A' 


say  /    {w  —  w')dz. 

J  c 

In  general  this  integral  is  not  zero,  so  that  the  values  of  the  integral 

round  B  and  round  B  are  not  equal  to  one  another :  and  therefore  the  curve 

D  cannot  be  deformed  into  the  curve  B  without  affecting  the  value  of  Jwdz 

round  the  whole  curve,  even  when  the  deformation  does  not  cause  the  curve 

to  pass  over  a  critical  point  of  the  function. 

But  in  special  cases  it  may  vanish.  The  most  important  and,  as  a 
matter  of  fact,  the  one  of  most  frequent  occurrence  is  that  in  which  the 
description  of  the  curve  B  restores  at  A'  the  initial  value  of  w  at  ^.  It 
easily  follows,  by  the  use  of  §  90,  Cor.  II.,  that  the  description  of  D  (as- 
suming that  the  area  between  B  and  D  includes  no  critical  point)  restores 
at  G'  the  initial  value  of  w  at  G.  In  such  a  case,  w  =  w'  for  corresponding 
points  on  AG  and  A'G',  and  the  integral,  which  expresses  the  difference, 
is  zero :  the  value  of  the  integral  for  the  curve  B  is  then  the  same  as  that 
for  D.     Hence  we  have  the  proposition : — 

If  a  curve  he  such  that  the  description  of  it  by  the  independent  variable 
restores  the  initial  value  of  a  imdtiform  function  w,  then  the  value  of  jwdz 
taken  round  the  curve  is  unaltered  when  the  cm've  is  deformed  into  any  other 
curve,  provided  that  no  branch-point  or  point  of  discontinuity  of  w  is  crossed 
in  the  course  of  deformation. 


100.]  OF   MULTIFORM   FUNCTIONS  217 

This  is  the  generalisation  of  the  proposition  of  |  19  which  has  thus  far 
been  used  only  for  uniform  functions. 

Note.  Two  particular  cases,  which  are  very  simple,  may  be  mentioned 
here :  special  examples  will  be  given  immediately. 

The  first  is  that  in  which  the  curve  B,  and  therefore  also  D,  encloses 
no  branch-point  or  infinity;  the  initial  value  of  lu  is  restored  after  a 
description  of  either  curve,  and  it  is  easy  to  see  (by  reducing  5  to  a 
point,  as  may  be  done)  that  the  value  of  the  integral  is  zero. 

The.  second  is  that  in  which  the  curve  encloses  more  than  one  branch- 
point, the  enclosed  branch-points  being  such  that  a  circuit  of  all  the  loops, 
into  which  (by  Corollary  V.,  §  90)  the  curve  can  be  deformed,  restores  the 
initial  branch  of  w.  This  case  is  of  especial  importance  when  w  is  two-valued : 
the  curves  then  enclose  an  even  number  of  branch-points. 

101.  It  is  important  to  know  the  value  of  the  integral  of  a  multiform 
function  round  a  small  curve  enclosing  a  branch-point. 

Let  c  be  a  point  at  which  m  branches  of  an  algebraic  function  are  equal 
and  interchange  in  a  single  cycle ;  and  let  c,  if  an  infinity,  be  of  only  finite 
order,  say  kini.  Then  in  the  vicinity  of  c,  any  of  the  branches  w  can  be 
expressed  in  the  form 

s=  —k 

where  A;  is  a  finite  integer. 

The  value  of  fwdz  taken  round  a  small  curve  enclosing  c  is  the  sum  of 
the  integrals 

gsj(z-crdz, 

the  value  of  which,  taken  once  round  the  curve  and  beginning  at  a  point  Zi,  is 

where  a  is  a  primitive  mth.  root  of  unity,  provided  m  -j-  s  is  not  zero. 
If  then  wi  4-  s  be  positive,  the  value  is  zero  in  the  limit  when  the  curve 
is  infinitesimal :    if  m  -f  s  be  negative,  the  value  is  oo  in  the  limit. 

But,  if  m  +  s  he  zero,  the  value  is  ^arigs. 

Hence  we  have  the  proposition :  If,  in  the  vicinity  of  a  hranch-point  c, 
where  m  branches  w  are  equal  to  one  another  and  interchange  cyclically,  the 
expression  of  one  of  the  branches  be 

_k                           fc-i 
,     gic{z-cy'"'  +  gic-^{z-c)    "*  + 

then  Jwdz,  taken  once  round  a  small  curve  enclosing  c,  is  zero,  if  k<m ;  is 
infinite,  if  k>  m ;  and  is  2'7rigk,  if  k  =  m. 


218  MULTIPLICITY   OF   VALUE  [101. 

It  is  easy  to  see  that,  if  the  integral  be  taken  m  times  round  the  small 
curve  enclosing  c,  then  the  value  of  the  integral  is  ^mirigm  when  k  is  greater 
than  m,  so  that  the  integral  vanishes  unless  there  be  a  term  involving  {z  —  c)~^ 
in  the  expansion  of  a  branch  w  in  the  vicinity  of  the  point.  The  reason  that 
the  integral,  which  can  furnish  an  infinite  value  for  a  single  circuit,  ceases  to 

A. 

do  so  for  m  circuits,  is  that  the  quantity  {z-^  —  c)  "%  which  becomes  indefi- 
nitely great  in  the  limit,  is  multiplied  for  a  single  circuit  by  a^— 1,  for  a 
second  circuit  by  a?^  —  a^,  and  so  on,  and  for  the  mth  circuit  by  ct"*^  — a*™'"^'^ 
the  sum  of  all  of  which  coefficients  is  zero. 

Ex.  The  integral  \{{z -a)  {z  —  b)...{z  —  f)}~idz  taken  round  an  indefinitely  small  curve 
enclosing  a  is  zero,  provided  no  one  of  the  quantities  6,  . . . ,  /  is  equal  to  a. 

102.  Some  illustrations  have  already  been  given  in  Chapter  II.,  but 
they  relate  solely  to  definite,  not  to  indefinite,  integrals  of  uniform 
functions.  The  whole  theory  will  not  be  considered  at  this  stage ;  we  shall 
merely  give  some  additional  illustrations,  which  will  shew  how  the  method 
can  be  applied  to  indefinite  integrals  of  uniform  functions  and  to  integrals 
of  multiform  functions,  and  which  will  also  form  a  simple  and  convenient 
introduction  to  the  theory  of  periodic  functions  of  a  single  variable. 

We  shall  first  consider  indefinite  integrals  of  uniform  functions. 


/dz 
— ,  and  denote*  it  by  f{z). 


The  function  to  be  integrated  is  uniform,  and  it  has  an  accidental  singularity  of  the  first 
order  at  the  origin,  which  is  its  only  singularity.  The  value  of  jz~''-dz  taken  positively 
along  a  small  curve  round  the  origin,  say  round  a  circle  with  the  origin  as  centre,  is  2ni ; 
but  the  value  of  the  integral  is  zero  when  taken  along  any  closed  curve  which  does  not 
include  the  origin. 

Taking  2  =  1  as  the  lower  limit  of  the  integral,  and  any  point  z  as  the  upper  limit,  we 
consider  the  possible  paths  from  1  to  z.  Any  path  from  1  to  2  can  be  deformed,  without 
crossing  the  origin,  into  a  path  which  circumscribes  the  origin  positively  some  number  of 
times,  say  mj,  and  negatively  some  number  of  times,  say  m^,  all  in  any  order,  and  then 
leads  in  a  straight  line  from  1  to  z.     For  this  path  the  value  of  the  integral  is  equal  to 

(27ri)  mi  +  (  -  Stti)  WI2  +1     — , 
J  1   2 

[^dz 
that  is,  to  2«^73■^^-  I    — , 


where  m  is  an  integer,  and  in  the  last  integral  the  variation  of  z  is  along  a  straight 
line  from  1  to  z.     Let  the  last  integral  be  denoted  by  u;  then 

f{z)  —  u  +  'i'>mTi, 
and  therefore,  inverting  the  function  and  denoting  /"i  by  (^,  we  have 

0  =  ^(%  +  2m7r2). 
Hence  the  general  integral  is  a  function  of  z  with  an  infinite  number  of  values ;  and  s  is  a 
periodic  function  of  the  integral,  the  period  being  ^Tri. 

*  See  Chrystal,  ii,  pp.  288—297,  for  the  elementary  properties  of  the  function  and  its  inverse, 
when  the  variable  is  complex. 


102.]  OF   INTEGRALS  219 

f    dz 
Ex.  2.     Consider  the  function   I  t—^;  and  again  denote  it  by  f{z). 

The  one-valued  function  to  be  integrated  has  two  accidental  singularities  ±  i,  each  of 
the  first  order.  The  value  of  the  integral  taken  positively  along  a  small  curve  round  i  is 
IT,  and  along  a  small  curve  round  —  i  is  -  tt. 

We  take  the  origin  0  as  the  lower  limit  aod  any  point  z  as  the  upper  limit.  Any  path 
from  0  to  z  can  be  deformed,  without  crossing  either  of  the  singularities  and  therefore 
without  changing  the  value  of  the  integral,  into 

(i)    any  numbers  of  positive  {m^,  m-i)  and  of  negative  (m/,  m{)  circuits  round  i  and 
round  —  ?',  in  any  order,  and 

(ii)   a  straight  line  from  0  to  z. 
Then  we  have 

/(2)  =  TOi7r  +TOi'  (  -  7r)  +  m2  (  -  ir)^m{  {-  (  -  7r)}+  [^  ^ 

[^     dz  ^'''"  •       ' 

=  Utt  +  u, 

where  ?i  is  an  integer  and  the  integral  on  the  right-hand  side  is  taken  along  a  straight  line 
from  0  to  z. 

Inverting  the  function  and  denoting  f~^  by  0,  we  have 

z  =  (f)  (u  +  7nr). 

The  integral,  as  before,  is  a  function  of  z  with  an  infinite  number  of  values ;  and  0  is  a 
periodic  function  of  the  integral,  the  period  being  n. 


Ex.  3.     Denoting  by  /  the  value  of  the  integral 

livl  J  : 


i^ *, 


2'rzj:s„  {z-a){z-h){z-c) 
taken  along  a  straight  line  from  z^,  to  z  on  which  no  one  of  the  points  a,  6,  c  lies,  find  the 
general  value  of  the  integral  for  a  path  from  Z(,  that  goes  I  times  round  a,  m  times  round  6, 
Qi  times  round  c. 

What  is  the  form  of  the  result,  when  a  and  h  coincide  ? 

103.  Before  passing  to  the  integrals  of  multiform  functions,  it  is  con- 
venient to  consider  the  method  in  which  Hermite*  discusses  the  multiplicity 
in  value  of  a  definite  integral  of  a  uniform  function. 

Taking  a  simple  case,  let         ^  {z)=  \    -z ^ 

.and  introduce  a  new  variable  t  such  that  Z  =  zt;  then 

'1   zdt 


When  the  path  of  t  is  assigned,  the  integral  is  definite,  finite  and  unique  in 
value  for  all  points  of  the  plane  except  for  those  for  which  1  +zt  =  0\  and, 
according  to  the  path  of  variation  of  t  from  0  to  1,  there  will  be  a  ^^-curve 
which  is  a  curve  of  discontinuity  for  the  subject  of  integration.     Suppose  the 

*  Crelle,   t.    xci,    (1881),   pp.    62—77;     Gours   a   la  Faculte   des   Sciences,   4'"^   6d.    (1891), 
pp.  76 — 79,  154 — 164,  and  elsewhere. 


220  hermite's  [103. 

path  of  t  to  be  the  straight  line  from  0  to  1 ;  then  the  curve  of  discontinuity- 
is  the  axis  of  x  between  —  1  and  —  oc  .  In  this  curve  let  any  point  —  |^  be 
taken  where  ^  >  1 ;  and  consider  a  point  ^^j  =  —  |  +  ^e  and  a  point  2^  =  —  ^  —  ie, 
respectively  on  the  positive  and  the  negative  sides  of  the  axis  of  x,  both 
being  ultimately  taken  as  infinitesimally  near  the  point  —  ^.     Then 


=  2i 


t- 


tan 


Let  6  become  infinitesimal ;  then,  when  t  is  infinite,  we  have 

tan~^ =  ^ir, 

for  e  is  positive ;  and,  when  t  is  unity,  we  have 

tan~^ =  —  ^TT, 

for  ^  is  >  1.     Hence  0  (z^)  —  cf)  {z^  =  2^%. 

The  part  of  the  axis  of  x  from  —  1  to  —  oo  is  therefore  a  line  of  discon- 
tinuity in  value  of  (f)  (s),  such  that  there  is  a  sudden  change  in  passing  from 
one  edge  of  it  to  the  other.  If  the  plane  be  cut  along  this  line  so  that 
it  cannot  be  crossed  by  the  variable  which  may  not  pass  out  of  the  plane, 
then  the  integral  is  everywhere  finite  and  uniform  in  the  modified  surface. 
If  the  plane  be  not  cut  along  the  line,  it  is  evident  that  a  single  passage 
across  the  line  from  one  edge  to  the  other  makes  a  difference  of  ^iri  in  the 
value,  and  consequently  any  number  of  passages  across  will  give  rise  to  the 
multiplicity  in  value  of  the  integral. 

Such  a  line  is  called  a  section*  by  Hermite,  after  Riemann  who,  in  a 
different  manner,  introduces  these  lines  of  singularity  into  his  method  of 
representing  the  variable  on  surfaces  f. 

When  we  take  the  general  integral  of  a  uniform  function  of  Z  and  make  • 
the  substitution  Z  =  zt,  the  integral  that  arises  for  consideration  is  of  the  form 


^{z)=j 


^'  F  (t,  z) 

t.G{t,z) 


dt. 


We  shall  suppose  that  the  path  of  variation  of  t  is  the  axis  of  real  quantities : 
and  the  subject  of  integration  will  be  taken  to  be  a  general  function  of  t  and 
z,  without  special  regard  to  its  derivation  from  a  uniform  function  of  Z. 

*  Goupure  ;  see  Crelle,  t.  xci,  p.  62.  t  See  Chapter  XV. 


103.]  SECTIONS  221 

It  is  easy,  after  the  special  example,  to  see  that  <l>  is  a  continuous  function 
of  z  in  any  space  that  does  not  include  a  ^-point  which,  for  values  of  t  included 
within  the  range  of  integration,  would  satisfy  the  equation 

G  {t,  z)  =  0. 

But  in  the  vicinity  of  a  2^-point,  say  ^,  corresponding  to  the  value  t  =  6  m 
the  range  of  integration,  there  will  be  discontinuity  in  the  subject  of 
integration  and  also,  as  will  now  be  proved,  in  the  value  of  the  integral 

Let  Z  be  the  point  ^,  and  draw  the  curve  through  Z  corresponding  to 
t  =  real  constant ;  let  iV^  be  a  point  on  the  positive  side  and  N^, 
a  point  on  the  negative  side  of  this  curve  positively  described, 
both  points  being  on  the  normal  at  Z ;  and  let  ZN-^  =  ZN^  —  e, 
supposed  small.     Then  for  N-^  we  have 

x^  =  P—e  sin  -v/r,         y  =n  +  e  cos  -v|r, 

SO  that  2-1  =  ^+^V  (cos -v/r  +  {sin -v/r), 

where  -v/r  is  the  inclination  of  the  tangent  to  the  axis  of  real  quantities.  But, 
if  da  be  an  arc  of  the  curve  at  Z, 

dar  ,    ,   ■  ■     ,  X     d^      .dt)     dt 

for  variations  along  the  tangent  at  Z,  that  is, 

~rr  (cos  y  +  *  sm  y)  =  —  -:: . 

•  f^G(0,O 

Thus,  since  -y-  may  be  taken  as  finite  on  the  supposition  that  Z  is  an 
ordinary  point  of  the  curve,  we  have 

where  ^  =  ^'|^        P=~G{e,^),        Q  =  l^G(d,0. 

.  P 
Similarly  Z2=^+ie 


Q' 


Hence  ^  (z,)  =  f' ^.^dt 


Ta rp(^t^ 


^a^)--|aj^ao}Q 


222  HERMITE'S  [103. 

with  a  similar  expression  for  <E>  (z^)  ;  and  therefore 


(^^. 


The  subject  of  integration  is  infinitesimal,  except  in  the  immediate  vicinity 
of  ^  =  ^ ;  and  there 

powers  of  small  quantities  other  than  those  retained  being  negligible.  Let 
the  limiting  values  of  t,  that  need  be  retained,  be  denoted  hy  6  +  v  and 
0  —  fi;   then,  after  reduction,  we  have 


,  J  e-ij. 


=  2'7Ti 


F{e,  0 


i{G(0,o} 


dd 

in  the  limit  when  e  is  made  infinitesimal. 

Hence  a  line  of  discontinuity  of  the  subject  of  integration  is  a  section 
for  the  integral ;  and  the  preceding  expression  is  the  magnitude,  by 
numerical  multiples  of  which  the  values  of  the  integral  differ*. 

Ex.  1.     Consider  the  integral 

zdt 


We  have 


Fie,c)        c       1      1 


|(^(.,C)}    ''''    '''    ''' 


so  that  TT  is  the  period  for  the  above  integral. 
Ux.  2.     Shew  that  the  sections  for  the  integral 

t"  sin  z 


i: 


dt. 


0    l  +  2tcosz  +  fi 

where  a  is  positive  and  less  than  1,  are  the  straight  lines  ^  =  (2^  +  1)  n,  where  k  assumes  all 
integral  values ;  and  that  the  period  of  the  integral  at  any  section  at  a  distance  tj  from  the 
axis  of  real  quantities  is  27r  cosh  (arj).  (Hermite.) 

*  The  memoir  and  the  Cours  d' Analyse  of  Hermite  should  be  consulted  for  further  develop- 
ments ;  and,  in  reference  to  the  integral  treated  above,  Jordan,  Cours  d' Analyse,  t.  ii,  pp.  293 — 
296,  may  be  consulted  with  advantage.  See  also,  generally,  for  functions  defined  by  definite 
integrals,  Goursat,  Acta  Math.,  t.  ii,  (1883),  pp.  1—70,  and  ib.,  t.  v,  (1884),  pp.  97—120;  and 
Pochhammer,  Math.  Ann.,  t.  xxxv,  (1890),  pp.  470 — 494,  495 — 526.  Goursat  also  discusses 
double  integrals. 


103.]  ~  SECTIONS  223 

Ex.  3.     Prove  that  the  function  defined  by 


l+^^+p  +  .p+, 


has  a  logarithmic  singularity  at  x=l  and  no  other  finite  singularity.  If  the  plane  be 
divided  by  a  cut  extending  along  the  positive  part  of  the  real  axis  extending  from  1  to  qo  , 
shew  that  in  the  divided  plane  the  function  defined  by  the  above  series  and  its  con- 
tinuations is  one-valued,  and  that,  at  corresponding  points  on  opposite  sides  of  the  cut,  its 
values  difier  by  2^1  log  x.  (Math.  Trip.,  Part  II.,  1899.) 

Ex.  4.     Shew  that  the  integral 

J  0 

where  the  real  parts  of  0  and  -y  -  /3  are  positive,  has  the  part  of  the  axis  of  real  quantities 
between  1  and  +  oo  for  a  section. 

Shew  also  that  the  integral 


<^  (2)=    I       «,P-1  (1  _  9.l)y-^-^  (1  -  ZU^du, 


where  the  real  parts  of  (3  and  I  — a  are  positive,  has  the  part  of  the  axis  of  real  quantities 
between  0  and  1  for  a  section :  but  that,  in  order  to  render  0  (2)  a  uniform  function  of  2, 
it  is  necessary  to  prevent  the  variable  from  crossing,  not  merely  the  section,  but  also  the 
part  of  the  axis  of  real  quantities  between  1  and  -l-oo  .  (Goursat.) 

(The  latter  line  is  called  a  section  of  the  second  kind.) 

Ex.  5.      Discuss  generally  the  effect  of  changing  the  path  of  ^  on  a  section  of  the 

integral ;  and,  in  particular,  obtain  the  section  for  I  „  when,  after  the  substitution 

J  0  i-  +  ^ 

Z=zt,  the  path  of  t  is  made  a  semi-circle  on  the  line  joining  0  and  1  as  diameter. 

Ex.  6.     Shew  that,  for  the  function  /(z)  defined  by  the  definite  integral 

e  -f-e  t         t    e  , 

-00    g2nnt_^^-2nnt      ^-27rt_^2nzi   '^^' 

where  %  is  a  positive  integer  and  the  integration  is  for  real  values  of  t,  while  z=x  +  i^,  the 
sections  are  the  lines 

a;=0,   ±1,   ±2,..., 

and  that  the  increment  oi  f{z)  in  crossing  the  section  .r=0  in  the  positive  direction  is  2". 

(Appell.) 

Note.  It  is  manifestly  impossible  to  discuss  all  the  important  bearings  of  theorems 
and  principles,  which  arise  from  time  to  time  in  our  subject;  we  can  do  no  more  than 
mention  the  subject  of  those  definite  integrals  involving  complex  variables,  which  first 
occur  as  solutions  of  the  better-known  linear  differential  equations  of  the  second  order. 

Thus  for  the  definite  integral  connected  with  the  hypergeometric  series,  memoirs  by 
Jacobi*  and  Goursatt  should  be  consulted;  for  the  definite  integral  connected  with 
Bessel's  functions,  memoirs  by  HankelJ  and  Weber  §  should  be  consulted;  and  Heine's 
Handhuch  der  Kugelfunctionen  for  the  definite  integrals  connected  with  Legendre's 
functions. 

*  Crelle,  t.  Ivi,  (1859),  pp.  149 — 165  ;  the  memoir  was  not  published  until  after  his  death, 
t  Sur  I'equation  differentielle   lineaire  qui  adviet  pour  integrale  la  serie   hypergeometrique, 
(Th^se,  Gauthier-Villars,  Paris,  1881). 

t  Math.  Ann.,  t.  i,  (1869),  pp.  467—501. 

§  Math.  Ann.,  t.  xxxvii,  (1890),  pp.  404—416. 


224 


EXAMPLES 


[104. 


104.     We  shall  now  consider  integrals  of  multiform  functions. 
Ex.  1.     To  find  the  integral  of  a  multiform  function  round  one  loop;  and  round  a 
number  of  loops. 

Let  the  function  be  w  =  {{z  —  a-i)  {z-a^ ...{z  —  «„)}»», 

where  m  may  be  a  negative  or  a  positive  integer,  and  the  quantities  a  are  unequal  to  one 
another ;  and  let  the  loop  be  from  the  origin  round  the  point  a-^ .  Then,  if  /  be  the  value 
of  the  integral  with  an  assigned  initial  branch  w,  we  have 


'"=  I       wdz+  I     wdz+  I      awdz, 
Jo  J  c  J  a, 


where  a  is  e  ™  and  the  middle  integral  is  taken  round  the  circle  at  a-^  of  infinitesimal  radius. 
But,  since  the  limit  of  (s-  ai)  w  when  z  =  aj^  is  zero,  the  middle  integral  vanishes  by  §  101 ; 
and  therefore 

/«,  =  (! -a)  I      wdz, 
Jo 

where  the  integral  may,  if  convenient,  be  considered  as  talcen  along  the  straight  line  from 

0  to  ai. 


(1)  (2)  (3) 

Fig.  25. 

Next,  consider  a  circuit  for  an  integral  of  w  which  (fig.  25)  encloses  two  branch-points, 

say  «!  and  a2,  but  no  others;  the  circuit  in  (1)  can  be  deformed  into  that  in  (2)  or  into 

that  in  (3)  as  well  as  into  other  forms.     Hence  the  integral  round  all  the  three  circuits 

must  be  the  same.     Beginning  with  the  same  branch  as  in  the  first  case,  we  have 

/"aj 
(1-a)  I      wdz, 

as  the  integral  after  the  first  loop  in  (2).  And  the  branch  with  which  the  second  loop 
begins  is  aw,  so  that  the  integral  described  as  in  the  second  loop  is 

f  Ul 

(I— a)  I      awdz; 
Jo 

and  therefore,  for  the  circuit  as  in  (2),  the  integral  is 

/=  (1  —  a)         wdz  +  a  (1  -  a)  /      wdz. 
Jo  Jo 

Proceeding  similarly  with  the  integral  for  the  circuit  in  (3),  we  find  that  its  expression  is 

/=(l-a)  I      wdz  +  a{l-a)  /      wdz, 
Jo  Jo 

and  these  two  values  must  be  equal. 

But  the  integrals  denoted  by  the  same  symbols  are  not  the  same  in  the  two  cases ;  the 

function  I      wdz  is  different  in  the  second  value  of  /  from  that  in  the  first,  for  the  deforma- 
Jo 

tion  of  path  necessary  to  change  from  the  one  to  the  other  passes  over  the  branch-point  a^^ . 
In  fact,  the  equality  of  the  two  values  of  /  really  determines  the  value  of  the  integral  for 
the  loop  Oai  in  (3). 


104.]  OF   PERIODICITY   OF'  INTEGRALS  225 

And,  in  general,  equations  thus  obtained  by  varied  deformations  do  not  give  relations 
(among  loop-integrals ;  they  define  the  values  of  those  loop-integrals  for  the  deformed  paths. 

We  therefore  take  that  deformation  of  the  circuit  into  loops  which  gives  the  simplest 
path.  Usually  the  path  is  changed  into  a  group  of  loops  round  the  hranch-points  as  they 
occur,  taken  in  order  in  a  trigonometrically  positive  direction. 

The  value  of  the  integral  roimd  a  circuit,  equivalent  to  any  number  of  loops,  is  obvious. 

Ex.  2.  To  find  the  value  of  \wdz,  taken  round  a  simple  curve  which  includes  all  the 
branch-points  of  iv  and  all  the  infinities. 

If  s  =  Qo  be  a  bra.nch -point  or  an  infinity,  then  all  .the  branch-points  and  all  the 
infinities  of  lo  lie  on  what  is  usually  regarded  as  the  exterior  of  the  curve,  or  the  curve 
may  in  one  sense  be  said  to  exclude  all  these  points.  The  integral  round  the  curve  is  then 
the  integral  of  a  function  round  a  curve,  such  that  over  the  area  included  by  it  the 
function  is  uniform,  finite  and  continuous ;  hence  the  integral  is  zero. 

If  2=  00  be  neither  a  branch-point  nor  an  infinity,  the  curve  can  be  deformed  until  it  is 
a  circle,  centre  the  origin  and  of  very  great  radius.  If  then  the  limit  of  zw,  when  j  ^  |  is 
infinitely  gTeat,  be  zero,  the  value  of  the  integral  again  is  zero,  by  II.,  §  24. 

Another  method  of  considering  the  integral,  is  to  use  Neumann's  sphere  for  the 
rej)resentation  of  the  variable.  Any  simple  closed  curve  divides  the  area  of  the  sphere 
into  two  parts ;  when  the  curve  is  defined  as  above,  one  of  those  parts  is  such  that  the 
function  is  uniform,  finite  and  continuous  throughout,  and  therefore  its  integral  round  the 
curve,  regarded  as  the  boundary  of  that  part,  is  zero.     (See  Corollary  III.,  §  90.) 

Ex.  3.  To  find  the  general  value  of  \{\-z'^)~'^dz.  The  function  to  be  integrated  is 
two-valued:  the  two  values  interchange  round  each  of  the  branch-points  ±  1,  which  are 
the  only  branch-points  of  the  function. 

Let  /  be  the  value  of  the  integral  for  a  loop  from  the  origin  round  -|- 1,  beginning  with 
the  branch  which  has  the  value  -1- 1  at  the  origin  ;  and  let  /'  be  the  corresponding  value 
for  the  loop  from  the  origin  round  -  1,  beginning  with  the  same  branch.     Then,  by  Ex.  1, 

/=2  Wl-z'^y^^dz,         /'  =  2  r\l-z'')-^^dz 
Jo  Jo' 

the  last  equality  being  easily  obtained  by  changing  variables. 

Now  consider  the  integral  when  taken  round  a  circle,  centre  the  origin  and  of  indefinitely 
great  radius  R ;  then  by  §  24,  II.,  if  the  limit  of  ziv  for  2=qo  be  k,  the  value  of  jwdz  round 

this  circle  is  2TTik.     In  the  present  case  w  =  {l-z-)~2  so  tha.t  the  limit  of  zw  is  +-;  hence 

J(l-22)-ic^^=27r, 
the  integral*  being  taken  round  the  circle.    But  since  a  description  of  the  circle  restores  the 
initial  value,  it  can  be  deformed  into  the  two  loops  from  0  q' 

to  A  and  from  Q  to  A'.     The  value  round  the  first  is  /;  and      ^  — — — y -D 

the  branch  with  which  the  second  begins  to  be  described  has 

the  value  -  1  at  the  origin,  so  that  the  consequent  value  ^ig-  26. 

round  the  second  is  —  /' ;  hence 

/-/'  =  27r* 

and  therefore  /=  -  /'  =  tt, 

verifying  the  ordinary  result  that 


/: 


when  the  integral  is  taken  along  a  straight  line. 

*  It  is  interesting  to  obtain  this  equation  when  0'  is  taken  as  the  initial  point,  instead  of  0. 
F.  F.  15 


226 


EXAMPLES   OF    PERIODICITY 


[104. 


To  find  the  general  value  of  w  for  any  path  of  variation  between  0  and  2,  we  proceed  as 
follows.  Let  S2  be  any  circuit  which  restores  the  initial  branch  of  (1— s^)-!.  Then  by 
§  100,  Corollary  II.,  9.  may  be  composed  of 

(i)    a  set  of  double  circuits  round  + 1,  say  m'  in  number, 
(ii)    a  set  of  double  circuits  round  - 1,  say  m"  in  number, 
and      (iii)    a  set  of  circuits  round  + 1  and  —  1  ; 

and  these  may  come  in  any  order  and  each  may  be  described  in  either  direction.  Now  for 
a  double  circuit  positively  described,  the  value  of  the  integral  for  the  first  description  is  / 
and  for  the  second  description,  which  begins  with  the  branch  —{l-z^)~^,  it  is  — /;  hence 
for  the  double  cii'cuit  it  is  zero  when  positively  described,  and  therefore  it  is  zero  also 
when  negatively  described.  Hence  each  of  the  m'  double  circuits  yields  zero  as  its 
nett  contribution  to  the  integral. 

Similarly,  each  of  the  m"  double  circuits  round  —  1  yields  zero  as  its  nett  contiubution 
to  the  integral. 

For  a  circuit  round  +1  and  —1  described  positively,  the  value  of  the  integral  has  just 
been  proved  to  be  /-/',  and  therefore  when  described  negatively  it  is  /'-/.  Hence,  if 
there  be  %i  positive  descriptions  and  7i2  negative  descriptions,  the  nett  contribution  of  all 
these  circuits  to  the  value  of  the  integral  is  {n-^  -  712)  {I- 1'),  that  is,  2%7r  where  n  is  an 
integer. 

Hence  the  complete  value  for  the  circuit  Q.  is  2«7r. 

Now  any  path  from  0  to  z  can  be  resolved  into  a  circuit  Q,  which  restores  the  initial 
branch  of  (1  — z^)"^,  chosen  to  have  the  value 
+  1  at  the  origin,  and  either  (i)  a  straight 
line  Oz ; 

or  (ii)  the  path  OA  Cz,  viz.,  a  loop  round 
+ 1  and  the  line  Oz  ; 

or  (iii)  the  path  OA'Cz,  viz.,  a  loop  round 
—  1  and  the  line  Oz. 
Let  u  denote  the  value  for  the  line  Oz,  so  that 

u=  f\l-z^)-idz. 


Hence,  for  case  (i),  the  general  value  of  the  integral  is 

2n7r  +  u. 

For  the  path  OACz,  the  value  is  /  for  the  loop  OAC,  and  is  {-u)  for  the  line  Cz,  the 
negative  sign  occurring  because,  after  the  loop,  the  branch  of  the  function  for  integration 
alono-  the  line  is  -(l-z^)"^;  this  value  is  I-u,  that  is,  it  is  tt-u.  Hence,  for  case  (ii), 
the  value  of  the  integral  is 

271 TT  +  77  -  ?J. 

For  the  path  OA'Cz,  the  value  is  similarly  found  to  be  -tt-u;  and  therefore,  for 
case  (iii),  the  value  of  the  integral  is 

2n7r  —  TT  —  II. 
If  f{z)  denote  the  general  value  of  the  integral,  we  have  either 

f{z)  =  2nTr  +  u, 
or  f{z)  =  {2m  +  l)7r-u, 

-where  n  and  m  are  any  integers,  so  that  f{z)  is  a  function  with  two  infinite  series  of  values. 


104.]  OF   INTEGRALS  227 

Lastly,  if  z=cj){6)  be  the  inverse  of  f{z)  =  6,  then  the  relation  between  u  and  z  given  by 

u=  r\l-z2)-hdz 

can  be  represented  in  the  form 

(f)  {u)  =  z  =  (j){2mr+ii)         "I 
and  0  (m)  =  2  =  (/)  (2??i7r  +  7r-?i)J  ' 

both  equations  being  necessary  for  the  full  representation.  Evidently  2  is  a  simply -periodic 
function  of  ic,  the  period  being  27r ;  and  from  the  definition  it  is  easily  seen  to  be  an  odd 
function. 

Let  y  =  (l  —  2^)^  =  x(^0)  so  that  3/  is  an  even  function  of  u;  from  the  consideration  of  the 
various  paths  from  0  to  z,  it  is  easy  to  prove  that 

=  —  X  (^wr  +  TT  -  ^()  J  " 

Ex.  4.  To  find  the  general  value  of  J{(1  -  z'^)  (1  —k-z-)}~2dz.  It  will  be  convenient  to 
regard  this  integral  as  a  special  case  of 

Z=j{{z-a){z-b){z-c){z-  d)}  -hdz= \wdz. 

The  two-valued  function  to  be  integrated  has  a,  6,  c,  d  (but  not  cc  )  as  the  complete 
system  of  branch-points ;  and  the  two  values  interchange  at  each  of  them.  We  proceed  as 
in  the  last  example,  omitting  mere  re-statements  of  reasons  there  given  that  are  applicable 
also  to  the  present  example. 

Any  circuit  O.  which  restores  an  initial  branch  of  w.,  can  be  made  up  of 

(i)    sets  of  double  circuits  round  each  of  the  branch-points, 

and     (ii)    sets  of  circuits  round  any  two  of  the  branch-points. 

The  value  of  \wdz  for  a  loop  from  the  origin  to  a  branch-point  k  (where  /t  =  a,  &,  c,  or  d)  is 

■ft 

wdz; 
0 

and  this  may  be  denoted  by  ^,  where  A' =  ^,  B,  C,  or  D. 

The  value  of  the  integral  for  a  double  circuit  round  a  branch-point  is  zero.  Hence  the 
amount  contributed  to  the  value  of  the  integral  by  all  the  sets  in  (i)  as  this  part  of  Q, 
is  zero. 

The  value  of  the  integral  for  a  circuit  round  a  and  b  taken  positively  is,  A—B;  for  one 
round  b  and  c  is  B  -C ;  for  one  round  c  and  d  is  C—  D ;  for  one  round  a  and  c  is  ^  —  0, 
which  is  the  sum  of  ^  -^  and  B  —  C;  and  similarly  for  circuits  round  a  and  d,  and  round 
b  and  d.  There  are  therefore  three  distinct  values,  say  A  —  B,  B—C,  C-B,  the  values 
for  circuits  round  a  and  b,  b  and  c,  c  and  d  respectively ;  the  values  for  circuits  round  any 
other  pair  can  be  expressed  linearly  in  terms  of  these  values.  Suppose  then  that  the  part 
of  Q,  represented  by  (ii),  when  thus  resolved,  is  the  nett  equivalent  of  the  description  of 
m'  circuits  round  a  and  b,  of  ?i'  circuits  round  b  and  c,  and  of  I'  circuits  round  c  and  d. 
Then  the  value  of  the  integral  contributed  by  this  part  of  Q,  is 

m'{A-B)  +  n'{B-C')  +  l'{C-I)), 
which  is  therefore  the  whole  value  of  the  integral  for  Q. 

But  the  values  of  A,  B,  C,  D  are  not  independent*.  Let  a  circle  with  centre  the  origin 
and  very  great  radius  be  drawn ;  then  since  the  limit  of  zw  for  |  2  |  =  oo  is  zero  and  since 

*  For  a  purely  analytical  proof  of  the  following  relation,  see  Greeuhill's  Elliptic  Functions, 
Chapter  II. 

15—2 


228  PERIODICITY  [104. 

z  =  co  is  not  a  branch-point,  the  value  of  \wdz  round  this  circle  is  zero  (Ex.  2).  The  circle 
can  be  deformed  into  four  loops  round  a,  6,  c,  d  respectively  in  order ;  and  therefore  the 
value  of  the  integral  is  A  —  B  +  C  —  D,  that  is, 

A-B  +  C-D  =  0. 

Hence  the  value  of  the  integral  for  the  circuit  Q,  is 

m{A-B)  +  n{B-C), 

where  m  and  n  denote  m'  —  I'  and  n'  respectively. 

Now  any  path  from  the  origin  to  z  can  be  resolved  into  Q.,  together  with  either 

(i)    a  straight  line  from  0  to  z, 

or        (ii)    a  loop  round  a  and  then  a  straight  line  to  z. 

It  might  appear  that  another  resolution  would  be  given  by  a  combination  of  O  with,  say, 
a  loop  round  b  and  then  a  straight  line  to  z ;  but  it  is  resoluble  into  the  second  of  the  above 
combinations.  For  at  G,  after  the  description  of  the  loop  B,  introduce  a  double  description 
of  the  loop  A,  which  adds  nothing  to  the  value  of  the  integral  and  does  not  in  the  end 
affect  the  branch  of  w  at  C ;  then  the  new  path  can  be  regarded  as  made  up  of  (a)  the 
circuit  constituted  by  the  loop  round  h  and  the  first  loop  round  a,  O)  the  second  loop 
round  a,  which  begins  with  the  initial  branch  of  to,  followed  by  a  straight  path  to  z.  Of 
these  (a)  can  be  absorbed  into  i2,  and  (/3)  is  the  same  as  (ii) ;  hence  the  path  is  not 
essentially  new.     Similarly  for  the  other  points. 

Let  u  denote  the  value  of  the  integral  with  a  straight  path  from  0  to  z;  then  the 
whole  value  of  the  integral  for  the  combination  of  Q.  with  (i)  is  of  the  form 

m{A-B)  +  n{B-C)  +  u. 

For  the  combination  of  Q,  with  (ii),  the  value  of  the  integral  for  the  part  (ii)  of 
the  path  is  A,  for  the  loop  round  a,  -\-{-u),  for  the  straight  path  which,  owing  to  the 
description  of  the  loop  round  a,  begins  with  -  w ;  hence  the  whole  value  of  the  integral 
is  of  the  form 

m{A-B)  +  n{B-C)  +  A-it^. 

Hence,  if  f{z)  denote  the  general  value  of  the  integral,  it  has  two  systems  of  values,  each 
containing  a  doubly-infinite  number  of  terms;  and,  if  z  =  (j}{u)  denote  the  inverse  of 
u=f{z),  we  have 

(j){tc)  =  (f){m{A-B)  +  n{B-C)  +  u} 

=  ct>{m{A-B)  +  n{B-0)  +  A-u}, 

where  m  and  n  are  any  integers.  Evidently  2  is  a  doubly -periodic  function  of  u,  with 
periods  A-B  and  B—C. 

Ex.  5.  The  case  of  the  foregoing  integral  which  most  frequently  occurs  is  the  elliptic 
integral  in  the  form  used  by  Legendre  and  Jacobi,  viz. : 

tt  =  J{(l  -z^)  {\-F'z^)}-^idz==jwdz, 

where  k  is  real.      The  branch-points  of  the  function  to  be  integrated  are   1,    —  1,  y, 

*  The  value  for  a  loop  round  h  and  then  a  straight  line  to  z,  just  considered,  is  i>  -  u, 

=  -{A-B)  +  A-u, 
giving  the  value  in  the  text  with  m  changed  to  hi  -  1. 


104.] 


OF   ELLIPTIC   INTEGRALS 


229 


and  —  -r 


^,  and  the  values  of  the  integral  for  the  corresponding  loops  from  the  origin  are 


i: 


2  I     wdz, 
J  0 

ivdz=  -2  /     wdz, 


and 


2  I     tvdz, 
J  0 

'k  fk 

2  I       wdz=  -2  /     w(^2. 


Now  the  values  for  the  loops  are  connected  by  the  equation 

A-B  +  G-D  =  0, 
and  so  it  will  be  convenient  that,  as  all  the  points  lie  on  the  axis  of  real  variables,  we 
arrange  the  order  of  the  loops  so  that  this  relation  is  identically'  satisfied.     Otherwise, 
the  relation  will,  after  Ex.  1,  be  a  definition  of  the  paths  of  integration  chosen  for  the 
loops. 

Among  the  methods  of  arrangement,  which  secure  the  identical  satisfaction  of  the 


Fig.   28. 


relation,  the  two  in  the  figure  are  the  simplest,  the  curved  lines  being  taken  straight  in 
the  limit ;  for,  by  the  first  arrangement  when  k  <  1,  we  have 

1  _1 

and,  by  the  second  when  X'  >  1,  we  have 


2-2        +2 

JO  ./  0 


2  /      \wdz  =  0, 


both  of  which  are  identically  satisfied.     We  may  therefore  take  either  of  them ;  let  the 

former  be  adopted. 

The  periods  are  A-B,  B-C,  (and  G -  D,  which  is  equal  to  B  -  A),  and  any  linear 

combination  of  these  is  a  period  :   we  shall  take  ^4  —  B,  and  B~  D.     The  latter,  B  -  D, 

is  equal  to 

2  I     iodz-2  I      ivdz, 
Jo  Jo 

which,  being  denoted  by.  4A'',  gives 

/■I                ^2 
4^=4 J 

Jo    {(l-02)(l_^2-2^}4 

as  one  period.     The  former,  A-B,  is  equal  to 

2 

2  /     wdz —  2  1    ivdz, 
Jo  Jo 


which  is 


wd,z: 


230  PERIODICITY  [104. 

this,  being  denoted  by  2^A'',  gives 


2 
2iA'  =  2 


1    {(1-S2)(l-Ps2)}4 
Jo  {(l-2'2)(l-/^'2/2)|4' 

where  X-'2  +  F  =  l,  and  the  relation  between  the  variables  of  the  integrals  is  k'^z^  +  k''^z''^  =  \. 

}_ 

[^ 
Hence  the  periods  of  the  integral  are  AK  and  '2,iK'.     Moreover,  ^  is  2  I    ivdz,  which  is 

J  n 
\_ 

[1  fk 

2  I     wch  +  2  I     todz  =  2K+2iK'. 

Hence  the  general  value  of  |    {(l—z^)(l—Pz-)]~2dz  is  either 
J  0 

or  2K+2iK'-u  +  4mK+2niK', 

that  is,  2K—u  +  4:mK+2niK', 

where  m  is  the  integral  taken  from  0  to  z  along  an  assigned  path,  often  taken  to  be 
a  straight  line ;  so  that  there  are  two  systems  of  values  for  the  integral,  each  containing 
a  doubly-iniinite  number  of  terms. 

If  z  be  denoted  by  (p  (u) — evidently,  from  the  integral  definition,  an  odd  function 
of  M-^-,  then 

cj)  {u)  =  (f)  {u  +  4raK+  2niK') 

=  (^  (2Z  -u  +  4mK+  2niK'\ 

so  that  2  is  a  doubly-periodic  function  of  ?«,  the  periods  being  4K  and  2iK'. 

Now  consider  the  function  Si  =  (l  —  z^)a.     A  0-path  round  y  does  not  affect  ^j  by  way  of 

change,  provided  the  curve  does  not  include  the  point  1 ;  hence,  if  2i=x  (■")»  ^^  have 

But  a  3-path  round  the  point  1  does  change  z^  into  —  % ;  so  that 

Hence  x.  (^0?  which  is  an  even  function,  has  two  periods,  viz.,  AlK  and  2A'-f  21^',  whence 

-^  (u)  =  ^  (^(  +  4mK+  2nK+  2mA'). 
Similarly,  taking  22  ==(1  -  k^z^)^  —  -^  (u),  it  is  easy  to  see  that 
^|.(M)  =  ^/.(tt+2A), 
-yjr{u)=yfr{u  +  2K+2iK')  =  yl^{u  +  2iK'), 
SO  that  -v//-  (m),  which  is  an  even  function,  has  two  periods,  viz.,  2A'  at:id  4iK' ;  whence 

yl^(u)=^^{u  +  2mK+4mK'). 
The  functions  (p  (m),  x  (u),  yj/  (k)  are  of  course  sn  u,  cu  n,  dn  to  respectively. 

Ex.  6.     If  in  a  single  infinite  sheet,  representing  the  values  of  z,  three  cuts  be  made 

along  the  real  axis  joining  respectively  (  —  00 ,   —  t))  (-1?  1)?  (7.5  ^  j?   shew  that  the 

integral  (in  the  notation  of  elliptic  functions,  0<^-<l,   J z^  —  1=  +i  J 1  —  z'^) 

K-E-  BKx'^ 


0  {(l-^2)(l_p^2)J^ 


dx 


104.]  OF   ELLIPTIC   INTEGRALS  231 

becomes  a  one-valued  function  of  z.     And  shew  that  ^  is  a  uniform  function  of  u  for 
the  values  of  u  which  arise. 

If  the  cut  joining  ( - 1,  1)  do  not  lie  along  the  real  axis,  describe  the  values  of  2  as  a 
function  of  u.  (Math.  Trip.,  Part  II.,  1894.) 

Ex.  7.     To  find  the  general  value  of  the  integral* 

I      {4{z  —  ei)  (2-62)  {s-e^)}~id2  =  iv. 

The  function  to  be  integrated  has  e^,  62,  63,  and  qo  for  its  branch -points ;  and  for 
paths  round  each  of  them  the  two  branches  interchange. 

A  circuit  O,  which  restores  the  initial  branch  of  the  function  to  be  integrated,  can 
be  resolved  into : — 

(i)    Sets  of  double  circuits  round  each  of  the  branch-points  alone:  as  before,  the 
value  of  the  integral  for  each  of  these  double  circuits  is  zero. 

(ii)  Sets  of  circuits,  each  enclosing  two  of  the  branch-points :  it  is  convenient  to 
retain  circuits  including  qo  and  ej,  qo  and  62,  qo  and  63,  the  other  thi'ee 
combinations  being  reducible  to  these. 

The  values  of  the  integral  for  these  three  retained  are  respectively 

/oo 
{4  (2-ei)  (z-e.,)  {z-e3)}-idz  =  2<o„ 

^2  =  2  [     {4:{z-e,){z-e2)(z-e,)}-idz  =  2co2, 

J    ^2 

^3  =  2/      {4{z-ei){z-e2){z-e3)}-idz  =  2(Os, 

and  therefore  the  value  of  the  integral  for  the  circuit  Q,  is  of  the  form 

m'Ei  +  n'Ei  +  l'E^,. 

But  El,  E-it  Er^  are  not  linearly  independent.     The  integral  of  the  function  round  any 

curve  in  the  finite  part   of  the  plane,  which  does  not 

include  ei,  62  or  63  within  its  boundary,  is  zero,  by  Ex.  2 ; 

and  this  curve  can  be  deformed  to  the  shape  in  the  figure, 

until  it  becomes  infinitely  large,  without  changing  the 

value  of  the  integral. 


Since  the  limit  of  ziv  for  1 2 1  =  00  is  zero,  the  value  of 

the  integral  from  00 '  to  qo  is  zero,  by  §  24,  II.;  and  if  the 

description  begin  with  a  branch  ■?<;,  the  branch  at  qo  is  —  w. 

The  rest  of  the  integral  consists  of  the  sum  of  the  values  e-  ""• 

round  the  loops,  which  is 

-E^+E^-E^, 

because  a  path  round  a  loop  changes  the  branch  of  lu  and  the  last  branch  after  describing 
the  loop  round  63  is  -fw  at  qo  ',  the  proper  value  (§  90,  III.).     Hence,  as  the  whole  integral 

is  zero,  we  have 

-Ei  +  Eo^-E^  =  0, 

or  say  £"2  =  ^1 4-  -£3 . 

*  The  choice  of  oo  for  the  upper  limit  is  made  on  a  ground  which  will  subsequently  be 
considered,  viz.,  that,  when  the  integral  is  zero,  z  is  infinite. 


232  UNIFORM  [104. 

Thus  the  value  of  the  integral  for  any  circuit  fl,  which  restores  the  initial  branch  of  w, 
can  be  expressed  in  any  of  the  equivalent  forms  viEi  +  nE^,  m'Ei  +  n'E2,  m"B2  +  n"Es, 
where  the  m's  and  n's  are  integers. 

Now  any  path  from  cc  to  2  can  be  resolved  into  a  circuit  Si,  which  restores  at  go  the 
initial  branch  of  w,  combined  with  either 

(i)    a  straight  path  from  cc  to  z, 
or      (ii)    a  loop  between  oo  and  e^,  together  with  a  straight  path  from  co  to  z. 
(The  apparently  distinct  alternatives,  of  a  loop  between  co  and  eo,  together  with  a  straight 
j)ath  from  co  to  z,  and  of  a  similar  path  round  e^,  are  inclusible  in  the  second  alternative 
above;  the  reasons  are  similar  to  those  in  Ex.  5.) 

If  «  denote  I     {4:{z-ei)  (z-ez)  {z  —  es)}~iclz  when  the  integral  is  taken  in  a  straight 

line,  then  the  value  of  the  integral  for  part  (i)  of  a  path  is  u;  and  the  value  of  the 
integral  for  part  (ii)  of  a  path  is  E^  —  u,  the  initial  branch  in  each  case  for  these  parts 
being  the  initial  branch  of  tv  for  the  whole  path.  Hence  the  most  genei'al  value  of  the 
integral  for  any  path  is  either 

or  Smfflj  +  2?iw3  +  2coi  —  M, 

the  two  being  evidently  included  in  the  form 

2mcoi  +  2«w3  ±  M. 
If,  then,  we  denote  by  z=^{;u)  the  relation  which  is  inverse  to 


u=  \     {4(s-ei)(s-e2)(2-e3)}    ^^dz, 


we  have  ^  {u)  =  ^  (2wicui  +  2Ma)3  ±  tt). 

In  the  same  way  as  in  the  preceding  example,  it  follows  that 

^'  {ii)  =  ^'  (2mQ)i  +  2?2cB3  +  w)=  -  ^i>'(2raa)i  +  2wco3-?<), 
where  ^'  (w)  is  -  {4  {z-e^)  iz-e.^  [z-e^^. 

Ex.  8.     Prove  that,  when  m  is  a  positive  integer  ^2,  and  when  q  is  a  positive  quantity 
such  that  0  <  9"  <  m, 

?/3-l        ,  TT  1 

•^         ait— 


0    1+?/™    -      m    .         o' 

"         "^  sni  TT  - 

m 


drawing  the  deformed  figure  of  the  loops. 
From  this  relation,  deduce  the  results 


(i)   /  --j.^=i.3,  .      (ii)  j^^^c^.=A.^ 


ao8-)^rf,_i,3  .      (ii)   [^'(logf)* 

(iii)  /;^w^^,=,...(^_^), 


where  A  and  B  are  constants. 

Shew  also  how  to  deduce  the  value  of 

i^(y)§(log3/) 


/ 


0  1+/"         ^^' 


where  Q  (log  y)  is  any  polynomial  in  log  y,  and  P  {y)  is  a  polynomial  in  y  of  degree  not 
greater  than  m  -  2. 


104.]  PERIODIC   FUNCTIONS  233 

The  foregoing  simple  examples  are  sufficient  illustrations  of  the  multi- 
plicity of  value  of  an  integral  of  a  uniform  function  or  of  a  multiform 
function,  when  branch-points  or  discontinuities  occur  in  the  part  of  the  plane 
in  which  the  path  of  integration  lies.  They  also  shew  one  of  the  modes  in 
which  singly-periodic  and  doubly-periodic  functions  arise,  the  periodicity 
consisting  in  the  addition  of  arithmetical  multiples  of  constant  quantities 
to  the  argument. 

To  the  properties  of  such  periodic  functions,  especially  of  uniform  periodic 
functions,  we  shall  return  in  Chapter  X.  It  will  there  appear  that  each  of 
the  special  functions,  which  have  been  considered  in  the  preceding  examples 
3,  4,  5,  7,  expresses  ^  as  a  uniform  function  of  its  argument. 

Meanwhile,  it  is  not  difficult  to  prove  directly  that  the  functions  of  u  in 
Ex.  5  and  of  lu  in  Ex.  7  are  uniform  functions  of  their  arguments. 

Consider  the  quantity  2  and  the  integral  u  connected  by  the  relation 


0 
or  by  the  differential  equation 


J  n 


with  the  condition  that  u  =  0  when  z  =  0  and  the  further  property  as  to  the 
periods  of  ti.  Evidently  the  vicinities  of  the  respective  critical  points 
1,  —1,  1/^',  —  l/k  must  be  taken  into  account;  likewise  the  vicinity  of  any 
other  finite  value  of  z ;  likewise  very  large  values  of  z.  We  take  them 
in  turn. 

In  the  vicinity  of  z=  l,\et  z  ^1  +  ^.     At  2^  =  1,  we  can  take  u  =  K  (subject 
to  periods) ;  so 

u-K=\  {-n  +  f)'''  {k'-' - 2kH - kH") "^dt 

J  0 


=  f  (-  2k''-)    'U    ''P  (t)  dt, 

J  0 


where  P  (t)  is  a  regular   function  of  t  in  the  vicinity  of  ^  =  0  such  that 
P(0)  =  1.     Thus 

where  E  (^)  is  a  regular  function  of  ^  such  that  R  (0)  =  1.     Consequently, 
z-l  =  ^=-^k''{u-KyS{u-K), 

where    S  (u  -  Z)  is    a   regular   function    of  («  -  Kf,  such    that   ^  (0)  =  1. 
Clearly  ^  is  a  regular  function  of  u  in  the  vicinity  of  the  place  z=l. 


234  UNIFORM  [104. 

Exactly  similar  analysis  shews  that  ^r  is  a  regular  function  of  u  in  the 
vicinity  of  the  place  z  =  —  l,  the  substitution  being  z  =  —  1  +  ^;  we  find 

^  +  1  =  _  1-  /c'2  (^a  +  Kf  8{n.  +  K), 
where  S  (u  +  K)  is  a  regular  function  of  (tt  +  Kf,  such  that  8  (0)  =  1. 

Again,  for  the  vicinity  of  ^^  =  Ijk,  we  take  z  -ljk=i:^'  \  we  find 

z-\=^'  ^^{u-  K  -iKJ  8  {u-  K  -iKJ, 

where  S  is  &  regular  function  of  its  argument  such  that  S  (0)  =  1. 
For  the  vicinity  of  2^  =  —  1/^,  we  find 

^  +  ^  =  ^  (^,  +  A^  +  iKy  S(u+K  +  iKJ, 

where  again  >S  is  a  regular  function  of  its  argument  such  that  8  {0)  —  \. 
Next,  for  a  value  of  |  ^  j  <  1,  we  have 

u=W{\-P){l-kH'')]~^dt 
J  0 

=  zP(z), 
where  P  (z)  is  a  regular  even  function  of  z  such  that  P  (0)  =  1.    Consequently 
^  is  a  regular  function  and  an  uneven  function  of  u  for  values  of  |  2^  |  <  1. 

For  any   ordinary  place   for  z,  given   by  ^  =  a,   let  a  value  of  u  be  a. 
Taking  z  =  a  +  Z,  we  have 

ra+Z  _i 

u-a=\        1(1  -  f)  (1  -  kH')]    ''  dt 

J  a 

= T  R  {Z), 

\{\-o?){\-k\i%-^ 
where  R  {Z)  is  a  regular  function  of  Z  such  that  P  (0)  =  1.     As  before,  Z  is 
a  regular  function  of  u  —  a  in  the  vicinity ;  that  is,  ^  is  a  regular  function  of 
u  in  the  vicinity  of  any  ordinary  place. 

Finally,  for  large  values  of  z,  say  z,  we  have 


u  = 

1 
rk 

J  0 

+  I 

J  1 

k 

{(1- 

t^)  (1  - 

kH-)]-^^ 

dt. 

In  the  integral. 

write 

kt. 

1 

then 

u  = 

1 
•k 

.   0 

kz' 

=/: 

1 
rkz' 

{(1- 

■p){i- 

-¥^^r 

■'dt' 

=  iK 

'^f: 

''{(1- 

-t'^-){i 

-k^^^yi' 

'h,'. 

104]  PERIODIC   FUNCTIONS  235 

Thus  -j—>  ^s  a  regular  function  of  u  in  the  vicinity  of  u  =  iK'  and  it  vanishes 

to  the  first  order  at  that  place.  Therefore  5  is  a  uniform  function  of  u  in 
the  vicinity  oiu  =  iK'  \  and  it  has  a  simple  pole  at  that  value. 

Hence,  in  every  case,  5  is  a  uniform  function  of  u  ;  and  this  uniform 
function  has  simple  poles  at  u  =  iK'  and  at  all  places  reducible  to  this  place 
by  multiples  of  2A'and  2iK'. 

As  already  stated,  we  shall  give  full  references  at  a  later  stage  to  the 
cases  when  a  differential  equation 

defines  ^^  as  a  uniform  function  of  u. 

Ex.  9.  Shew  that,  for  the  relation  just  discussed,  the  functions  {l  —  z^)^  and 
(1— Ps^)^  fire  uniform  functions  of  u. 

Ex.  10.     Shew  that,  when  u  and  z  are  connected  by  the  relation 

of  Ex.  7,  when  we  denote  s  as  a  function  of  m  by  ^  (u),  each  of  the  functions 

^{u),    {<p{u)-e,]h,    {<^{u)-e.,}h,    {iHy)~<^z)^ 
is  a  uniform  function  of  ?<. 

105.  We  proceed  to  the  theory  of  uniform  periodic  functions,  some 
special  examples  of  which  have  just  been  considered  ;  and  limitation  will 
be  made  here  to  periodicity  of  the  linear  additive  type,  which  is  only  a  very 
special  form  of  periodicity. 

A  function  f{z)  is  said  to  be  periodic  when  there  is  a  quantity  w  such 

that  the  equation 

f{z  +  co)=f{z) 

is  an  identity  for  all  values  of  z.  Then  f{z  +  nw)=f{z),  where  n  is  any 
integer  positive  or  negative ;  and  it  is  assumed  that  w  is  the  smallest 
quantity  for  which  the  equation  holds,  that  is,  that  no  submultiple  of  «  will 
satisfy  the  equation.     The  quantity  w  is  called  a  period  of  the  function. 

A  function  is  said  to  be  simply -periodic  when  there  is  only  a  single 
period:  to  be  doubly -periodic  when  there  are  two  periods;  and  so  on,  the 
periodicity  being  for  the  present  limited  to  additive  modification  of  the 
argument.  Moreover,  we  exclude  the  possibility  of  periods  that  can  be 
made  less  than  any  finite  quantity,  however  small.  If  such  infinitesimal 
periods  were  admissible  for  a  uniform  function,  then  within  a  finite  region 
(however  small)  round  any  point  the  function  would  acquire  the  same  value 
an  unlimited  number  of  times.  Then  the  uniform  function  would  either  be 
constant  everywhere  within  that  finite   region  and   so  would  be   constant 


236  UNIFORM   PERIODIC    FUNCTIONS  [105. 

everywhere :  or  it  would  possess  an  unlimited  number  of  constant  values 
within  that  region  :  or  an  unlimited  number  of  infinities  within  the  region. 
In  the  second  case,  its  derivative  would  possess  an  unlimited  number  of  zeros 
in  the  region,  which  is  any  small  region  round  any  point :  as  at  the  end  of 
§  37,  the  point  would  be  an  essential  singularity.  Similarly,  in  the  third  case, 
the  point  would  be  an  essential  singularity.  Each  of  the  alternatives,  conse- 
quent upon  the  possession  of  an  infinitesimal  period,  is  to  be  excluded  :  hence 
we  also  exclude  the  possibility  of  infinitesimal  periods. 

It  is  convenient  to  have  a  graphical  representation  of  the  periodicity  of  a 
function. 

(i)  For  simply-periodic  functions,  we 
take  a  series  of  points  0,  A^,  A^,..., 
A_i,  A^2,---  representing  0,  co,  2a>,..., 
—  CO,  —  2(0, ...;  and  through  these  points 
we  draw  a  series  of  parallel  lines,  dividing 
the  plane  into  bands.  Let  P  be  any 
point  z  in  the  band  between  the  lines 
through  0  and  through  A^;  through  P 
draw  a  line  parallel  to  OA^  and  measure 
off  PPi-P^Ps  =  . . .  =  PP-,  =P-,P-,== . . ., 
each  equal  to  OA^ ;  then  all  the  points 

Pi,  Pg,...,  P-i,  P-2,...  are  represented  ^ig-  30. 

hy  z  +  nco  for  positive  and  negative  integral  values  of  n.  But  f{z  +  no))  =f{z); 
and  therefore  the  value  of  the  function  at  a  point  P„  in  any  of  the  bands  is 
the  same  as  the  value  at  P.  Moreover,  to  a  point  in  any  of  the  bands  there 
corresponds  a  point  in  any  other  of  the  bands ;  and  therefore,  owing  to  the 
periodic  resumption  of  the  value  at  the  points  corresponding  to  each  point  P, 
it  is  sufficient  to  consider  the  variation  of  the  function  for  points  within  one 
band,  say  the  band  between  the  lines  through  0  and  through  A^^.  A  point  P 
within  the  band  is  sometimes  called  irreducible,  the  corresponding  points  P 
in  the  other  bands  reducible. 

If  it  were  convenient,  the  boundary  lines  of  the  bands  could  be  taken 
through  points  other  than  A-^,  A^_,  ...\  for  example,  through  points  (m  -f  ■^)  cd 
for  positive  and  negative  integral  values  of  m.  Moreover,  they  need  not  be 
straight  lines.  The  essential  feature  of  the  graphic  representation  is  the 
division  of  the  plane  into  bands. 

(ii)  '  For  doubly-periodic  functions  a  similar  method  is  adopted.  Let  w 
and  w'  be  the  two  periods  of  such  a  function  f{z),  so  that 

/(5  +  a))=/»=/(5  +  a.'); 

then  f{z  -h  nu)  +  n'co')  =f{z), 

where  n  and  n  are  any  integers  positive  or  negative. 


.105.] 


DOUBLE   PERIODICITY 


237 


For  graphic  purposes,  we  take  points  0,  ^i,  J.,,, ...,  J._i,  J._2, ...  representing 
0,  &),  2co, . . . ,  —  oj,  —  2&), . . .  ;  and  we  take 
another  series  0,B-^,B.2, ... ,  B_^,  B^n,  . . . 
representing  0,  w,  2<w', . .  ,  —  a>',  —  2(o', . . . ; 
through  the  points  A  we  draw  lines 
parallel  to  the  line  of  points  B,  and 
through  the  points  B  we  draw  lines 
parallel  to  the  line  of  points  A.  The 
intersection  of  the  lines  through  An 
and  Bn'  is  evidently  the  point  nco  +  n'co', 
that  is,  the  angular  points  of  the 
parallelograms  into  which  the  plane  is 
divided  represent  the  points  nco  +  n'w' 
for  the  values  of  n  and  n. 

Let  P  be  any  point  z  in  the  paral-  Fig.  31. 

lelogram  OA-JJ-^B^ ;  on  lines  through  P, 

parallel  to  the  sides  of  the  parallelogram,  take  points  Qi,  Q^, ... ,  Q-i,  Q-2,  ••• 
such  that  PQi  =  QiQo=  ...  =  <w,  and  points  R^,  R^, ...,  R-i,  R-2,  •••  such  that 
PRi  =  RiRo  =  ...  =0)' ;  and  through  these  new  points  draw  lines  parallel  to 
the  sides  of  the  parallelogram.  Then  the^variables  of  the  points  in  which 
these  lines  intersect  are  all  represented  hj  2  +  mco  +  m'co'  for  positive  and  nega- 
tive integral  values  of  in  and  m' ;  and  the  point  represented  by  5  -f  mco  +  mw 
is  situated  in  the  parallelogram,  the  angular  points  of  which  are  mw-\-mw, 
(m  +  1)  CO  +  m'ct)',  mw  +  {m  +!)&)',  and  (m  +  1)  co  +  {m  +  1)  w  ,  exactly  as  P  is 
situated  in  OA^C-^B^.     But 

f  {z  -\-  viu)  +  m'&)')  =f{z), 
and  therefore  the  value  of  the  function  at  such  a  point  is  the  same  as  the 
value  at  P.  Since  the  parallelograms  are  all  equal  and  similarly  situated, 
to  any  point  in  any  of  them  there  corresponds  a  point  in  OAiCiB^^;  and  the 
value  of  the  function  at  the  two  points  is  the  same.  Hence  it  is  sufficient  to 
consider  the  variation  of  the  function  for  points  ivithin  one  parallelogram,  say, 
that  which  has  0,  &),  &>  +  m,  w  for  its  angular  points.  A  point  P  within 
this  parallelogram  is  sometimes  called  irreducible,  the  corresponding  points 
within  the  other  parallelograms  reducible  to  P  ;  the  whole  aggregate  of  the 
points  thus  reducible  to  any  one  are  called  homologous  points.  And  the 
parallelogram  to  which  the  reduction  is  made  is  called  the  parallelogram  of 
periods. 

As  in  the  case  of  simplj^-periodic  functions,  it  may  prove  convenient  to 
choose  the  position  of  the  fundamental  parallelogram  so  that  the  origin  is 
not  on  its  boundary;  thus  it  might  be  the  parallelogram  the  middle  points  of 
whose  sides  are  +  \co,  ±  ^-w'. 

Ex.  Shew  how  to  reduce  a  given  point  numerically ;  for  instance,  find  the  irreducible 
point  homologous  to  730  +  ■''i82^■  for  periods  l  +  9i,  3  +  2z'. 


238  RATIO   OF   THE    PERIODS   OF  [106. 

106.  In  the  preceding  representation  it  has  been  assumed  that  the  line 
of  points  A  is  ditferent  in  direction  from  the  line  of  points  B.  If  w  =  m  +  iv 
and  (o'  =  u+iv',  this  assumption  implies  that  v'/u'  is  unequal  to  v/u,  and 
therefore  that  the  real  part  of  w'/ico  does  not  vanish.  The  justification  of 
this  assumption  is  established  by  the  proposition,  due  to  Jacobi*  : — 

The  ratio  of  the  periods  of  a  uniform  doubly-periodic  function  cannot  he 
real. 

Let/(2)  be  a  function,  having  w  and  w  as  its  periods.  If  the  ratio  w'/fw 
be  real,  it  must  be  either  commensurable  or  incommensurable. 

If  it  be  commensurable,  let  it  be  equal  to  n'jn,  where  n  and  n  are 
integers,  neither  of  which  is  unity  owing  to  the  definition  of  the  periods  w 
and  ct)'. 

Let  n'jn  be  developed  as  a  continued  fraction,  and  let  yn'lm  be  the  last 
convergent  before  nfn,  where  m  and  m  are  integers.     Then 

n'      m  _    1 
n      m      inn ' 

that  is,  *'»^'  ~  '>n'n  =  1, 

so  that  m'co  ^moi  =--  {m  n  ~  mn  )  =  -. 

to  Iti 

Therefore  f{z)  =f{z  +  m'o)  ~  mco'), 

since  m  and  m'  are  integers ;  so  that 

contravening  the  definition  of  w  as  a  period,  viz.,  that  no  submultiple  of  m  is  a 
period.  Hence  the  ratio  of  the  periods  is  not  a  conniiensurable  real  quantity. 
If  it  be  incommensurable,  we  express  co'/w  as  a  continued  fraction.  Let 
pjq  and  p  jq^  be  two  consecutive  convergents :  their  values  are  separated  by 
the  value  of  w/o),  so  that  we  may  write 

w     q      \q     q/ 

where  l>h>0. 

Now  pq  -p'q  =  1,  so  that 


o)       q      qq" 


where  e  is  real  and  lej  <  1 ;  hence 


,  e 

qo)  —po}  =  —,  CO. 

Ges.  Werke,  t.  ii,  pp.  25,  26. 


106.]  A   UNIFORM   DOUBLY-PERIODIC    FUNCTION  239 

Therefore  f{z)  =f{z  +  qw'  -  pco), 

since  p  and  q  are  integers ;  so  that 


f(z)=f[z  +  ^^co 

Now  since  co'/co  is  incommensurable,  the  continued  fraction  is  unending.  We 
therefore  can  take  an  advanced  convergent,  so  that  q'  is  very  large ;  and  we 

choose  it  so  that    —  6)    is  less  than  any  assigned  positive  quantity,  however 

a  ■ 

small.     But  -co  is   equal  to   qco'  —pco,  where  q  and  p  are  integers,  and  it 

therefore  is  a  period  of  the  function  f{z).  Hence,  on  the  assumption  that 
ft)7«  is  real  and  incommensurable,  it  follows  that  the  function  possesses  an 
infinitesimal  period :  the  possibility  of  which  was  initially  excluded  (§  105). 

The  ratio  of  the  periods  is  thus  not  an  incommensurable  real  quantity. 

We  therefore  infer  Jacobi's  theorem  that  the  ratio  of  the  periods  cannot 
be  real.  In  general,  the  ratio  is  a  complex  quantity;  it  may,  however,  be  a 
pure  imaginary*. 

Corollary.  If  a  uniform  function  have  two  periods  a^  and  m^,  such  that 
a  relation 

??2-iC0i  +  nu(02  =  0 

exists  for  integral  values  of  m^  and  niz,  the  function  is  only  simply-periodic. 
And  such  a  relation  cannot  exist  between  two  periods  of  a  simply-periodic 
function,  if  mj  and  m2  be  real  and  incommensurable ;  for  then  the  function 
would  have  an  infinitesimal  period. 

Similarly,  if  a  uniform  function  have  three  periods  Wj,  Wg,  cog,  connected 
by  two  relations 

7?ii&)i  -I-  m20).2  +  m^cos  —  0, 

?li&)i  -I-  722(1)2  +  "^Isf^s  =  0, 

where  the  coefficients  m  and  n  are  integers,  then  the  function  is  only  simply- 
periodic. 

107.  The  two  following  propositions,  also  due  to  Jacobif,  are  important 
in  the  theory  of  uniform  periodic  functions  of  a  single  variable  : — 

If  a  uniform  function  have  three  periods  w^,  w^,  co-i,  such  that  a  relation 

niiCOi  +  ni2002  +  ms&jg  =  0 

is  satisfied  for  integral  values  of  n^,  m^,  m^,  then  the  function  is  only  a  doubly- 
periodic  function. 

*  It  was  proved,  in  Ex.  5  and  Ex.  7  of  §  104,  that  certain  uniform  functions  are  doubly -periodic. 
A  direct  proof,  that  the  ratio  of  the  distinct  periods  of  the  functions  there  obtained  is  not  a  real 
quantity,  is  given  by  Falk,  Acta  Math.,  t.  vii,  (1885),  pp.  197 — 200,  and  by  Pringsheim,  Math. 
Ann.,  t.  xxvii,  (1886),  pp.  151—157. 

t  Ges.  Werke,  t.  ii,  pp.  27—32. 


240  JACOBl'S  THEOREMS  ON  THE  PERIODS  OF  [107. 

What  has  to  be  proved,  in  order  to  establish  this  proposition,  is  that  two 
periods  exist  of  which  coi,  coo,  Wj  are  integral  multiple  combinations. 

Evidently  we  may  assume  that  ???i,  mg,  vu  have  no  common  factor:  let  / 
be  the  common  foctor  (if  any)  of  m^  and  nig,  which  is  prime  to  m^.  Then 
since 

and  the  right-hand  side  is  an  integral  combination  of  periods,  it  follows  that 
-^  &)i  IS  a  period. 

]SI'ow  ^  is  a  fraction  in  its  lowest  terms.      Change  it  into  a  continued 

fraction  and  let  -  be  the  last  convergent  before  the  proper  value;  then 

"^  _  1?  =  +  2l 

/    q    ~f<i 

Wlj  1 

so  that  <1^-P=±f- 

But  (Oj  is  a  period  and  -j  a\  is  a  period;  therefore  g  — r^  Wj  —  pcoi  is  a  period, 
or  coi/f  is  a  period,  =  o)/  say. 

Let  ^2//=  W,  m^lf=m^,  so  that  m^w^  +  vuw.i  + nh  (0-^  =  0.  Change 
7*12 /nis  into  a  continued  fraction,  taking  -  to  be  the  last  convergent  before  the 

proper  value,  so  that 

ma'      r  _        1 

nis       s      "  snis  ' 

Then  ?'&)„  + sojg,  being  an  integral  combination  of  periods,  is  a  period.     But 
±  (1)2=  0)2 (snh  —  rm-i) 

=  —  roi.m^  —  s  (miCOi  +  M^'co-i) 

=  —  miScoi  —  m^  (rcoo  +  sws) ; 
also  +  W3  =  &.>3 (snh  - rm^) 

--^  smJwz  +  ^  (?n](Wi'  +  "io'wo) 

=  7?iir&)i'  +  m^  (rft).2  +  sw-^  ; 
and  &)i  =f(Oi. 

Hence  two  periods  &)/  and  rco^  +  soos  exist  of  which  co^,  coo,  w^  are  integral 
multiple  combinations  ;  and  therefore  all  the  periods  are  equivalent  to  o)i  and 
rcoo  +  scos,  that  is,  the  function  is  only  doubly-periodic. 


107.]  UNIFORM  PERIODIC  fun;ctions  241 

Corollary.  If  a  function  have  four  periods  to^,  Wo,  (o-,,  w^  connected  by 
two  relations 

iiiWi  -\-  HqCOo  +  ?i3&>3  +  n^w^  =  0, 

where  the   coefficients  m  and  n  are   integers,  the   function  is  only  doubly- 
periodic. 

108.  If  a  uniforiu  function  of  one  variable  have  three  periods  w,,  Wo,  w., 
then  a  relation  of  the  form 

nijO)^  +  7n.2a)2  +  "jg&jo  =  0 

must  be  satisfied  for  some  integral  values  of  vii,  uio,  ni-^. 

Let  Wr  =  «;•  +  ^/3r,  for  r  =  1,  2,  3  ;  in  consequence  of  §  106,  we  shall  assume 
that  no  one  of  the  ratios  of  Wy,  &>2,  w.^  in  pairs  is  real,  for,  otherwise,  either 
the  three  periods  reduce  to  two  immediately,  or  the  function  has  an  infini- 
tesimal period.     Then,  determining  twp/quantities  \  and  jx  by  the  equations 

OTg  =  XCKj  +  fji^.^,       l3s  =  X/8i  +  1X^.2, 

so  that  A,  and  /x  are  real  quantities  and  neither  zero  nor  infinity,  we  have 

Wo  =  A,&)i  -f  jjiw.T, 
for  real  values  of  X  and  jx. 

Then,  first,  if  either  X  or  yu,  be  commensurable,  the  other  is  also  commen- 
surable.    Let  X  =  ajb,  where  a,  and  b  are  integers;  then 

6/iCt)o  =  6(1)3  ~  b\(o-y 

=  bcos  —  aa)i, 

so  that  6/xft»2  is  a  period.  Now,  if  bfx  be  not  commensurable,  change  it  into  a 
continued  fraction,  and  let  pjq,  p'/q'  be  two  consecutive  convergents,  so  that, 
as  in  §  106, 

T  p  X 

bfx  =  -^+~, 
q      qq 

where  1  >  ^-  >  —  1.     Then  -  &>.,  -I y  is  a  period,  and  so  is  w, ;  hence 

q     '      qq  ^ 

(p  xw.,\ 

q[^  w^-] A- po).2 

\q  qq  J 

is  a  period,  that  is,  —  w^  is  a  period.     We  may  take  q  indefinitely  large,  and 

then  the  function  has  an  infinitesimal  quantity  for  a  period,  which  has  been 
excluded  by  our  initial  argument.  Hence  bp,  (and  therefore  /x)  cannot  be 
incommensurable,  if  X  be  commensurable;  and  thus  X  and  p.  are  simul- 
taneously commensurable  or  simultaneously  incommensurable. 

F.  P.  16 


242  JACOBI'S  THEOREMS  ON  THE  PERIODS  OF  [108. 


that 


If  \  and  fi  be  simultaneously  commensurable,  let  A,  =  t-  ,  /x  =  -7  ,  so 

a  c 

and  therefore  bdcos  =  adw-^  +  hcu)^, 

a  relation  of  the  kind  required. 

If  \  and  fx  be  simultaneously  incommensurable,  express  X  as  a  continued 
fraction;  then  by  taking  any  convergent  rjs,  we  have 


X- 

r 

X 

0  ) 

s 

s- 

s\  - 

-  r 

X 

"J' 

where  1  >  ^  >  —  1,  so  that 

by  taking  the  convergent  sufficiently  advanced  the  right-hand  side  can  be 
made  infinitesimal. 

Let  Vy  be  the  nearest  integer  to  the  value  of  Syu.,  so  that,  if 

s^i  -r,  =  /\, 
we  have  A  numerically  not  greater  than  ^.     Then 

X 

so)s  —  rwi  —  7\oi.->  —  —  (01  +  Zifi).2, 
s 

and  the  quantity  -  coi  can  be  made  so  small  as  to  be  negligible.  Hence 
•integers  r,  r-^^,  s  can  be  chosen  so  as  to  give  a  new  period  cog'  (=  Awa),  such 
that  I  Wa'  j  ^  -2  I  &>2 1- 

We  now  take  Wi,  co^',  0)^:  ihej  will  be  connected  by  a  relation  of  the  form 

and  A,'  and  fi'  must  be  incommensurable :  for  otherwise  the  substitution  for 
&)./  of  its  value  just  obtained  would  lead  to  a  relation  among  &)i,  Wg,  0)3  that 
would  imply  commensurability  of  \  and  of  yu,. 

Proceeding  just  as  before,  we  may  similarly  obtain  a  new  period  Wg"  such 
that  I  Wg"  \^^\a>2\]  and  so  on  in  succession.     Hence  we  shall  obtain,  after  n 

such  processes,  a  period  0)3"*'  such  that  |  Wo""  |  ^  ^J  coa  1,  so  that  by  making  n 

sufficiently  large  we  shall  ultimately  obtain  a  period  less  than  any  assigned 
quantity.  Such  a  period  is  infinitesimal ;  and  infinitesimal  periods  were 
initially  excluded  (§  105)  for  reasons  there  given.  Thus  \  and  fi  cannot 
be  simultaneously  incommensurable. 

Hence  the  only  constructive  result  is  that  X  and  /i  are  simultaneously 
commensurable ;  and  then  there  is  a  period-equation  of  the  form 

where  wij,  m.j,  mj  are  integers. 


108.]  UNIFORM   PERIODIC    FUNCTIONS  243 

The  foregoing  proof  is  substantially  due  to  Jacobi  (I.e.).  The  result  can 
be  obtained  from  geometrical  considerations  by  shewing  that  the  infinite 
number  of  points,  at  which  the  function  resumes  its  value,  along  a  line 
through  z  parallel  to  the  Wg-line  will,  unless  the  condition  be  satisfied,  reduce 
to  an  infinite  number  of  points  in  the  w^,  co^  parallelogram  which  will  form 
either  a  continuous  line  or  a  continuous  area,  in  either  of  which  cases  the 
function  would  be  a  constant;  or  there  will  be  an  unlimited  number 
condensed  in  any  region  round  z,  however  small,  thus  making  the  point 
an  essential  singularity,  which  is  impossible  for  every  point  z.  But,  if  the 
condition  be  satisfied,  then  the  points  along  the  line  through  z  reduce  to  only 
a  finite  number  of  points*. 

Corollary  I.  Uniform  functions  of  a  single  variable  cannot  have  throe 
independent  periods ;  in  other  words,  triply -periodic  uniform  functions  of  a 
single  variable  do  not  exist  ^i  and,  a  fortiori,  uniform  functions  of  a  single 
variable  tuith  a  number  of  independent  periods  greater  than  tiuo  do  not  exist. 

But  functions  involving  more  than  one  variable  can  have  more  than  two 
periods,  e.g.,  Abelian  transcendents ;  and  a  function  of  one  variable,  having 
more  than  two  periods,  is  tiot  uniform. 

Corollary  II.  All  the  periods  of  a  uniform,  periodic  function  of  a 
single  variable  reduce  either  to  integral  multiples  of  one  period  or  to  linear 
combinations  of  integral  multiples  of  tivo  periods  whose  ratio  is  not  a  real 
quantity. 

109.  It  is  desirable  to  have  the  parallelogram,  in  which  a  doubly- 
periodic  function  is  considered,  as  small  as  possible.  If  in  the  parallelogram 
(supposed,  for  convenience,  to  have  the  origin  for  an  angular  point)  there  be 
a  point  &)",  such  that 

f{z+C0"):^f{z) 

for  all  values  of  z,  then  the  parallelogram  can  be  replaced  by  another. 

It  is  evident  that  w"  is  a  period  of  the  function;  hence  (§  108)  we  must 
have 

&) "  =  \(0  +  yu-o)   ; 

and  both  A,  and  fx,  which  are  commensurable  quantities,  are  less  than  unity 
since  the  point  is  within  the  parallelogram.  Moreover,  a  -\-o)'  —  w",  which 
is  equal  to  (1  —  X)  to  +  (1  —  /a)  to',  is  another  point  within  the  parallelogram  ; 
and 

f  {z -^^  w  +  w  -  Oi")  =  f  {z), 

since  w,  w',  w"  are  periods.     Thus  there  cannot  be  only  one  such  point  unless 

X  =  ^  =  /u.. 

*  For  another  proof,  see  Goursat,  Cours  d' analyse  n.athematique,  t.  ii,  §  324. 
t  This  theorem  is  also  due  to  Jacobi,  (I.e.,  p.  239,  note). 

16—2 


244  FUNDAMENTAL  [109. 

But  the  number  of  such  points  within  the  parallelogram  must  be  finite. 
If  there  were  an  infinite  number,  they  would  form  a  continuous  line  or  a 
continuous  area  where  the  uniform  function  had  an  unvarying  value,  and 
the  function  would  have  a  constant  value  everywhere ;  or  they  would 
condense  within  any  region  (however  small)  round  any  point,  and  so  would 
make  the  point  an  essential  singularity,  a  result  to  be  excluded  as  in  §  37. 

To  construct  a  new  parallelogram  when  all  the  points  are  known,  we  first 
choose  the  series  of  points  parallel  to  the  co-line  through  the  origin  0,  and  of 
that  series  we  choose  the  point  nearest  0,  say  A^.  We  similarly  choose  the 
point,  nearest  the  origin,  of  the  series  of  points  parallel  to  the  co-line  and 
nearest  to  it  after  the  series  that  includes  A^,  say  B^:  we  take  OA-^,  OB^  as 
adjacent  sides  of  the  parallelogram,  and  these  lines  as  the  vectorial  repre- 
sentations of  the  periods.  No  point  lies  within  this  parallelogram  where  the 
function  has  the  same  value  as  at  0  ;  hence  the  angular  points  of  the  original 
parallelograms  coincide  with  angular  points  of  the  new  parallelograms. 

When  a  parallelogram  has  thus  been  obtained,  containing  no  internal 
point  n  such  that  the  function  can  satisfy  the  equation 

f{z  +  ^)=f{z)         •' 

for  all  values  of  z,  it  is  called  a  fundamental,  or  a  primitive,  'parallelogram. 
The  parallelogram  of  reference  in  subsequent  investigations  will  be  assumed 
to  be  of  a  fundamental  character. 

But  a  fundamental  parallelogram  is  vot  unique. 

Let  (o  and  co'  be  the  periods  for  a  given  fundamental  parallelogram,  so 
that  every  other  period  &>"  is  of  the  form  \(o  +  /xco',  where  \  and  /j,  are 
integers.  Take  any  four  integers  a,  h,  c,  d  such  that  ad  —  bc  —  ±  1,  as  may 
be  done  in  an  infinite  variety  of  ways;  and  adopt  two  new  periods  coj  and  co,, 
such  that 

&>!  =  aa>  +  bo)',  o).i  =  Cay  +  dco'. 

Then  the  parallelogram  with  co^  and  coo  for  adjacent  sides  is  fundamental. 
For  we  have 

±  0)  =  da>i  —  bo)2,  ±  co' =  — ca)i  + aa)2, 

and  therefore  any  period  co" 

=  X(o  +  /jLco' 

=  {Xd  —  fxc)  coj  +  (—  X6  -t-  /Lta)  (jci.2,  save  as  to  signs  of  \  and  jx. 

The  coefficients  of  coj  and  w,,  are  integers,  that  is,  the  point  co'  lies  outside 
the  new  parallelogram  of  reference;  ther.e  is  therefore  no  point  in  it  such  that 

f{z  +  co")=f{z), 
and  hence  the  parallelogram  is  fundamental. 


109.]  PARALLELOGEAM  245 

CoROLT.AEY.  The  aggregate  of  the  angular  points  in  one  division  of  the 
plane  into  fundamental  parallelograms  coincides  with  their  aggregate  in 
any  other  division  into  fundamental  parallelograms ;  and  all  fundamental 
parallelograms  for  a  given  function  are  of  the  same  area. 

The  method  suggested  above  for  the  construction  of  a  fundamental  parallelogram  is 
geometrical,  and  it  assumes  a  knowledge  of  all  the  points  w"  within  a  gi\en  pai'allelograra 

for  which  the  equation  f{z  +  a)")  =  f{z)  is  satisfied. 

Such  a  point  0)3  within  the  wj,  C02  parallelogram  is  given  by 

mi  m, 

TO3  m-s    -' 

where  mj,  m^,  m^  are  integers.  We  may  assume  that  no  two  of  these  three  integers 
have  a  common  factor;  were  it  otherwise,  say  for  Wj  and  m^,  then,  as  in  §  107,  a 
submultiple  of  0)3  would  be  a  period — a  result  which  may  be  considered  as  excluded. 
Evidently  all  the  points  in  the  parallelogram  are  the  reduced  points  homologous  with 

<B3,  2a)3,  ,    (^3  — 1)003;    'when    these   are   obtained,  the   geometrical    construction   is 

possible. 

The  following  is  a  simple  and  practicable  analytical  method  for  the  construction. 

Change  mi/tn^  and  ?«2/'"3  i"^^'"  continued  fractions ;  and  let  p/g  and  r/s  be  the  last 
convergents  before  the  respective  proper  values,  so  that 


where 


where  X  and  fj.  are  taken  to  be  less  than  m^,  but  they  do  not  vanish  because  q  and  s  are 
less  than  '/H3.     Then 

qa^—pcoi  —  ^0)2  =  —  (;x&)2  +  e(Bi),         Soo^  —  rwo  —(f)a>i  =  —  ('^'"i  +  e'&)2)  > 

the  left-hand  sides  are  periods,  say  Oj  and  Q2  respectively,  and  since  ja  +  e  is  not  >  ms  and 
A  +  e'  is  not  >  ms,  the  points  Qj  and  0,2  determine  a  parallelogram  smaller  than  the  initial 
parallelogram. 

Thus  ecBl  +  /^(B2  =  H^3Ql,  Xcui -f- c'a)2  =  OT3Q2 ) 

are  equations  defining  new  periods  Hi,  i22.     Moreover 

J   .    X        TO,        p        es  .  ,    fjL         m^        r       e'q 

^     m^       ms       q      qm^  m^     ^  m^     -'  s      sm^ 

so  that,  multiplying  the  right-hand  sides  together  and  likewise  the  left-hand  sides,  we 
at  once  see  that  X^-ee'  is  divisible  by  7n.^  if  it  be  not  zero:  let 

\fi  -  ee'  =  Wig  \. 

Then,  as  X  and  fj.  are  less  than  m^,  they  are  greater  than  A;  and  they  are  prime  to  it, 
because  ee'  is  +  1.     Hence  we  have 

Acoi  =  jLifi2~  «'^])         AQ)2  =  Xi2|  -  6Q2- 


»ll       p         e 
7713      q  ~  qyiis ' 

5?l2         ^'  _     f 

ms      s      sni;, 

them  ±  1.     Let 

TO3              Wis 

5  — =  6-1-  — 

TO3     ^     ms 

246  FUNDAMENTAL   PARALLELOGRAM  [109. 

Since  X  and  /x  arc  both  greater  than  A,  let 

X=XiA  +  X',         ij.  =  ^i^+fj.', 

where  X'  and  /x'  are  <  A.     Then  X'/x'  -  ee'  is  divisible  by  A  if  it  be  not  zero,  say 

X'/x'  — 66'  =  AA' ; 

then  X'  and  fi'  are  >  A'  and  are  prime  to  it.     And  now 

A  {o)^  -  fjnSl2)  — 1^'^2—  f'^i)         ^  ('"2  ~  XiOj)  =  X'Qi  —  6^2  ; 

thus,  if  a>i-  fjLiSl2=^3,  co2-XjOi  =  f24,  which  are  periods,  we  have 

AQ,o=ij.'Q.o-€'Sli,  An4  =  X'Oi-6i22- 

With  Q3  and  ^4  we  can  construct  a  parallelogram  smaller  than  that  constructed 
with  Oj  and  Qn.     We  now  have 

A'£2]  =  612,3 +  /x'i24,  A'02  =  ^'^3  +  f'^4) 

that  is,  equations  of  the  same  form  as  before.  We.  proceed  thus  in  successive  stages : 
each  quantity  A  thus  obtained  is  distinctly  less  than  the  preceding  A,  and  so  finally  we 
shall  reach  a  stage  when  the  succeeding  A  would  be  unity,  that  is,  the  solvition  of  the  pair 
of  equations  then  leads  to  periods  that  determine  a  fundamental  jaarallelogram.  It  is 
not  difficult  to  prove  that  wi,  0)2,  CB3  are  combinations  of  integral  multiples  of  these 
periods.  ^ 

If  one  of  the  quantities,  such  as  X'/x'-«')  be  zei'o,  then  X'  =  yx'  =  l,  e  =  e'=±l;  and 
then  123  and  i24  are  identical.  If  €  =  e'=  +  l,  then  AQ3=Q2-i2i,  and  the  fundamental 
parallelogram  is  determined  by 

r  n,'  =  Sl^-\ (Qg-^l),  04'  =  02--  (02-i2i). 

If  €  =  €  =  —  1,  then  A123=r02-|-i2j,  so  that,  as  A  is  not  unity  in  this  case,  the  fimdamental 
parallelogram  is  determined  by  O2  and  SI3.        , 

Sx.     If  a  function  be  periodic  in  a>i,  0)2,  and  also  in  a>3  where 

29a)3=  17coi  +  lla)2, 

periods  for  a  fundamental  parallelogram  are 

Qj' =  5ci)i  +  3cB2  ~  8^3 ,  i22'  =  '^<"i  +  2ci)2—  5w3, 

and  the  values  of  coi,  wo,  013  in  terms  of  O/  and  Qo'  are 

coj  =  fi/ +  3l2i',         (B2  =  9fi2'-2fii',         a)3  =  4fi2'  +  fii'. 

Further  discussion  relating  to  the  transformation  of  periods  and  of  fundamental 
parallelograms  will  be  found  in  Briot  and  Bouquet's  Theorie  des  fonctions  ellipdques, 
pp.  234,  235,  268—272. 

110.  It  has  been  proved  that  uniform  periodic  functions  of  a  single 
variable  cannot  have  more  than  two  periods,  independent  in  the  sense  that 
their  ratio  is  not  a  real  quantity.  If  then  a  function  exist,  which  has  two 
periods  with  a  real  incommensurable  ratio  or  has  more  than  two  independent 
periods,  either  it  is  not  uniform  or  it  is  a  function  (whether  uniform  or 
multiform)  of  more  variables  than  one. 


110.]  MULTIPLE   PERIODICITY  247 

When  restriction  is  made  to  uniform  functions,  the  only  alternative  is 
that  the  function  should  depend  on  more  than  one  variable. 

In  the  case  when  three  periods  co^,  w^,  w^  (each  of  the  form  a  +  i^)  were 
assigned,  it  was  proved  that  the  necessary  condition  for  the  existence  of  a 
uniform  function  of  a  single  variable  is  that  finite  integers  nii,  m^,  m^  can 
be  found  such  that 

m-^a-^ -{- iiincn^ -\- m-^az  =  0 , 

nij  /3i  +  wia/^a  +  in.,  183  =  0  ; 

and  that,  if  these  conditions  be  not  satisfied,  then  finite  integers  m^,  m^,  m^ 
can  be  found  such  that  both  Swa  and  -mj3  become  infinitesimally  small. 

This  theorem  is  purely  algebraical,  and  is  only  a  special  case  of  a  more 
general  theorem  as  follows  : — 

Let  ail,  «i2,  •••  >  ^\,r+i',  «2i:  «22.  •••  ,  «2,r+i;  ••• ;  ot-n,  oi.ro,  ...  ,  w,-, r+i  ^&  ^  sets  of 
real  quantities  such  that  a  relation  of  the  form 

?li  ttsi  +  71.20(^2+  ...  -r  '«r+i  "s,  r+i  =  0 

is  not  satisfied  among  any  one  set.  Then  finite  integers  m^,  ...,  Trir+i  can  he 
determined  such  that  each  of  the  sums 

m  1  a^i  +  7?i  o  ofgo  +  . . .  +  ???,;.+i  a,^  ,.+1 

(for  s=l,  2,  ...,  r)  can  be  made  less  than  any  assigned  quantity,  however 
small.  And,  a  fortiori,  if  fewer  than  r  sets,  each  containing  r  +  1  quantities 
be  given,  the  r  +  1  integers  can  be  determined  so  as  to  lead  to  the  result 
enunciated ;  all  that  is  necessary  for  the  purpose  being  an  arbitrary  assign- 
ment of  sets  of  real  quantities  necessary  to  make  the  number  of  sets  equal  to 
r.     But  the  result  is  not  true  if  more  than  r  sets  be  given. 

We  shall  not  give  a  proof  of  this  general  theorem  * ;  it  would  follow  the 
lines  of  the  proof  in  the  limited  case,  as  given  in  §  108.  But  the  theorem 
can  be  used  to  indicate  how  the  value  of  an  integral  with  more  than  two 
periods  is  affected  by  the  periodicity. 

Let  /  be  the  value  of  the  integral  taken  along  some  assigned  path  from 
an  initial  point  z^  to  a  final  point  z;  and  let  the  perio*ds  be  rui,  Wo,  ...,  &>,., 
(where  r  >  2),  so  that  the  general  value  is 

I  +  nil  &)i  +  m2  &)2  +  . . .  +  nir  coy , 

where  mi,  ma,...,??^,.  are  integers.  Now  if  a)s  =  o(s  +  i^s,  for  s  =  l,  2,...,r, 
when  it  is  divided  into  its  real  and  its  imaginary  parts,  then  finite  integers 
/ii ,  Wo ,  . . . ,  Ur  can  be  determined  such  that  * 

n^cti  +  n^Oc,  +  . . .  +  n,.0(r 

n^l3^  +  n.2/32+  ...+nr^r 

*  A  proof  will  be  found  in  Clebsch  and  Gordan's  Theorie  der  AbeVschen  Ftmctmnen,  §  38. 
See  also  Baker's  Abelian  Functions,  chapters  ix,  xix,  where  full  references  will  be  found. 


248  MULTIPLE    PERIODICITY  [110. 

can  be  made  infinitesimal,  that  is,  less  than  any  assigned  quantity,  however 

small;  and  then     2??s&)s|  is  infinitesimal.     But  the  addition  of  '^yigCOg  still 

gives  a  value  of  the  integral ;  hence  the  value  can  be  modified  by  infinitesimal 
quantities,  and  the  modification  can  be  repeated  indefinitely.  The  modifica- 
tions of  the  value  correspond  to  modifications  of  the  path  from  z^  to  z;  and 
hence  the  integral,  regarded  as  depending  on  a  single  variable,  can  be  made, 
by  modifications  of  the  path  of  the  variable,  to  assume  any  value.  The 
integral,  in  fact,  has  not  a  definite  value  dependent  solely  upon  the  final 
value  of  the  variable ;  to  make  the  value  definite,  the  path  by  which  the 
variable  passes  from  the  lower  to  the  upper  limit  must  be  specified. 

It  will  subsequently  (§  239)  be  shewn  how  this  limitation  is  avoided  by 
making  the  integral,  regarded  as  a  function,  depend  upon  a  proper  number 
of  independent  variables — the  number  being  greater  than  unity. 

Ex.  1.     If  Fq  be  the  value  of   I -,  ,  {n  integral),  taken  along  an  assigned  path, 


and  if 


^     „  /"'       dx       ,         ,, 

P=2       -.  (.^■real)J 

Jo  {\-x^)-i 


0  (1  -  x^y^ 

then  the  general  value  of  the  integral  is 

(-i)«r„+p  " 


!{!_(_  1)9}+    2  m^e 


where  q  is  any  integer  and  m.p  any  positive  or  negative  integer  such  that  2  TOp=0. 

(Math.  Trip.,  Part  II.,  1889.) 

Ex.  2.  If,  in  an  integration  in  regard  to  the  complex  variable  5,  (a,.&g...)  denote  a 
contour  enclosing  the  "critical"  points  a,.,  6^,  ...  ;  and,  for  two  points,  (a,.;  6s)  denote  the 
triple  contour  (a,,  6s)  (a,.) - 1  (6s) ~\  prove  that  in  the  integrals 

2/1=  f        zv-\{z-\yi-\{z-xy-^dz,         3/2=/         zP-^{z-\Y-^\z-xy-'^dz, 

J(cc;0)  7  {:<-•;  1) 

where  jo,  q,  r  are  not  rational  integers,  if  x  describe  a  closed  curve  round  ^  =  0,  the  s-loops 
being  deformed  so  as  not  t6  be  intersected  by  this  x-closed  curve,  the  new  values  of 
?/i,  2/2  are 

and  determine  the  similar  changes  in  y^,  y^  when  x  moves  round  z  =  \. 

Deduce  without  direct  calculation,  that  if  |)  +  r  be  a  rational  integer,  y^  is  uniform  in 
the  neighbourhood  of  ^  =  0,  and,  also  in  this  neighbourhood, 

y,  =  ^  {x)  + ^-^^ yx  log  X, 

<j)  (x)  being  also  unifona  in  this  neighbourhood. 

Calculate  yi  and  (ft  (x)  from  the  integi'als,  as  ordinary  power-series  in  x,  when 
p  =  q  =  r  =  ^.  (Math.  Trip.,  Part  II.,  1893.) 


110.]  EXAMPLES  249 

Ex.  3.     Prove  that  v=  \    ii.dz,  where 

is  an  algebraic  function  satisfying  the  equation 

8  (t^  +  f)3-  12  (^•  +  §)2- 12^3  (v+f)  +  £6+ 1623  =  0; 

and  obtain  the  conditions  necessary  and  sufficient  to  ensure  that 

V  =  j  udz 

should  be  an  algebraic  function,  when  ?i  is  an  algebraic  function  satisfying  an  equation 

/(^,  «)=o. 

(Liouville,  Briot  and  Bouquet.) 


CHAPTER   X. 


Uniform  Simply-Periodic  and  Doubly-Periodic  Functions. 


111.  Only  a  few  of  the  properties  of  simply-periodic  functions  will  be 
given*,  partly  because  some  of  them  are  connected  with  Fourier's  series  the 
detailed  discussion  of  which  lies  beyond  our  limits,  and  partly  because,  as 
will  shortly  be  explained,  many  of  them  can  at  once  be  changed  into 
properties  of  uniform  non-periodic  functions  v^hich  have  already  been 
considered. 

When  we  use  the  graphical  method  of  §  105,  it  is  evident  that  we  need 
consider  the  variation  of  the  function  within  only  a  single  band.  Within 
that  band  any  function  must  have  at  least  one  infinity,  for,  if  it  had  not,  it 
would  not  have  an  infinity  anywhere  in  the  plane  and  so  would  be  a  constant ; 
and  it  must  have  at  least  one  zero,  for,  if  it  had  not,  its  reciprocal,  also  a 
simply-periodic  function,  would  not  have  an  infinity  in  the  band.  The 
infinities  may,  of  course,  be  accidental  or  essential :  their  character  is  repro- 
duced at  the  homologous  points  in  all  the  bands. 

For  purposes  of  analytical  representation,  it  is  convenient  to  use  a 
relation 

so  that,  if  the  point  Z  in  its  plane  have  R  and  @ 
for  polar  coordinates, 

ft)    ,       „       @ 

Z  = ;  log  it  -1-  7;—  ft). 

If  we  take  any  point  A  in  the  .^-plane  and  a 
corresponding  point  a  in  the  ^■-plane,  then,  as  Z 
describes  a  complete  circle  through  A  with  the 
origin  as  centre,  z  moves  along  a  line  aaj,  where 
tJi  is  a  -f  ft).  A  second  description  of  the  circle 
makes  z  move  from  a-^  to  a^,  where  ao  =  a-i^  +  (o  ;    and   so   on   in  succession. 


Fig.  32. 


For  a  fuller  discussion,  see  Chessin,  Amer.  Journ.  Math.,  t.  xix,  (1897),  pp.  217 — 258. 


111.]  SIMPLE   PERIODICITY  251 

For  various  descriptions,  positive  and  negative,  the  point  a  describes  a  line, 
the  inclination  of  which  to  the  axis  of  real  quantities  is  the  argument  of  eo. 

Instead  of  making  Z  describe  a  circle  through  A,  let  us  make  it 
describe  a  part  of  the  straight  line  from  the  origin  through  A,  say  from  A, 
where  OA  =  R,  to  C,  where  00  =  R'.  Then  z  describes  a  line  through  a 
perpendicular  to  aa^,  and  it  moves  to  c  where 

Similarly,  if  any  point  A'  on  the  former  circumference  move  radially  to  a 
point  (7  at  a  distance  R'  from  the  ^-origin,  the  corresponding  ^^-point  a' 
moves  through  a  distance  a'c,  parallel  and  equal  to  ac :  and  all  the  points  c 
lie  on  a  line  parallel  to  aai.  Repeated  description  of  a  Z-circumference  with 
the  origin  as  centre  makes  z  describe  the  whole  line  cCiCg. 

If  then  a  function  be  simply-periodic  in  &>,  we  may  conveniently  take 
any  point  a,  and  another  point  a^  =  a  +  a),  through  a  and  a^  draw  straight 
lines  perpendicular  to  aa^,  and  then  consider  the  function  within  this  band. 
The  aggregate  of  points  within  this  band  is  obtained  by  taking 

(i)      all  points  along  a  straight  line,  perpendicular  to  a  boundary  of 
the  band,  as  aa^ ; 

(ii)     the  points  along  all  sti'aight  lines,  which  are  drawn  through  the 
points  of  (i)  parallel  to  a  boundary  of  the  band. 

In  (i),  the  value  of  z  varies  from  0  to  w  in  an  expression  a  +  z,  that  is,  in 
the  Z-plane  for  a  given  value  of  R,  the  angle  0  varies  from  0  to  27r. 

In  (ii),  the   value   of  logR  varies   from   —  oo   to   +  go  in   an  expression 

(O  © 

7i — .  log  R  +  7,—  (o,  that  is,  the  radius  R  must  vary  from  0  to  oo  . 
ZTTi     °  ZTT  -^ 

Hence  the  band  in  the  ^-plane  and  the  whole  of  the  .^-plane  are  made 
equivalent  to  one  another  by  the  transformation 


Now  let  Zq  be  any  special  point  in  the  finite  part  of  the  band  for  a  given 
simply-periodic  function,  and  let  Zq  be  the  corresponding  point  in  the  .^-plane. 
Then  for  points  z  in  the  immediate  vicinity  of  Zo  and  for  points  Z  which 
are  consequently  in  the  immediate  vicinity  of  Z^,  we  have 

Z-Zo  =  e^"-e^"' 

2rri  2vi 

=  e"     {e"'  -1} 

=  \ e  "    (z  —  Zo), 

CO 

where  |  A,  |  differs  from  unity  only  by  an  infinitesimal  quantity. 


252  Fourier's  [111. 

If  then  w,  a  function  of  z,  be  changed  into  W  a  function  of  Z,  the  following 
relations  subsist : — 

When  a  point  z^  \'s,  2.  zero  of  w,  the  corresponding  point  Z^  is  a  zero 
of  F. 

When  a  point  z^  is  an  accidental  singularity  of  xv,  the  corresponding 
point  Zq  is  an  accidental  singularity  of  W . 

When  a  point  z^  is  an  essential  singularity  of  %v,  the  corresponding- 
point  ^0  is  an  essential  singularity  of  W. 

When  a  point  z^  is  a  branch-point  of  any  order  for  a  function  w,  the 
corresponding  point  -^0  is  a  branch-point  of  the  same  order  for  W. 

And  the  converses  of  these  relations  also  hold. 

Since  the  character  of  any  finite  critical  point  for  iv  is  thus  unchanged  by  the 
transformation,  it  is  often  convenient  to  change  the  variable  to  Z  so  as  to  let 
the  variable  range  over  the  whole  plane,  in  which  case  the  theorems  already 
proved  in  the  preceding  chapters  are  applicable.  But  special  account  must 
be  taken  of  the  point  z=  ao  . 

112.  We  can  now  apply  Laurent's  theorem  to  deduce  what  is  practically 
Fourier's  series,  as  follows. 

Let  f{z)  he  a  simply-periodic  function  having  co  as  its  period,  and  suppose 
that  in  a  portion  of  the  z-pkme  bounded  by  any  two  parallel  lines,  the  inclina- 
tion of  which  to  the  axis  of  real  quantities  is  equal  to  the  argument  of  (o,  the 
function  is  uniform  and  has  no  singularities;  then,  at  points  tuithin  that 
portion  of  the  plane,  the  function  can  be  expressed  in  the  form  of  a  converging 

series  of  positive  and  of  negative  integral  powers  of  e  '^  . 

In  figure  32,  let  aa^a^...  and  cc-^c^...  be  the  two  lines  which  bound  the 
portion  of  the  plane :  the  variations  of  the  function  will  all  take  place  within 
that  part  of  the  portion  of  the  plane  which  lies  within  one  of  the  repre- 
sentative bands,  say  within  the  band  bounded  by  ...ac...  and  . . .ttiCj. . . : . that  is, 
we  may  consider  the  function  within  the  rectangle  acc^a^a,  where  it  has  no 
singularities  and  is  uniform. 

Now  the  rectangle  acc-ia^a  in  the  5-plane  corresponds  to  a  portion  of  the 
.^-plane  which,  after  the  preceding  explanation,  is  bounded  by  two  circles 

tin  2tu 

with  the  origin  for  common  centre  and  of  radii  \e'^  '"'  j  and  |  e  ^  "'  j ;  and  the 
variations  of  the  function  within  the  rectangle  are  given  by  the  variations  of 
a  transformed  function  within  the  circular  ring.  The  characteristics  of  the 
one  function  at  points  in  the  rectangle  are  the  same  as  the  characteristics  of 
the  other  at  points  in  the  circular  ring :  and  therefore,  from  the  character 
of  the  assigned  function,  the  transformed  function  has  no  singularities  and  it 


112.]  THEOREM  253 

is  uniform  within  the  circular  ring.     Hence,  by  Laurent's  Theorem  (§  28), 
the  transformed  function  is  expressible  in  the  form 

F{Z)=    2     anZ^\ 

n=  —  00 

a  series  which  converges  within  the  ring :    and  the  value  of  the  coefficient  a,i 
is  given  by 

27rr  '  Z'"+' 

taken  along  any  circle  in  the  ring  concentric  with  the  boundaries. 

Retransforming  to  the  variable  z,  the  expression  for  the  original  function 
is 

f{z)=     2     <ine  '-    . 

11=  -  ^ 

The  series  converges  for  points  within  the  rectangle  and  therefore,  as  it 
is  periodic,  it  converges  within  the  portion  of  the  plane  assigned.  And  the 
value  of  «,i  is 

-1     -  'limiz 

an  =  -  \f{2;)e     "    dz, 


taken  along  a  path  which  is  the  equivalent  of  any  circle  in  the  ring  concentric 
with  the  boundaries,  that  is,  along  any  line  perpendicular  to  ac  and  a^Ci,  and 
therefore  parallel  to  the  lines  which  bound  the  assigned  portion  of  the  plane. 

The  expression  of  the  function  can  evidently  be  changed  into  the  form 

1  r'^=-  ^^^iz-i) 


f(z)  =  -\    i    e-    '''"'/(Odt, 


■n=  -  00 


where  the  integral  is  taken  along  the  piece  of  a  line,  perpendicular  to  the 
boundaries  and  intercepted  between  them. 

If  one  of  the  boundaries  of  the  portion  of  the  plane  be  at  infinity,  (so  that 
the  periodic  function  has  no  singularities  within  one  part  of  the  plane),  then 
the  corresponding  portion  of  the  Z-plane  is  either  the  part  within  or  the  part 
without  a  circle,  centre  the  origin,  according  as  the  one  or  the  other  of  the 
boundaries  is  at  go  .  In  the  former  case,  the  terms  with  negative  indices 
n  are  absent ;  in  the  latter,  the  terms  with  positive  indices  are  absent. 

113.  On  account  of  the  consequences  of  the  relation  subsisting  between 
the  variables  z  and  Z,  many  of  the  propositions  relating  to  general  uniform 
functions,  as  well  as  of  those  relating  to  multiform  functions,  can  be  changed, 
merely  by  the  transformation  of  the  variables,  into  propositions  relating  to 
simply-periodic  functions.  One  such  proposition  occurs  in  the  preceding 
section ;  the  following  are  a  few  others,  the  full  development  being  un- 
necessary here,  in  consequence  of  the  foregoing  remark  The  band  of 
reference  for  the  simply-periodic  functions  considered  will   be  supposed   to 


254  SIMPLY-PERIODIC  [113. 

include  the  origin:  and,  when  any  point  is  spoken  of,  it  is  that  one  of  the 
series  of  homologous  points  in  the  plane,  which  lies  in  the  band. 

We  know  that,  if  a  uniform  function  of  Z  have  no  essential  singularity, 
then  it  is  a  rational  function,  which  is  integral  if  Z  =  qc  be  the  only 
accidental  singularity  and  is  meromorphic  if  there  be  accidental  singularities 
in  the  finite  part  of  the  plane ;  and  every  such  function  has  as  many  zeros 
as  it  has  accidental  singularities. 

Hence  a  uniform  simply -periodic  function  with  z=  co  as  its  sole  essential 
singularity  has  as  many  zeros  as  it  has  infinities  in  each  band  of  the  plane  ; 
the  number  of  points  at  which  it  assumes  a  given  value  is  e*qual  to  the  number 
of  its  zeros ;    if  this  common  number  be  finite,  and  if  the  period  be  &>,  the 

function  is  a  rational  f  miction  of  e  '^  ,  which  is  integral  if  all  the  singidarities 
be  at  an  infinite  distance  and  is  meromorphic  if  some  (or  all)  of  them  be  in 
a  finite  part  of  the  plane.  (But  any  number  of  zeros  and  any  number  of 
infinities  may  be  absorbed  in  the  essential  singularity  at  ^  =  oo  .) 

The  simplest  function  of  Z,  thus  restricted  to  have  the  same  number 
of  zeros  as  of  infinities,  is  one  which  has  a  single  zero  and  a  single  infinity 
in  the  finite  part  of  the  plane;  the  possession  of  a  single  zero  and  a 
single  infinity  will  therefore  characterise  the  most  elementary  simply-periodic 
function.     Now,  bearing  in  mind  the  relation 

Z  =  e~^, 
the  simplest  ^-point  to  choose  for  a  zero  is  the  origin,  so  that  Z=l;  and 
then  the  simplest  ^-point  to  choose  for  an  infinity  at  a  finite  distance  is  ^  w, 
(being  half  the  period),  so  that  Z=  —  l.     The  expression  of  the  function  in 
the  Z-plane  with  1  for  a  zero  and  —  1  for  an  accidental  singularity  is 

^Z+l' 

and  therefore  assuming  as  the  most  elementary  simply-periodic  function  that 
which  in  the  plane  has  a  series  of  zeros  and  a  series  of  accidental  singularities 
all  of  the  first  order,  the  points  of  the  one  being  midway  between  those  of  the 
other,  its  expression  is 


A 


•Iiviz 

e  "    -  1 

^itiz 

e^-f  1 


irz  .  ~ — 

which  is  a  constant  multiple  of  tan  —  .     Since  e  '"    is  a  rational  fractional 


function  of  tan  —  ,  part  of  the  foregoing  theorem  can  be  re-stated  as  follows : — • 
If  the  period  of  the  f  motion  be  w,  the  function  is  a  rational  function 


ttz 
of  tan  —  , 


113.]  FUNCTIONS  255 

Moreover,  in  the  general  theory  of  uniform  functions,  it  was  found 
convenient  to  have  a  simple  element  for  the  construction  of  products,  there 
(§  53)  called  a  primary  factor :    it  was  of  the  type 


Z-c 

where  the   function   G\-y J   could  be  a  constant;    and   it   had  only  one 

infinity  and  one  zero. 

Hence  for  simply-periodic  functions  we  may  regard  tan  —  as  a  typical 

ft) 

primary  factor  when  the  number  of  irreducible  zeros  and  .the  (equal)  number 

of  irreducible  accidental  singularities  are  finite.     If  these  numbers  should 

tend   to   an   infinite    limit,   then   an   exponential    factor   might    have   to  be 

TTZ 

associated  with  tan  —  ;  and  the  function  in  that  case  mig'ht  have  essential 

O)  ^ 

singularities  elsewhere  than  at  ^  =  oo  . 

Ex.     Prove  that  a  rational  function  of  z  cannot  be  simply-periodic. 

114.  We  can  now  prove  that  every  uniform  function,  luhich  has  no 
essential  singularities  in  the  finite  part  of  the  plane  and  is  such  that  all 
its  accidental  singularities  and  its  zeros  are  arranged  in  groups  equal  and 
finite  in  number  at  equal  distances  along  directions  parallel  to  a  given 
direction,  is  a  simjjly -periodic  function,  save  as  to  a  possible  factor  of  the 
form  e^<^',  tvhere  g  (z)  is  a  uniform  function  of  z  regular  everyiuhere  in 
the  finite  part  of  the  plane. 

Let  (o  be  the  common  period  of  the  groups  of  zeros  and  of  singularities : 
and  let  the  plane  be  divided  into  bands  by  parallel  lines,  perpendicular  to 
any  line  representing  co.  Let  a,b,  ...  be  the  zeros,  a,  /3,  ...  the  singularities 
in  any  one  band. 

Take  a  uniform  function  (f){z),  simply -periodic  in  w,  and  having  a  single 

zero  and  a  single  singularity  in  the  band  :  we  might  take  tan  —  as  a  value 

of  </)  (z).     Then 

(f>(2!)-(f>  (a) 
cfi\z)  -  cf)  {a) 

is  a  simply-periodic  function  having  only  a  single  zero,  viz.,  z  =  a  and  a  single 
singularity,  viz.,  z  =  a:  for  as  (f)(z)  has  only  a  single  zero,  there  is  only  a 
single  point  for  which  (l){z)  =  (f)  (a),  and  a  single  point  for  which  (f}(z)  =  <p  (a). 
Hence 

{<f>(z)-cf>(a)]\cj>(z)-cf>(b)]... 
{<P{z)-cf>{^u)}[cl>{z)-4>{l3)\... 


256  SIMPLY-PERIODIC  [114. 

is  a  simply-periodic  function  with  all  the  zeros  and  with  all  the  infinities  of 
the  given  function  within  the  band.  But  on  account  of  its  periodicity  it  has 
all  the  zeros  and  all  the  infinities  of  the  given  function  over  the  whole  plane; 
hence  its  quotient  by  the  given  function  has  no  zero  and  no  singularity  over 
the  whole  plane.  Hence,  by  Corollary  I.  in  §  52,  this  quotient  is  of  the  form 
e^'^',  where  g(2)  is  a  uniform  function  of  z,  finite  everywhere  in  the  finite 
part  of  the  plane  :  and  it  may  be  a  constant.  Consequently,  the  expression 
for  the  given  function  is  known.  It  is  thus  a  simply-periodic  function,  save 
as  to  the  factor  specified ;  and  this  factor  may  be  a  constant,  in  which  case 
the  function  is  actually  simply-periodic. 

This  method  can  evidently  be  used  to  construct  simply-jaeriodic  functions,  having 
assigned  zeros  and  assigned  singularities.  Thus  if  a  function  have  a  +  ma  as  its  zeros  and 
G  +  7n'co  as  its  singularities,  where  m  and  in'  have  all  integral  values  from  —  oo  to  -f-oo, 
the  simplest  form  is  obtained  by  taking  a  constant  multiple  of 

tan tan  — 


TTZ 

tan  — 


£x.     Construct  a  function,  simply-periodic  in  w,  having  zeros  given  by  (?n  +  J)co  and 
(m  +  |)a)  and  singularities  by  (m-t-J)cB  and  {m  +  ^)co. 

The  irreducible  zeros  are  |cb  and  |<b  ;  the  irreducible  singularities  are  Jw  and  -!|<b.    Now 


( tan  —  —  tan  ^n  j  (  tan  — -  —  tan  fir 
tan tan  ivr  )  (  tan  ^^  —  tan  In- 


A 


is  evidently  a  function,  initially  satisfying  the  required  conditions.  But,  as  tan  jtt  is 
infinite,  we  divide  out  by  it  and  absorb  it  into  A'  as  a  factor;  the  function  then  takes 
the  form 

1  +  tan  — 


3  —  tan-   - 


We  shall  not  consider  simply-periodic  functions,  which  have  essential 
singularities  elsewhere  than  at  z  =  cc  ;  adequate  investigation  will  be  found 
in  the  second  part  of  Guichard's  memoir,  (I.e.,  p.  176).  But  before  leaving 
the  consideration  of  the  present  class  of  functions,  one  remark  may  be  made. 
It  was  proved,  in  our  earlier  investigations,  that  uniform  functions  can  be 
expressed  as  infinite  series  of  functions  of  the  variable  and  also  as  infinite 
products  of  functions  of  the  variable.  This  general  result  is  true  when  the 
functions  in  the  series  and  in  the  products  are  simply-periodic  in  the  same 
period.  But  the  function,  so  represented,  though  periodic  in  that  common 
period,  may  also  have  another  period :  and,  in  fact,  many  doubly-periodic 
functions  of  different  kinds  (§136)  are  often  conveniently  expressed  as  infinite 
converging  series  or  infinite  converging  products  of  simply-periodic  functions. 


114.]  FUNCTIONS  257 

Any  detailed  illustration  of  this  remark  belongs  to  the  theory  of  elliptic  functions :  one 
simple  example  must  suffice. 

Let  the  real  part  of be  negative,  and  let  q  denote  e  "   ;  then  the  function 

CO 

71=— 00 

being  an  infinite  converging  series  of  powers  of  the  simply-periodic  function  e  "  ,  is 
finite  everywhere  in  the  plane.     Evidently  6  (z)  is  periodic  in  a,  so  that 

d{z  +  co)  =  d{z). 

71=00  2wtV(z  +  tt)') 

Again,  0{z  +  co')=     2    (-l)"^"'e         '" 

)l=-oo 

_  'iniirz 

=     2    i-lYq'^^e    '^     ^2" 
m=-tio  • 

%inz 

=  -~e~  -   e{z), 

the  change  in  the  summation  so  as  to  give  6  {z)  being  permissible,  because  the  con- 
vergence of  6  (s)  is  absolute  on  account  of  the  assumption  with  regai'd  to  q.  There  is 
thus  a  pseudo-periodicity  for  6  (z)  in  a  period  m'. 

•iniirz 

n  —  cc  

Similarly,  if  6^  (z)  =     2     9""  e     "    , 


we' can  prove  that  6^{z-\-co)  =  d  (z), 

'iiiTZ 
e^{z  +  a>')  =  -e~^6{z). 

Then  ^3(2)^^(2)  is  douhly-|  eriodic  in  a  and  2co',  though  constructed  only  from 
functions  simply-periodic  in  u :  it  is  a  function  with  an  infinite  number  of  irreducible 
accidental  singularities  in  a  band. 

115.  We  now  pass  to  doubly-periodic  functions  of  a  single  variable,  the 
periodicity  being  additive.  The  properties,  characteristic  of  this  important 
class  of  functions,  will  be  given  in  the  form  either  of  new  theorems  or 
appropriate  modifications  of  theorems,  already  established ;  and  the  develop- 
ment adopted  will  follow,  in  a  general  manner,  the  theory  given  by  Liouville*. 
It  will  be  assumed  that  the  functions  are  uniform,  unless  multiformity  be 
explicitly  stated,  and  that  all  the  singularities  in  the  finite  part  of  the  plane 
are  accidental  •}■. 

*  In  his  lectures  of  1847,  edited  by  Borchardt  and  published  in  Crelle,  t.  Ixxxviii,  (1880), 
pp.  277 — 310.  They  are  the  basis  of  the  researches  of  Briot  and  Bouquet,  the  most  com- 
plete exposition  of  which  will  be  found  in  their  Theorie  des  fonctions  elliptiques,  (2nd  ed.), 
pp.  239—280. 

t  For  doubly-periodic  functions,  which  have  essential  singularities,  reference  should  be 
made  to  Guichard's  memoir,  (the  introductory  remarks  and  the  third  part),  already  quoted 
on  p.  176,  note. 

F.    F.  17 


258  PROPERTIES  [115. 

The  geometrical  representation  of  double-periodicity,  explained  in  §  105, 
will  be  used  concurrently  with  the  analysis ;  and  the  pai^allelogram  of 
periods,  to  which  the  variable  argument  of  the  function  is  referred,  is  a 
fundamental  parallelogram  (§  109)  with  periods*  2od  and  2&)'.  An  angular 
point  Zq  for  the  parallelogram  of  reference  can  be  chosen  so  that  neither  a 
zero  nor  a  pole  of  the  function  lies  on  the  perimeter ;  for  the  number 
of  zeros  and  the  number  of  poles  in  any  finite  area  must  be  finite,  as 
otherwise  they  would  form  a  continuous  line  or  a  continuous  area,  or  they 
would  be  in  the  vicinity  of  an  essential  singularity.  This  choice  will,  in 
general,  be  made ;  but,  in  particular  cases,  it  is  convenient  to  have  the  origin 
as  an  angular  point  of  the  parallelogram  and  then  it  not  infrequently  occurs 
that  a  zero  or  a  pole  lies  on  a  side  or  at  a  corner.  If  such  a  point  lie  on  a  side, 
the  homologous  point  on  the  opposite  side  is  assigned  to  the  parallelogram 
which  has  that  opposite  side  as  homologous;  and  if  it  be  at  an  angular  point, 
the  remaining  angular  points  are  assigned  to  the  parallelograms  which  have 
them  as  homologous  corners. 

The  parallelogram  of  reference  will  therefore,  in  general,  have  Zq,  Zo+  2&j, 
Z(,  +  2(1)',  Zo-¥2o)  +  2ft)'  for  its  angular  points;  but  occasionally  it  is  desirable 
to  take  an  equivalent  parallelogram  having  z^  ±  co  ±  o)'  SiS  its  angular  points. 

When  the  function  is  denoted  by  cf)  (z),  the  equations  indicating  the 
periodicity  are 

(t>{z  +  2w)  =  ({>  (z)  =  </)  (^  +  2ft)'). 

116.  We  now  proceed  to  the  fundamental  propositions  relating  'to 
doubly-periodic  functions, 

I.  Every  doubly-periodic  function  must  have  zeros  and  infinities  within 
the  fundamental  parallelogram. 

For  the  function,  not  being  a  constant,  has  zeros  somewhere  in  the  plane 
arid  it  has  infinities  somewhere  in  the  plane ;  and,  being  doubly-periodic,  it 
experiences  within  the  parallelogram  all  the  variations  that  it  can  have  over 
the  plane. 

Corollary.      The  function  cannot  be  a  roMonal  function  of  z. 

A  rational  function  of  z  possesses  only  a  limited  number  of  zeros  in  the 
plane.  Within  the  fundamental  parallelogram,  a  doubly-periodic  function 
possesses  zeros:  and  therefore  the  number  of  zeros  which  it  possesses  in  the 
plane  is  unlimited.     The  two  functions  therefore  cannot  be  equivalent. 

An  analytical  form  for  <^{z)  can-be  obtained  which  will  put  its  singu- 
larities in  evidence.  Let  a  be  such  a  pole,  of  multiplicity  n\  then  we  know 
that,  as  the  function  is  uniform,  coefficients  A  can  be  determined  so  that  the 
function 

(h  (7\  —         ''^      _      -^n—\      _       _      Aj^ Ai 

^^  ^     (z-a)"'     (z  -  af^'      '"     (z-ay     J^  a 

*  The  factor  2  is  introduced  merely  for  the  sake  of  convenience. 


116.]  OF   DOUBLY-PERIODIC    FUNCTIONS  259 

is  finite  in  the  vicinity  of  a;  but  the  remaining  poles  of  <^{z)  are  singularities 
of  this  modified  function.  Proceeding  similarly  with  the  other  singularities 
h,  c,...,  which  are  finite  in  number  and  each  of  which  is  finite  in  degree,  we 
have  coefficients  A,  B,  C, ...  determined  so  that 

K  =  a,b,...   [r=l  \Z  —  Kf 

is  finite  in  the  vicinity  of  every  pole  of  0  {z)  within  the  parallelogram  and 
therefore  is  finite  everywhere  within  the  parallelogram.  Let  its  value  be 
x{^)\  then  for  points  lying  within  the  parallelogram,  the  function  ^{z)  is 
expressed  in  the  form 

,  ,         Ji-,  Ao  Art 


z  —  a     (z  —  a)-      '"      (z  -  ay 

+  S-  +  / Tv^+---  + 


-h     {z-hf  iz-by- 

+ 

-"  1  jEt  2  Hi 

+  — V  +  / — v^  +  ...+ 


{z-hf    ■■■    {z-hy' 

But  though  (^  {z)  is  periodic,  x  (^)  ^^  ^^o^  periodic.  It  has  the  property  of 
being  finite  everywhere  within  the  parallelogram;  if  it  were  periodic,  it 
would  be  finite  everywhere,  and  therefore  could  have  only  a  constant  value ; 
and  then  0  {z)  would  be  a  rational  meromorphic  function,  which  is  not 
periodic.  The  sum  of  the  fractions  in  0  {z)  may  be  called  the  fractional 
part  of  the  function :  owing  to  the  meromorphic  character  of  the  function, 
it  cannot  be  evanescent. 

The  analytical  expression  can  be  put  in  the  form 

{z  _  a)-^  (z  -  h)-^  ...{z-  h)-^ F (z), 

where  F (z)  is  finite  everywhere  within  the  parallelogram.  li  a,  /3, ... ,  rj  he 
all  the  zeros,  of  degrees  v,  fi,  .,.,  A,,  within  the  parallelogram,  then 

F{z)  =  (z-  ay  {z-l3f  ...{z-  ijy  G  (zy 

where  G  (z)  has  no  zero  within  the  parallelogram ;  and  so  the  function  can 
be  expressed  in  the  form 

(2-ay(z-^Y...{z-vy 

{z  -  aY{z  -h)^  ...{z  -  hy     ^  ^' 

where  G{z)  has  no  zero  and  no  infinity  within  the  parallelogram  or  on  its 
boundary ;  and  G  {z)  is  not  periodic. 

The  oi^der  of  a  doubly-periodic  function  is  the  sum  of  the  multiplicities 
of  all  the  poles  which  the  function  has  within  a  fundamental  parallelogram ; 

17—2 


260  GENERAL   PROPERTIES  [116. 

and,  the  sum  being  n,  the  function  is  said  to  be  of  the  nth.  order.  All 
these  singularities  are,  as  already  assumed,  accidental ;  it  is  convenient 
to  speak  of  any  particular  singularity  as  simple,  double,  ...  according  to 
its  multiplicity. 

If  two  doubly-periodic  functions  u  and  v  be  such  that  an  equation 

Au+Bv  +  C=0 

is  satisfied  for  constant  values  of  A,  B,  C,  the  functions  are  said  to  be 
equivalent  to  one  another.  Equivalent  functions  evidently  have  the  same 
accidental  singularities  in  the  same  multiplicity. 

II.     The  integral  of  a  doubly-periodic  function  round  the  houndary  of  a 
fundamental  parallelogram  is  zero. 

Let  A  BCD  be  a  fundamental  parallelogram,  the  boundary  of  it  being 
taken  so  as  to  pass   through   no   pole   of  the 
function.     Let  A   he  Zq,  B  he   Zo  +  2q},  and* 
D  he  Zo+  2&)' ;   then  any  point  in  AB  is 

Zo  A-  2a)t, 

where  ^  is  a  real    quantity  lying   between  0 
and  1 ;  and  therefore  the  integral  along  AB  is 

•1 
(f)  (zo  +  ^(ot)  2codt. 


Any  point  in  BC  is  Zq  +  2q)  +  2oi't,  where  t  is  a,  real  quantity  lying  between  0 
and  1 ;  therefore  the  integral  along  BC  is 

<j){z,  +  2(o  +  2(o't)2o)'dt 

0 

=  I    (f)(zo  +  2(o't)2(o'dt, 

J  0 

since  0  is  periodic  in  2&). 

Any  point  in  BG  is   z^  +  2&)'  +  2(ot,  where   ^   is   a  real    quantity  lying 
between  0  and  1 ;    therefore  the  integral  along  CD  is 

•0 

(j)  (zo  +  2w  +  2wt)  2aidt 


=  1    </)  (^0  +  2^0  ^<^dt 

=  _  1    (/)  (^0  +  2wt)  2o)dt. 
J  0 


*  The  figure  implies  that  the  argument  of  w'  is  greater  than  the  argument  of  w,  a 
hypothesis  which,  though  unimportant  for  the  present  proposition,  must  be  taken  account 
of  hereafter  (e.g.,  §  129). 


116.]  OF   DOUBLY-PERIODIC    FUNCTIONS  261 

Similarly,  the  integral  along  DA  is 

I'l 
^    =  -      (^  (^0  +  2&)7)  2co'dt. 

Hence  the  complete  value  of  the  integral,  taken  round  the  parallelogram,  is 
=  <^  (^0  +  2^0  2&)C?^  +      (f)(zo  +  2&)'0  ^oi' dt 

'  0  .0 

-  I     </)  (^0  +  2a)0  2ajc?^  -  I     (/)  (2^0  +  2ft)'0  2&)'c^i, 

.0  .0 

which  is  manifestly  zero,  since  each  of  the  integrals  is  the  integral  of  a 
continuous  function. 

Corollary.  Let  -yjr  (z)  be  any  uniform  function  of  2,  not  necessarily 
doubly-periodic,  but  without  singularities  on  the  boundary.  Then  the 
integTal  f\{r  (z)  dz  taken  round  the  parallelogram  of  periods  is  easily  seen 
to  be 


■s\r  {z,  -f  2a)t)  2Q)dt  +\    ylr(zo  +  2co  +  2co't)  2a)'dt 


-      yjr(z,  +  2Q)'  +  2'vt)2codt-      y\r{z,  +  2o3't)2wdt; 
.0  .'0 

or,  if  we  write  i/r  ( ^  +  2 &>)  -  o/r  ( ^)  =  i/tj  ( ^), 

then  \-^{z)dz=      i/r,  (^'o  +  2(o't)  2(o'dt  -      yjr^  {z^  +  2ait)  2a)dt, 

■  .'0  .'0 

where  on  the  left-hand  side  the  integral  is  taken  positively  round  the 
boundary  of  the  parallelogram  and  on  the  right-hand  side  the  variable  t 
in  the  integrals  is  real. 

The  result  may  also  be  written  in  the  form 

j'yjr{z)dz=  I    -v/tj  (z)  dz—  I    i/r^  (z)  dz, 

the  integrals  on  the  right-hand  side  being  taken  along  the  straight  lines  AD 
and  AB  respectively. 

Evidently  the  foregoing  main  proposition  is  established,  when  yjr^  (^)  and 
1/^2  (^)  vanish  for  all  values  of  ^. 

III.  If  <^  doubly-periodic  function  <^{z)  have  infinities  a^,  az,  ...  within 
the  parallelogram,  and  if  A^,  A^,  ...  be  the  coefficients  of  (z  —  ai)"^,  (z  —  a2)~^,  . . . 
respectively  in  the  fractional  part  of  (f)  (z)  when  it  is  expanded  in  the  paral- 
lelogram, then 

^1  +  ^2+. ..  =  0. 


262  INTEGRAL  RESIDUE  [116. 

As  the  function  0  (z)  is  uniform,  the  integral  /<^  (z)  dz  is,  by  §  19,  II.,  the 
sum  of  the  integrals  round  a  number  of  curves  each  including  one  and  only 
one  of  the  infinities  within  that  parallelogram. 

Taking  the  expression  for  ^{z)  on  p.  259,  the  integral  A,-^l{z  -  a)-'"^  dz 
round  the  curve  enclosing  a  is  0,  if  m  be  not  unity,  and  is  ^iriA-^,  if  m  be 
unity;  the  integral  KmSi^  —  Ky^dz  round  that  curve  is  0  for  all  values  of  m 
and  for  all  points  k  other  than  a ;  and  the  integral  Jx  (^)  dz  round  the  curve 
is  zero,  since  x  (•^)  i^  uniform  and  finite  everywhere  in  the  vicinity  of  a. 
Hence  the  integral  of  <^  {z)  round  a  curve  enclosing  ai  alone  of  all  the 
infinities  is  27riA-^. 

Similarly  the  integral  round  a  curve  enclosing  a^  alone  is  ^ttiA^  ',  and  so 
on,  for  each  of  the  curves  in  succession. 

Hence  the  value  of  the  integral  round  the  parallelogram  is 

27ri%A. 

But  by  the  preceding  proposition,  the  value  of  /^  (z)  dz  round  the  parallelo- 
gram is  zero ;  and  therefore 

A,  +  A,+  ...  =  0. 

This  result  can  be  expressed  in  the  form  that  the  sum  of  the  residues*  of  a 
douhly -periodic  function  relative  to  a  fundamental  parallelogram  of  periods 
is  zero. 

Corollary  1.  A  doubly -periodic  function  of  the  first  order  does  not 
exist. 

Let  such  a  function  have  a  for  its  single  simple  infinity.  Then  an 
expression  for  the  function  within  the  parallelogram  is 

where  x  (^)  i^  everywhere  finite  in  the  parallelogram.  By  the  above  propo- 
sition, A  vanishes ;  and  so  the  function  has  no  infinity  in  the  parallelogram. 
It  therefore  has  no  infinity  anywhere  in  the  plane,  and  so  is  merely  a 
constant :    that  is,  qua  function  of  a  variable,  it  does  not  exist,  . 

Corollary  2.  Doubly-periodic  functions  of  the  second  order  are  of  tiuo 
classes. 

As  the  function  is  of  the  second  order,  the  sum  of  the  degrees  of  the 
infinities  is  two.  There  may  thus  be  either  a  single  infinity  of  the  second 
degree  or  two  simple  infinities. 

In  the  former  case,  the  analytical  expression  of  the  function  is 

*  See  p.  48. 


116.]  FUNCTIONS   OF   THE   SECOND   ORDER  263 

where  a  is  the  infinity  of  the  second  degree  and  xi^)  ^s  holomorphic  within 
the  parallelogram.  But,  by  the  preceding  proposition,  J.i  =  0 ;  hence  the 
analytical  expression  for  a  doubly-periodic  function  with  a  single  irreducible 
infinity  a  of  the  second  degree  is 

within  the  parallelogram.  Such  functions  of  the  second  order,  which  have 
only  a  single  irreducible  infinity,  may  be  called  the  first  class. 

In  the  latter  case,  the  analytical  expression  of  the  function  is 

</>  (^)  =  — -  +  — —  +  X  i^\ 

Z  —  Ci        Z  —  C2 

where  Cj  and  Co  are  the  two  simple  infinities  and  %  {z)  is  finite  within  the 
parallelogram.     Then 

so  that,  if  C^  =  —  62=  C,  the  analytical  expression  for  a  doubly-periodic 
function  with  two  simple  irreducible  infinities  ttj  and  ttg  is 


\z-a,     z-  a  J      ^ 


within  the  parallelogram.  Such  functions  of  the  second  order,  which  have 
two  irreducible  infinities,  may  be  called  the  second  class. 

Corollary  3.  If  within  any  parallelogram  of  periods  a  function  is 
only  of  the  second  order,  the  parallelogram  is  ftmdamental. 

Corollary  4.  A  similar  division  of  douhly -periodic  functions  of  any 
order  into  classes  can  be  effected  according  to  the  variety  in  th^  constitution  of 
the  order,  the  number  of  classes  being  the  number  of  partitions  of  the  order. 

The  simplest  class  of  functions  of  the  nth  order  is  that  in  which  the  func- 
tions have  only  a  single  irreducible  infinity  of  the  nth  degree.  Evidently 
the  analytical  expression  of  the  function  within  the  parallelogram  is 

(z  —  of     (z  —  af  (z  —  ap     "^ 

where  x  (^)  is  holomorphic  within  the  parallelogram.  Some  of  the  coefficients 
G  may  vanish  ;  but  all  may  not  vanish,  for  the  function  would  then  be  finite 
everywhere  in  the  parallelogram. 

It  will  however  be  seen,  from  the  next  succeeding  propositions,  that  the 
division  into  classes  is  of  most  importance  for  functions  of  the  second  order. 

IV.  Two  functions,  which  are  doubly-periodic  in  the  same  pei^iods*,  and 
which  have  the  same  zeros  and  the  same  infinities  each  in  the  same  degrees 
respectively,  are  in  a  constant  ratio. 

*  Such  functions  will  be  called  homoper iodic. 


264  *  HOMOPERIODIC  [116. 

Let  (f)  and  ^jr  be  the  functions,  having  the  same  periods ;  let  a  of  degree  v, 
^  of  degree  /x, . . .  be  all  the  irreducible  zeros  of  0  and  yp- ;  and  let  a  of 
degree  /?,  b  of  degree  m,...  be  all  the  irreducible  infinities  of  (f>  and  of  ^/r. 
Then  a  function  G  (2),  without  zeros  or  infinities  within  the  parallelogram, 
exists  such  that 

^  {z  —  ay^  {z  —  by^. . . 

and  another  function  H  (z),  without  zeros  or  infinities  within  the  parallelo- 
gram, exists  such  that 

(z-ay{z-By...   ^^^^ 

^('^  =  (,-ariz-by^...^^'^- 

Hence  ^,  /--x  =  u  /  \- 

-yjr  {z)      H  [z) 

Now  the  function  on  the  right-hand  side  has  no  zeros  in  the  parallelogram, 

for  G  has  no  zeros  and  H  has  no  infinities ;  and  it  has  no  infinities  in  the 

parallelogram,  for  G  has  no  infinities  and  H  has  no   zeros :    hence  it   has 

neither  zeros  nor  infinities  in  the  parallelogram.     Since  it  is  equal  to  the 

function  on  the  left-hand  side,  which  is  a  doubly-periodic  function,  it  has  no 

zeros  and  no  infinities  in  the  whole  plane;    it  is  therefore  a  constant,  say 

A.     Thus* 

(l)(z)  =  Ayjr(z\ 

V.  Tivo  functions  of  the  second  order,  doubly -periodic  in  the  same  periods 
and  having  the  same  infinities,  are  equivalent  to  one  another. 

If  one  of  the  functions  be  of  the  first  class  in  the  second  order,  it  has  one 
irreducible  double  infinity,  say  at  a;  so  that  we  have 

'^W  =  (^)= +  ?<:«■ 

where  ^  {^)  i^  finite  everywhere  within  the  parallelogram.  Then  the  other 
function  also  has  ^  =  a  for  its  sole  irreducible  infinity  and  that  infinity  is  of 
the  second  degree ;  therefore  we  have 

where  %i  {z)  is  finite  everywhere  within  the  parallelogram.     Hence 

H<^  {z)  -  Gf  {z)  =  Hx  (^)  -  Gx.  {zy 

Now  X  and  xi  are  finite  everywhere  within  the  parallelogram,  and  therefore 
so  is  Hx  -  Gxi-  But  Hx  -  Gxi,  being  equal  to  the  doubly-periodic  function 
H(^-G^\r,  is  therefore  doubly-periodic;    as  it  has  no  infinities  within  the 

*  This  proposition  is  the  modified  form  of  the  proposition  of  §  52,  when  the  generalising 
exponential  factor  has  been  determined  so  as  to  admit  of  the  periodicity. 


116.]  FUNCTIONS  265 

parallelogram,  it  consequently  can  have  none  over  the  plane  and  therefore  it 
is  a  constant,  say  /.     Thus 

H(f>  (z)  -  G^jr  (z)  =  /, 

proving  that  the  functions  (f)  and  yfr  are  equivalent. 

If  on  the  other  hand  one  of  the  functions  be  of  the  second  class  in  the 
second  order,  it  has  two  irreducible  simple  infinities,  say  at  b  and  c,  so  that 
we  have 

where  6  (z)  is  finite  everywhere  within  the  parallelogram.  Then  the  other 
function  also  has  z  =  b  and  z  =  c  for  its  irreducible  infinities,  each  of  them 
being  simple;    therefore  we  have 

where  d^  (z)  is  finite  everywhere  within  the  parallelogram.     Hence 

D(f>  (z)  -  Of  (z)  =  De  (z)  -  ce,  (z). 

The  right-hand  side,  being  finite  everywhere  in  the  parallelogram,  and  equal 
to  the  left-hand  side  which  is  a  doubly-periodic  function,  is  finite  everywhere 
in  the  plane ;  it  is  therefore  a  constant,  say  B,  so  that 

Dcf>  (z)  -  Cyfr  (z)  =  B, 

proving  that  (f)  and  yjr  are  equivalent  to  one  another. 

It  thus  appears  that  in  considering  doubly-periodic  functions  of  the  second 
order,  horaoperiodic  functions  of  the  same  class  are  equivalent  to  one  another 
if  they  have  the  same  infinities ;  so  that,  practically,  it  is  by  their  infinities 
that  homoperiodic  functions  of  the  second  order  and  the  same  class  are 
discriminated. 

CoROLLAEY  1.  If  two  equivalent  functions  of  the  second  order  have  one 
zero  the  same,  all  their  zeros  are  the  same. 

For  in  the  one  class  the  constant  /,  and  in  the  other  class  the  constant  B, 
is  seen  to  vanish  on  substituting  for  z  the  common  zero  ;  and  then  the  two 
functions  always  vanish  together. 

Corollary  2.  If  two  functions,  doubly -periodic  in  the  same  periods  but 
not  necessarily  of  the  second  order,  have  the  same  infinities  occurring  in  such  a 
way  that  the  fractional  parts  of  the  two  functions  are  the  same  except  as  to  a 
constant  factor,  the  functions  are  equivalent  to  one  another.  And  if  in 
addition,  they  have  one  zero  common,  then  all  their  zeros  are  common,  so 
that  the  functions  are  then  in  a  constant  ratio. 


266  '      IRREDUCIBLE   ZEROS  [116. 

Corollary  3.  If  two  f mictions  of  the  second  order,  doubly-periodic  in 
the  same  periods,  have  their  zeros  the  same,  and  one  infinity  common,  they  are 
in  a  constant  ratio. 

VI.  Every  doubly-periodic  function  has  as  ynany  irreducible  zeros  as  it 
has  irreducible  infinities. 

Let  ^  {z)  be  such  a  function.     Then 

(}>(z  +  h)-  (f)  (z) 
z  +  h  —  z 

is  a  doubly-periodic  function  for  any  value  of  h,  for  the  numerator  is  doubly- 
periodic  and  the  denominator  does  not  involve  z ;  so  that,  in  the  limit  when 
k  =  0,  the  function  is  doubly-periodic,  that  is,  (j)'  (z)  is  doubly-periodic. 

Now  suppose  ^(z)  has  irreducible  zeros  of  degree  m^  at  a^,  mg  at  tto, ..., 
and  has  irreducible  infinities  of  degree  yUj  at  «!,  fj,2  at  0I2,...',  so  that  the 
number  of  irreducible  zeros  is  nii  +  m<>+  ...,  and  the  number  of  irreducible 
infinities  is  /ij  -f  /^a  -H  . . . ,  both  of  these  numbers  being  finite.  It  has  been 
shewn  that  (f>(z)  can  be  expressed  in  the  form 

•    (z-a,)^^(z-a2)"'^...  ^ 

{z  -  Oj)'^'  (z  -  a^Y^  ...      ^  ^'      ■ 

where  F(z)  has  neither  a  zero  nor  an  infinity  within,  or  on  the  boundary  of, 
the  parallelogram  of  reference. 

Since  F(z)  has  a  value,  which  is  finite,  continuous  and  different  from  zero 

everywhere  within  the  parallelogram  or  on  its  boundary,  the  function    „    . 

is  not  infinite  within  the  same  limits.     Hence  we  have 

(f){z)      ^  ^  ^     z  —  tti      z  —  a.2 

z  —  Ui      z  —  a2 

where  g  (z)  has  no  infinities  within,  or  on  the  boundary  of,  the  parallelogram 
of  reference.  But,  because  4>'(z)  and  <f)  (z)  are  doubly-periodic,  their  quotient 
is  also  doubly-periodic ;  and  therefore,  applying  Prop.  II.,  we  have 

mj  -t-  /Tla  +  . . .  —  yU-i  —  ytio  —  . . .  =  0, 

that  is,  mi  +  m2+  ...  =  fJi'i  +  t^i  +  ■■-, 

or  the  number  of  irreducible  zeros  is  equal  to  the  number  of  irreducible 
infinities. 

Corollary  1.  The  number  of  irreducible  points  for  luhich  a  doubly- 
periodic  function  assumes  a  given  value  is  equal  to  the  number  of  irreducible 
zeros. 


116.]  AND   IRREDUCIBLE   INFINITIES  267 

For  if  the  value  be  A,  every  infinity  of  (p{z)  is  an  infinity  of  the  doubly- 
periodic  function  (f)(z)  —  A;  hence  the  number  of  the  irreducible  zeros  of  the 
latter  is  equal  to  the  number  of  its  irreducible  infinities,  which  is  the  same 
as  the  number  for  (f>(2:)  and  therefore  the  same  as  the  number  of  irreducible, 
zeros  of  (p  (z).  And  every  irreducible  zero  of  <p(z)  —  A  is  an  irreducible 
point,  for  which  <f>{z)  assumes  the  value  A. 

Corollary  2.  A  doubly -periodic  function  luith  only  a  single  zero  does 
not  exist;  a  douhly-periodic  function  of  the  second  order  has  two  zeros  ;  and, 
generally,  the  order  of  a  function  can  he  measured  by  its  number  of  irreducible 
zeros. 

Note.  It  may  here  be  remarked  that  the  doubly-periodic  functions 
(§  115),  that  have  only  accidental  singularities  in  the  finite  part  of  the 
plane,  have  2^=00  for  an  essential  singularity.  It  is  evident  that  for  infinite 
values  of  z,  the  finite  magnitude  of  the  parallelogram  of  periods  is  not 
recognisable ;  and  thus  for  z=  qc  the  function  can  have  any  value,  shewing 
that  z='X}  is  an  essential  singularity. 

VII.  Let  ttj,  ttg,  ...  he  the  irreducible  zeros  of  a  function  of  degrees 
m-i,  ma,  ...  respectively ;  Oj,  0.3,  ...  its  irreducible  infinities  of  degrees  /u,i,  /X2,  ... 
respectively ;  and  z^,  z^,  ...  the  irreducible  points  tuhere  it  assumes  a  value  c, 
luhich  is  neither  zero  nor  infinity,  their  degrees  being  M^,  M.2,  ...  respectively. 
Then,  except  possibly  as  to  additive  multiples  of  the  periods,  the  quantities 
S  m,.ar,    S  /i,.a,.  and    2  MrZ,.  are  equal  to  one  another,  so  that 

r=l  r=l  r=l 

2  mrCir  =  S  3IrZ,.  =  2  yu.,.a,.  (mod.  2a),  2(o'). 

r=l  r=\  >•  =  ! 

Let  (f>  (z)  be  the  function.  Then  the  quantities  which  occur  are  the  sums 
of  the  zeros,  the  assigned  values,  and  the  infinities,  the  degree  of  each  being 
taken  accou];it  of  when  there  is  multiple  occurrence ;  and  by  the  last 
proposition  tiiese  degrees  satisfy  the  relations 

Sm,.  =  %Mr  =  "2  fir. 

The  function  (f)(z)  —  c  is  doubly-periodic  in  2&>  and  2&)';  its  zeros  are 
z^,  z^,  ...  of  degrees  il/j,  M„,  ...  respectively;  and  its  infinities  are  a^,  a.^,  ...  of 
degrees  /^i,  fx.^,  ...,  being  the  same  as  those  of  ^{z).  Hence  there  exists  a 
function  G  {z),  without  either  a  zero  or  an  infinity  lying  in  the  parallelogram 
or  on  its  boundary,  such  that  (f}(z)  —  c  can  be  expressed  in  the  form 

(z-z,)^^{z-z,)^^...^ 
{z  -  a^Y^  (z  -  a^Y^  . . .   ^  ^  ^ 

for  all  points  not  outside  the  parallelogram ;  and  therefore,  for  points  in  that 
region 

<i>'(z)       ^^        Mr  K       I'r        ^    0'{Z) 


(f){z)  —  C        y.  =  i  Z  —  Zr        "^  Z  —  C^r        ^  (z) 


268 

IRREDUCIBLE   ZEROS 

Hence 

zcf)' (z)         ^    MyZ       ^    fjbyZ     ^  zG' (z) 

<fi(z)  —  C        ,d  Z  —  z.y        '^  Z  —  Ur    '      G  (z) 

r=l                  j-=l  Z  —  Zr       r  =  l                     Z  —  (X^          Cr  {Z) 

_        ^         MyZy                    ^         fJ^rOy           ^        ZG'{z) 

r^i  Z  —  Zy        ^  Z  —  ay          G  (z)    ' 

because 

S    My    =X    /Jir- 
r  =  l                r=l 

[116. 


Integrate  both  sides  round  the  boundary  of  the  fundamental  parallelogram. 
Because  G{z)  has  no  zero  and  no  infinity  in  the  included  region  and  does  not 
vanish  along  the  curve,  the  integral 

vanishes.     But  the  points  Zi  and  a^  are  enclosed  in  the  area ;   and  therefore 
the  value  of  the  right-hand  side  is 

2'7ri'S,MrZr  —  27r*S/X,.0fr5 

zcf)'  (z) 


dz. 


so  that  27ri  (2ilf,.^,.  —  2yu,rCK,.)=  I -, 

J  (f){z)-C 

the  integral  being  extended  round  the  parallelogram. 

Denoting  the  subject  of  integration  by/(^),  we  have 

f{z+2<o)-f{z)^2<o,f^^^     , 

f{z  +  2a,')  -f{z)  =  2a)'  j^-P-  ; 

and   therefore,  by  the    Corollary  to   Prop.  II.,  the    value  of  the  foregoing 
integral  is 


the  integrals  being  taken  along  the  straight  lines  AD  and  AB  respectively 
(fig.  33,  p.  260). 

Let  tu  =  (f){z)  —  c;  then,  as  z  describes  a  path,  w  will  also  describe  a  single 
path  as  it  is  a  uniform  function  of  z.  When  z  moves  from  A  to  D,  w  moves 
from  (f)(A)-c  by  some  path  to  (f){D)  —  c,  that  is,  it  returns  to  its  initial 
position  since  0 (D)  =  (f){A);  hence,  as  z  describes  AD,  w  describes  a  simple 
closed  path,  the  area  included  by  which  may  or  may  not  contain  zeros  and 
infinities  of  lu.     Now 

diu  =  (f)'  (z)  dz, 


116.]  AND    IRREDUCIBLE   INFINITIES  269 

rB     (h' (z) 
and  therefore  the  integral         ,  ,  ,    —  dz  is  equal  to 

[dw 

taken  in  some  direction  round  the  corresponding  closed  path  for  w.  This 
integral  vanishes,  if  no  w-zero  or  w-infinity  be  included  within  the  area 
bounded  by  the  path ;  it  is  ±  27n7ri,  if  m'  be  the  excess  of  the  number  of 
included  zeros  over  the  number  of  included  infinities,  the  +  or  —  sign  being 
taken  with  a  positive  or  a  negative  description ;  hence  we  have 


I       ,  .  , dz  =  2vi7ri, 

JA<i>{z)-c 


where  in  is  some  positive  or  negative  integer  and  may  be  zero.     Similarly 

a4>{z)-c 

where  n  is  some  positive  or  negative  integer  and  may  be  zero. 

Thus  27rt  {l^MrZr  —  2/6t;.a,.)  =  2ft) .  2miri  —  2ft)' .  ^niri, 

and  therefore  "EM^Zr  —  '^P'r^r  =  2mo)  —  2nco' 

=  0(mod.  2ft),  2ft)'). 

Finally,  since  'EMrZr  =  -f^r^r  whatever  be  the  value  of  c,  for  the  right-hand 
side  is  independent  of  c,  we  may  assign  to  c  any  value  we  please.  Let  the 
value  zero  be  assigned;  then  271/^^,.  becomes  Intrttr,  so  that 

^nirar  =  Itii-cxr  (mod.  2ft),  2&)'). 

The  combination  of  these  results  leads  to  the  required  theorem*,  expressed 
by  the  congruences 

S  m,.a^  =  2  MrZy=  S  i^rOir  (mod.  2&),  2&)'). 

r=\  r=l  r=l 

Note.  Any  point  within  the  parallelogram  can  be  represented  in  the 
form  Zo  +  a2w  -\-  62&)',  where  a  and  h  are  real  positive  quantities  less  than 
unity.     Hence 

tMrZr  =  J.,2ft)  +  B,2(o'  +  z.XMr, 

where  A  and  B  are  real  positive  quantities  each  less  than  XM^,  that  is,  less 
than  the  order  of  the  function. 

In  particular,  for  functions  of  the  second  order,  we  have 

^1  +  ^2  =  A^2w  +  B^2(o'  +  2^0, 

*  The  foregoing  proof  is  suggested  by  Konigsberger,  Theorie  der  elliptischen  Functionen,  t.  i, 
p.  342  ;  other  proofs  are  given  by  Briot  and  Bouquet  and  by  Ijiouville,  to  whom  the  adopted 
form  of  the  theorem  is  due.  The  theorem  is  substantially  contained  in  one  of  Abel's  general 
theorems  in  the  comparison  of  transcendents. 


270  DOUBLY-PERIODIC    FUNCTIONS  [116. 

where  A,  and  B,  are  positive  quantities  each  less  than  2.     Similarly,  if  a  and 

b  be  the  zeros, 

a  +  h  =  Aa2o)  +  Ba2a}'  +  2z„, 

where  Aa  and  Ba  are  each  less  than  2;  hence,  if 

Zi  +  Z2  —  a  —  h  =  m2co  +  m'2Q)', 

then  m  may  have  any  one  of  the  three  values  —  1,  0,  1,  and  so  may  m',  the 
simultaneous  values  not  being  necessarily  the  same. 

Let. a  and  /3  be  the  infinities  of  a  function  of  the  second  class  ;  then 

a  +  ^  -  a  —  h  =  n2o)  +  n'2oi', 

where  n  and  n  may  each  have  any  one  of  the  three  values  —  1,  0,  1.  By 
changing  the  origin  of  the  fundamental  parallelogram,  so  as  to  obtain  a 
different  set  of  irreducible  points,  we  can  secure  that   n  and  n    are   zero, 

and  then 

a  +  /3  =  a  +  6. 

Thus,  if  n  be  1   with  an  initial  parallelogram,  so  that 

a  +  ^  =  a-\-h  +  2&), 

we  should  take  either  /3  —  2ft)  =  /3',  or  a—^w  —  a',  according  to  the  position  of 
a  and  /3,  and  then  have  a  new  parallelogram  such  that 

ot  +  jS'  =  a  +  &,    or    a'  +  /3  =  a  +  6. 

The  case  of  exception  is  when  the  function  is  of  the  first  class  and  has  a 
repeated  zero. 

VIII.     Let  4>{z)  he  a  doubly -periodic  function  of  the  second  order.     If  j 

be  the  one  double  infinity  ivhen  the  function  is  of  the  first  class,  and  if  ol  and  /3 

he  tl'.e  two  simple  infinities  when  the  function  is  of  the  second  class,  then  in  the 

former  case 

cf>{z)  =  cl>{2y-z), 

and  in  the  latter  case  (f>  (z)  =  (f)  (a  +  ^  —  z). 

Since  the  function  is  of  the  second  order,  so  that  it  has  two  irreducible 
infinities,  there  are  two  (and  only  two)  irreducible  points  in  a  fundamental 
parallelogram  at  which  the  function  can  assume  any  the  same  value :  let 
them  be  z  and  z'. 

Then,  for  the  first  class  of  functions,  we  have 

z  +  z''=  27 

=  2y  +  2m&)  +  2n(o', 

where  m  and  n  are  integers ;  and  then,  since  (^{z)  =  ^  {z')  by  definition  of  z 
and  /,  we  have 

(f)(z)  =  (f)  (2y  —  z  +  2niu)  +  2?i&)') 

=  (^{2r^-Z). 


116.]  OF   THE   SECOND   ORDER  271 

For  the  second  class  of  functions,  we  have 

z  +  z'  =  ci  +  13 

=  a  +  /3  +  2mQ}  +  27i(o'; 
so  that,  as  before, 

(f)(z)  =  (f>{a  +  /3  -z  +  2m(i}  +  2nco') 

=  <i>(a  +  ^-z). 

111.  Among  the  functions  which  have  the  same  periodicity  as  a  given 
function  (f){z),  the  one  which  is  most  closely  related  to  it  is  its  derivative 
<f>'  {z).  We  proceed  to  find  the  zeros  and  the  infinities  of  the  derivative  of  a 
function,  in  j^articular,  of  a  function  of  the  second  order. 

Since  <p  {z)  is  uniform,  an  irreducible  infinity  of  degree  n  for  ^  {z)  is  an 
irreducible  infinity  of  degree  n  +  \  for  ^'  {z).  Moreover  0'  {z),  being  uniform, 
has  no  infinity  which  is  not  an  infinity  of  <^(^);  thus  the  order  of  <^'  {z)  is 
S(w+1),  or  its  order  is  greater  than  that  of  <^{z)  by  an  integer  which 
represents  the  number  of  distinct  irreducible  infinities  of  <jf>  {z),  no  account 
being  taken  of  their  degree.  If,  then,  a  function  be  of  order  m,  the  order  of 
its  derivative  is  not  less  than  m  + 1  and  is  not  greater  than  2m. . 

Functions  of  the  second  order  either  possess  one  double  infinity,  so  that 
within  the  parallelogram  they  take  the  form 

<^  ^^)  =  {z  -  7f  ^  ^  ^^^' 

—  2A 
and  then  <^'  {£)  =  jj^Ty  +  X  (^l 

that  is,  the  infinity  of  (f){z)  is  the  single  infinity  of  (f)' (z)  and  it  is  of  the 
third  degree,  so  that  ^'  (z)  is  of  the  third  order ;  or  they  possess  two  simple 
infinities,  so  that  within  the  parallelogram  they  take  the  form 

cf>{z)  =  C[~ -^)  +  %(^), 

^  ^  \z  -  ai     z  -  OiJ 

and  then  <^'  {z)  =  -C  \^^^  -  ^-^j  +  x  (z), 

that  is,  each  of  the  simple  infinities  of  4>  {■^)  is  an  infinity  for  (p'  (z)  of  the 
second  degree,  so  that  cf)'  (z)  is  of  the  fourth  order. 

It  is  of  importance  (as  will  be  seen  presently)  to  know  the  zeros  of 
the  derivative  of  a  function  of  the  second  order. 

FQr  a  function  of  the  first  class,  let  y  be  the  irreducible  infinity  of  the 
second  degree;  then  we  have 

<f>{z)  =  cj,(2j-z), 
and  therefore  <f>'{^)  =  ~  ^  (^7  '"  ^)- 


272  DOUBLY-PERIODIC   FUNCTIONS  [117. 

Now  (/>'  (z)  is  of  the  third  order,  having  7  for  its  irreducible  infinity  in  the 
third  degree :  hence  it  has  three  irreducible  zeros. 

In  the  foregoing  equation,  take  z  =  ^:  then 

^  (7)  =  -i>'  (7), 
shewing  that  7  is  either  a  zero  or  an  infinity.     It  is  known  to  be  the  only 
infinity  of  <^'  {z). 

Next,  take  ^r  =  7  +  &) ;  then 

^'  (7  +  a>)  =  —  </)'  (7  —  &)) 

=  —  <^'  (7  —  o)  +  2o)) 
=  -(^'(7  +  0,), 

shewing  that  7  +  co  is  either  a  zero  or  an  infinity.     It  is  known  not  to  be  an 
infinity ;  hence  it  is  a  zero. 

Similarly  7  +  0;'  and  7  +  w  +  &)'  are  zeros.  Thus  three  zeros  are  obtained, 
distinct  from  one  another ;  and  only  three  zeros  are  required ;  if  they  be  not 
within  the  parallelogi-am,  we  take  the  irreducible  points  homologous  with 
them.     Hence : — 

IX.  The  three  zeros  of  the  derivative  of  a  function,  doiibly-periodic  in 
2a>  and  Iw    and  having  7  for  its  double  {and  only)  irreducible  infinity,  are 

J  +  0),      <y  +  oi' ,     7  +  &)  +  ft)'. 

For  a  function  of  the  second  class,  let  a  and  /S  be  the  two  simple 
irreducible  infinities;   then  we  have 

<^{z)  =  (\>{a-\-^-z), 
and  therefore  4>  {z)  =  -  (f)'  {a+  ^  -  z). 

Now  ^' {z)  is  of  the  fourth  order,  having  a  and  y8  as  its  irreducible 
infinities  each  in  the  second  degree;  hence  it  must  have  four  irreducible 
zeros. 

In  the  foregoing  equation,  take  ^  =  ^  (a  4-  ^);  then 

shewing  that  ^  (a  +  /3)  is  either  a  zero  or  an  infinity.     It  is  known  not  to  be 
an  infinity;  hence  it  is  a  zero. 

Next,  take  ^  =  ^  (q  +  yS)  +  &) ;  then 

</)'|-|-(a  +  ^)+ft)}=-f  li(«  +  ^)-^ 

=  -  (/)' {1  (a  + /8)  -  ft)  +  2ft.}  .    . 

=  -</>'{i(a  +  ^)+ft)}, 
shewing  that  h  (a  +  /8)  +  &)  is  either  a  zero  or  an  infinity.     As  before,  it  is 
a  zero. 


117.]  OF   THE   SECOND   ORDER  273 

Similarly  ^  (a  + /3;  4- ft)'  and  ^(a  + ^)  +  (o  + w  are  zeros.  Four  zeros  are 
thus  obtained.,  distinct  from  one  another ;  and  only  four  zeros  are  required. 
Hence : — 

X.  The  four  zeros  of  the  derivative  of  a  function,  douhly -periodic  in  2a) 
and  2a>'  and  having  a  and  ^  for  its  simple  (and  only)  irreducible  infinities,  are 

^  (a  +  /3),     ^  (a  +  /3)  +  w,     -^a  +  yS)  +  «',     ^  (a  +  /3)  +  «  +  w. 

The  verification  in  each  of  these  two  cases  of  Prop,  VII.,  that  the  sum  of 
the  zeros  of  the  doubly-periodic  function  ^'  {z)  is  congruent  with  the  sum  of 
its  infinities,  is  immediate. 

Lastly,  it  may  be  noted  that,  if  z^  and  z^  he  the  tivo  irreducil^e  points  for 
which  a  doubly -periodic  function  of  the  second  order  assumes  a  given  value, 
then  the  values  of  its  derivative  for  Zi  and  for  z.,  are  equal  and  opposite.     For 

(f>  (z)  =  (f)  (a  +  j3  -  z)  =  (f)  (z,  +  z.  -  z), 

since  z-l  +  Z2  =  a  +  ^;  and  therefore 

(f)'  (z)  =  -</)'  (^1  +  Z2  -  z), 

that  is,  (/)'  (^i)  =  -  (f)'  (zo), 

which  proves  the  statement. 

118.     We  now  come  to  a  different  class  of  theorems. 

XI.  Any  doubly-periodic  function  of  the  second  order  can  be  expressed 
rationally  in  terms  of  an  assigned  doubly -periodic  function  of  the  second  order, 
if  the  periods  be  the  same. 

The  theorem  will  be  sufficiently  illustrated  and  the  line  of  proof 
sufficiently  indicated,  if  we  express  a  function  <^  {z)  of  the  second  class,  with 
irreducible  infinities  a,  ^  and  irreducible  zeros  a,  b  such  that  a  +  ^8  =  a  +  6,  in 
terms  of  a  function  <1>  of  the  first  class  Avith  7  as  its  irreducible  double 
infinity. 

Consider  a  function  z— rr .   ; , .;, . 

^{z  +  h)  -  <J>  {h  ) 

A  zero  of  ^{z  +  h)  is  neither  a  zero  nor  an  infinity  of  this  function ;  nor 
is  an  infinity  oi  ^{z-^h)  a  zero  or  an  infinity  of  the  function.  It  will  have 
a  and  b  for  its  irreducible  zeros,  if 

a  -\rh  =  h', 

b-\-h  +  h'=2y; 

and  these  will  be  the  only  zeros,  for  O  is  of  the  second  order.     It  will  have  a 
and  /3  for  its  irreducible  infinities,  if 

a  +  h  =  h", 

^  +  h  +  h"  =  2y; 

F.  P.  18 


274  RELATIONS   BETWEEN  [118. 

and  these  will  be  the  only  infinities,  for  <[>  is  of  the  second  order.     These 
equations  are  satisfied  by 

/?/  =  ^  {2y  -  b  +  a), 

h  =  -h  {2ry-a-j3)  =  ^  (27  -  a-b). 
Hence  the  assigned  function,  with  these   values  of  h,  has  the  same    zeros 
and  the  same  infinities  as  <f>{z)',  and  it  is  doubly-periodic  in  the  same  periods. 
The  ratio  of  the  two  functions  is  therefore  a  constant,  by  Prop.  IV.,  so  that 

If  the  expression  be  required  in  terms  of  <I>  (z)  alone  and  constants,  then 
<i>{z  ^-h)  must  be  expressed  in  terms  of  <I>  (z)  and  constants  which  are  values 
of  $  (z)  for  special  values  of  z.     This  will  be  effected  later. 

The  preceding  proposition  is  a  special  case  of  a  more  general  theorem 
which  will  be  considered  later;  the  following  is  another  special  case  of  that 
theorem  :  viz. : — 

XII.  A  doubly -pei'iodic  function  with  any  number  of  simple  infinities  can 
be  expressed  either  as  a  sum  or  as  a  product,  of  functions  of  the  second  order 
and  tJte  second  class  which  are  doubly -periodic  in  the  same  periods. 

Let  tti,  flo,  ...,  a,i  be  the  irreducible  infinities  of  the  function  <I>,  and 
suppose  that  the  fractional  part  of  ^  (^)  is 

^'    +-^+  +  -^, 


z  —  OL-^     z  —  a^ 

with  the  condition  A^+  A2+ +An  =  0.  Let  ^^ (z)  be  a  function,  doubly- 
periodic  in  the  same  periods,  with  a^,  aj  as  its  only  irreducible  infinities, 
supposed  simple;  where  i  has  the  values  l,...,n  —  l,  and  j  =  i+l.  Then 
the  fractional  parts  of  the  functions  <^i2  (z),  (^os  (^),  •  •  ■  are 

G 


(J^ LV 

^\Z  —  Ui        Z  —  ttoj  ' 

\z-02      z-  a  J 


respectively;  and  therefore  the  fractional  part  of 

A,  ^    ,  ^     A,  +  A,  ^    ,  .  .  A^  +  A^+.-.-hAn-i^  .  , 

qT  (f>i2  KZ)  +  —^ Cf>,3  {z)+  ...  + g—^ 4>n~h  n  (^) 


IS 


118.]  DOUBLY-PERIODIC    FUNCTIONS  275 

**  .     . 

since  ^Ai  =  0.     This  is  the  same  as  the  fractional  part  of  <t>  (z);  and  therefore 

i  =  l 
^  (Z)  -  -^  (/)i2  (^) ^--  <f>os  (Z)-... ^- (l>n-i,n  (2) 

has  no  fractional  part.  It  thus  has  no  infinity  within  the  parallelogram ;  it 
is  a  doubly-periodic  function  and  therefore  has  no  infinity  anywhere  in  the 
plane  ;  and  it  is  therefore  merely  a  constant,  say  B.     Hence,  changing  the 

constants,  we  have 

4)  (z)  -  B^  <^i2  {z)  -  B^(^^  {z)'-  ...-  Bn-^  (i>n-i,n  (^)  =  A 

giving  an  expression  for  <P  (z)  as  a  linear  combination  of  functions  of  the 
second  order  and  the  second  class.  But  as  the  assignment  of  the  infinities  is 
arbitrary,  the  expression  is  not  unique. 

For  the  expression  in  the  form  of  a  product,  we  may  denote  the  n 
irreducible  zeros,  supposed  simple,  by  «!,...,  a„.  We  determine  n—2  new 
irreducible  quantities  c,  such  that 

Ci  =  «!  -f-  O2  —  0^1 . 

C2=a3  +  Ci-a2, 

Cs^a^  +  Co-  a-i, 


^11  =  ^n        >    C^i— 2       ^n—i  I 
n  n 

this  being  possible  because  2ar  =  2a,.;  and  we  denote  by  <^(^;  a,  ^;  e,/)  a 

function  of  z,  which  is  doubly-periodic  in  the  periods  of  the  given  function, 
has  o  and  /3  for  simple  irreducible  infinities,  and  has  e  and  /  for  simple 
irreducible  zeros.     Then  the  function 

4>{z;  «!,  a^;  aj,  c^)^{z;  a.^,  Ci ;  a^,  0.3)  ...  cj)(z;  «„,  c^-a;  a^-i,  ««) 

has  neither  a  zero  nor  an  infinity  at  Ci,  at  c^,  ...,  and  at  Cn-2',  it  has  simple 
infinities  at  a^,  cto,...,  a,i,  and  simple  zeros  at  a^,  a^,  ...,a„_i,  a^-  Hence  it 
has  the  same  irreducible  infinities  and  the  same  irreducible  zeros  in  the  same 
degree  as  the  given  function  ^  {z);  and  therefore,  by  Prop.  IV.,  ^{z)  is 
a  mere  constant  multiple  of  the  foregoing  product. 

The  theorem  is  thus  completely  proved. 

Other  developments  for  functions,  the  infinities  of  which  are  not  simple, 
are  possible  ;  but  they  are  relatively  unimportant  in  view  of  a  theorem. 
Prop.  XV.,  about  to  be  proved,  which  expresses  any  periodic  function  in 
terms  of  a  single  function  of  the  second  order  and  its  derivative. 

18—2 


276  RELATIONS   BETWEEN  [118. 

XIII.  If  two  doubly -periodic  functions  have  the  same  periods,  tliey  are 
connected  by  an  algebraic  equation. 

Let  u  be  one  of  the  functions,  having  n  irreducible  infinities,  and  v  be 
the  other,  having  m  irreducible  infinities. 

By  Prop.  VI.,  Corollary  1,  there  are  n  irreducible  values  of  z  for  a  value 
of  u ;  and  to  each  irreducible  value  of  z  there  is  a  doubly-infinite  series  of 
values  of  z  over  the  plane.  The  function  v  has  the  same  value  for  all  the 
points  in  any  one  series,  so  that  a  single  value  of  v  can  be  associated  uniquely 
with  each  of  the  irreducible  values  of  z,  that  is,  there  are  n  values  of  v  for 
each  value  of  u.  Hence  (§  99)  v  is  a  root  of  an  algebraic  equation  of  the  nt\\ 
degree,  the  coefficients  of  which  are  functions  of  u. 

Similarly  u  is  a  root  of  an  algebraic  equation  of  the  mth  degree,  the 
coefficients  of  which  are  functions  of  v. 

Hence,  combining  these  results,  we  have  an  algebraic  equation  between 
u  and  V  of  the  /ith  degree  in  v  and  the  mth  in  u,  where  m  and  n  are  the 
respective  orders  of  v  and  u. 

Corollary  1.  If  both  the  functions  be  even  functions  of  z,  then  n  and  m 
are  even  integers  ;  and  the  algebraic  relation  between  u  and  v  is  of  degree  \n  in 
V  and  of  degree  |-m  in  u. 

Corollary  2.  If  a  function  u  be  doubly-periodic  in  co  and  co',  and  a 
function  v  be  doubly -periodic  in  O  and  D,',  where 

n  =  mco  +  no)',     £1'  =  m'co  +  ii'co', 
m,  n,  III',  n  being  integers,  then  there  is  an  algebraic  relation  between  ii  and  v. 

119.  It  has  been  proved  that,  if  a  doubly-periodic  function  ii  be  of  order 
m,  then  its  derivative  du/dz  is  doubly-periodic  in  the  same  periods  and  is  of 
an  order  n,  which  is  not  less  than  m  + 1  and  not  greater  than  2m.  Hence,  by 
Prop.  XIII.,  there  subsists  between  u  and  u  an  algebraic  equation  of  order  m 
in  u  and  of  order  n  in  tt ;  let  it  be  arranged  in  powers  of  u,  so  that  it  takes 
the  form 

Uou'""  +  U,  u''''-^  +  . . .  +  U^-2u'^  +  U^^^u'  +  Uy,,  =  0, 

where  Uq,  Uj,  ...,  U^n  are  rational  integral  functions  of  u  one  at  least  of  which 
must  be  of  degree  n. 

Because  the  only  distinct  infinities  of  u  are  infinities  of  u,  it  is  iriipossible 
that  u  should  become  infinite  for  finite  values  of  u  :  hence  Uo=0  can  have  no 
finite  roots  for  u,  that  is,  it  is  a  constant  and  so  it  may  be  taken  as  unity. 

And  because  the  m  values  of  z,  for  which  u  assumes  a  given  value,  have 
their  sum  constant  save  as  to  integral  multiples  of  the  periods,  we  have 

Sz-i  +  Bz2-\-  ...  +  &z,n  =  0 
corresponding  to  a  variation  Bu ;  or 

dZ:i        dz2    ,  ,    dZm  _  Q 

du      du      '"      du 


119.]  DOUBLY-PERIODIC   FUNCTIONS  277 

dtt 

Now  ^y-  is  one  of  the  values  of  u'  corresponding  to  the  value  of  u,  and  so  for 

the  others ;  hence 

m      "I 

that  is,  by  the  foregoing  equation, 

and  therefore  f/m_i  vanishes.     Hence  : — 

XIV.  There  is  a  relation,  between  a  uniform  doubly-periodic  function  ti  of 
order  m  and  its  derivative,  of  the  form 

tuhere  Ui,  ...,  Um-2,  U.,„,  are  rational  integral  functions  of  u,  at  least  one  of 
which  must  be  of  degree  n,  the  order  of  the  derivative,  and  n  is  not  less  than 
m  +  1  and  not  greater  than  2m. 

Further,  by  taking  v  =  -  ,  which  is  a  function  of  order  ni  because  it  has  the 
m  irreducible  zeros  of  u  for  its  infinities,  and  substituting,  we  have 

The  coefficients  of  this  equation  must  be  integral  functions  of  v ;  hence  the 
degree  of  Ur  in  u  cannot  be  greater  than  2r*.- 

Corollary.  The  foregoing  equation  becomes  very  simple  in  the  case  of 
doubly-periodic  functions  of  the  second  order. 

Then  m  =  2. 

If  the  function  have  one  infinity  of  the  second  degree,  its  derivative  has 
that  infinity  in  the  third  degree,  and  is  of  the  third  order,  so  that  w  =  3  ;  and 
the  equation  is 

=  Xu"  4-  6fj,u-  +  6vu  -\-  p, 


^dz , 

where  \,  fx,  v,  p  are  constants.     If  6  be  the  infinity,  so  that 

A 

where  x  {^)  is  everywhere  finite  in  the  parallelogram,  then  -  =  ^A ;  and  the 

fill 
zeros  of  -r-  are  0  +  co,  6  +  o)' ,  6  +  co  +  co' :  so  that 
dz 


This  is  the  general  differential  equation  of  Weierstrass's  elliptic  functions. 

*  For  a  converse  proposition,  see  the  Note  on  differential  equations  of  the  first  order  having 
uniform  integrals,  at  the  end  of  this  chapter. 


278  DIFFERENTIAL   EQUATIONS   SATISFIED   BY  [119. 

'If  the  function  have  two  simple  infinities  a  and  /3,  its  derivative  has  each 
of  them  as  an  infinity  of  the  second  degree,  and  is  of  the  fourth  order,  so  that 
?i  =  4  ;  and  the  equation  is 

( ^-  I  =  CqW^  +  4<CiH^  +  Gc.u^  +  4>CsU  +  c^, 
\dzj 

where  Co,  Ci,  c.,,  c^,  c^  are  constants.     Moreover 

1  1 


.=  *(.)  =  g(_-^_)  +  ^(.). 


where  x  {^)  is  finite  everywhere  in  the  parallelogram.  Then  Co=  G  - ;  and 
the  zeros  of  j-  are  ^(a+/S),  ^  (a  +/3)  +  o),  ^{a  +/3)  +  w,  |  (a  +yS)  +  w  +  a, 
so  that  the  equation  is 

^'{^J=\-^(')-^{'2{^+/3)]m{^)-<f>{H^+/3)  +  co]] 

X  [<^  (z)  -  (^  |i  (a  +/8)  +  ft)'}]  [0  (z)-cf>{^ia+/3)  +  co  +  (o']]. 

This  is  the  general  differential  equation  of  Jacohi's  elliptic  functions. 

The  canonical  forms  of  both  of  these  equations  will  be  obtained  in 
"Chapter  XI,,  where  some  properties  of  the  functions  are  investigated  as 
special  illustrations  of  the  general  theorems. 

Note.  All  the  derivatives  of  a  doubly-periodic  function  are  doubly- 
periodic  in  the  same  periods,  and  have  the  same  infinities  as  the  function 
but  in  different  degrees.  In  the  case  of  a  function  of  the  second  order,  which 
must  satisfy  one  or  other  of  the  two  foregoing  equations,  it  is  easy  to  see  that 
a  derivative  of  even  rank  is  a  rational  integral  function  of  u,  and  that  a 
derivative  of  odd  rank  is  the  product  of  a  rational  integral  function  of  u  by 
the  first  derivative  of  u. 

It  may  be  remarked  that  the  form  of  these  equations  confirms  the  result 
at  the  end  of  §  117,  by  giving  two  values  of  u'  for  one  value  of  u,  the  two 
values  being  equal  and  opposite. 

Ex.  1.  If  z6  be  a  doubly-periodic  function  having  a  single  irreducible  infinity  of  the 
third  degree  so  as  to  be  expressible  in  the  form 

— 5  +  ^  +  integral  function  of  s 

within  the  parallelogram  of  periods,  then  the  differential  equation  of  the  first  order  which 
determines  u  is 

2<'3  +  (a-l-3(9«)?f'^=t^4, 

where  ^7^  is  a  quartic  function  of  u  and  where  a  is  a  constant  which  does  not  vanish  with  6. 

(Math.  Trip.,  Part  II.,  1889.) 

Ex.  2.  A  doubly-periodic  function  u  has  three  irreducible  poles  a^,  02,  ag,  such  that  in 
the  immediate  vicinity  of  each 

n  =  — ^  -H  v^  {z  -  aj  +  powers  of  2  -  a,, 
2  — a* 


119.]  DOUBLY-PERIODIC   FUNCTIONS  279 

'  for  s=l,  2,  3.     Prove  that 
where 

\Ai   A2   A3/         A1A2  +  A2A3  +  A3A1 

and    V  is  a  sextic  polynomial  in  u  of  which  the  highest  terms  are 

(Math.  Trip.,  Part  II.,  1895.) 

XV.  Every  doubly -periodic  function  can  he  expressed  rationally  in  ter7ns 
of  a  /miction  of  the  second  order,  doubly -periodic  in  the  same  periods,  and  its 
derivative. 

Let  II  be  a  function  of  the  second  order  and  the  second  class,  having  the 
same  two  periods  as  v,  a  function  of  the  ??ith  order;  then,  by  Prop.  XIII., 
there  is  an  algebraic  relation  between  u  and  v  which,  being  of  the  second 
degree  in  v  and  the  ???-th  degree  in  u,  may  be  taken  in  the  form 

Lv'  -  2Mv  +  P  =  0, 

where  the  quantities  L,  M,  P  are  rational  integral  functions  of  u  and  at  least 
one  of  them  is  of  degree  m.     Taking 

Lv-  M  =  w, 

we  have  w^  =  M^  —  LP, 

a  rational  integral  function  of  u  of  degree  not  higher  than  2m. 

Thus  lu  cannot  be  infinite  for  any  finite  value  of  u :  an  infinite  value  of  u 
makes  w  infinite,  of  finite  multiplicity.  To  each  value  of  u  there  correspond 
two  values  of  to  equal  to  one  another  but  opposite  in  sign. 

Moreover  w,  being  equal  to  Lv  —  M,  is  a  uniform  function  of  z,  say  F{z), 
while  it  is  a  two-valued  function  of  u.  A  value  of  u  gives  two  distinct 
values  of  z,  say  z-^  and  z., ;  hence  the  values  of  lu,  which  arise  from  an  assigned 
value  of  u,  are  values  of  w  arising  as  uniform  functions  of  the  two  distinct 
values  of  z.  Hence  as  the  two  values  of  w  are  equal  in  magnitude  and 
opposite  in  sign,  we  have 

F{z,)  +  F{z,)  =  0, 
tljat  is,  since  z-^  +  z^^ol-^- ^  where  a  and  ^  are  the  irreducible  infinities  of  u, 

F{z,)  +  F{ci  +  ^-z,)  =  0, 

so  that  ^(a  4-/3),  ^(a  +  /?)  +  «,  H«  +/S)  +  «'.  and  i(a  -I-  ^)  -|-  &>  +  a  are  either 
zeros  or  infinities  of  iv.  They  are  known  not  to  be  infinities  of  m,  and  w  is 
infinite  only  for  infinite  values  of  u ;  hence  the  four  points  are  zeros  of  w. 

But  these  are  all  the  irreducible  zeros  of  u' ;  hence  the  zeros  of  u'  are 
included  among  the  zeros  of  w. 


280  EELATIONS   BETWEEN  [119- 

Now  consider  the  function  tuju'.  The  numerator  has  two  values  equal 
and  opposite  for  an  assigned  value  of  u  ;  so  also  has  the  denominator.  Hence 
wju'  is  a  uniform  function  of  u. 

This  uniform  function  of  u  may  become  infinite  for 

(i)     infinities  of  the  numerator, 

(ii)     zeros  of  the  denominator. 

But,  so  far  as  concerns  (ii),  we  know  that  the  four  irreducible  zeros  of  the 
denominator  are  all  simple  zeros  of  it  and  each  of  them  is  a  zero  of  w ;  hence 
wju  does  not  become  infinite  for  any  of  the  points  in  (ii).  And,  so  far  as 
concerns  (i),  we  know  that  all  of  them  are  infinities  of  u.  Hence  wju,  a 
uniform  function  of  u,  can  become  infinite  only  for  an  infinite  value  of  u,  and 
its  multiplicity  for  such  a  value  is  finite ;  hence  it  is  a  rational  integral 
function  of  u,  say  N,  so  that 

w  =  Nu' . 

Moreover,  because  w-  is  of  degree  in  u  not  higher  than  '2m,  and  ?./"- 
is  of  the  fourth  degree  in  u,  it  follows  that  N  is  of  degree  not  higher 
than  m  —  2. 

We  thus  have        ■'  Lv  -  M  =  Nu', 

M  +  Nu'     M     N   , 

where  L,  M,  N  are  rational  integral  functions  of  u ;  the  degrees  of  L  and  M 
are  not  higher  than  m,  and  that  of  N  is  not  higher  than  m  —  2. 

Note  1.  The  function  u,  which  has  been  considered  in  the  preceding 
proof,  is  of  the  second  order  and  the  second  class.  If  a  function  u  of  the 
second  order  and  the  first  class,  having  a  double  irreducible  infinity,  be 
chosen,  the  course  of  proof  is  similar;  the  function  iv  has  the  three  irreducible 
zeros  of  u'  among  its  zeros  and  the  result,  as  before,  is 

w  =  Nil. 

But,  now,  vf  is  of  degree  in  u  not  higher  than  2m  and  u'-  is  of  the  third 
degree  in  u ;  hence  N  is  of  degree  not  higher  than  m  -  2,  and  the  degree  of 
vf'  in  u  cannot  be  higher  than  2m  —  1. 

Hence,  if  L,  M,  P  be  all  of  degree  m,  the  terms  of  degi-ee  2m  in  LP  -  M- 
disappear.  If  all  of  them  be  not  of  degree  in,  the  degree  of  M  must  not  be 
higher  than  m-1  ;  the  degree  of  either  X  or  P  must  be  m,  but  the  degree 
of  the  other  must  not  be  greater  than  m  —  1,  for  otherwise  the  algebraic 
equation  between  u  and  v  would  not  be  of  degree  m  in  u. 

We  thus  have 

Xv-  -  2Mv  +  P  =  Q,      Lv-M  =  Nu, 


119.]  DOUBLY-PERIODIC   FUNCTIONS  281 

where  the  degree  of  N  in  u  is  not  higher  than  m  —  2.  If  the  degree  of  L  be 
less  than  m,  the  degree  of  M  is  not  higher  than  m  —  1  and  the  degree  of  P  is 
m.  If  the  degree  of  L  be  m,  the  degree  of  M  may  also  be  m  provided  that 
the  degree  of  P  be  m  and  that  the  highest  terms  be  such  that  the  coefficient 
of  iv^  in  LP  —  ]\P  vanishes. 

Note  2.  The  theorem  expresses  a  function  v  rationally  in  terms  of  u  and 
w' :  but  w'  is  an  irrational  function  of  u,  so  that  v  is  not  expressed  rationally 
in  terms  of  u  alone. 

But,  in  Propositions  XI.  and  XII.,  it  was  indicated  that  a  function  such 
as  V  could  be  rationally  expressed  in  terms  of  a  doubly-periodic  function,  such 
as  u.  The  apparent  contradiction  is  explained  by  the  fact  that,  in  the  earlier 
propositions,  the  arguments  of  the  function  u  in  the  rational  expression  and 
of  the  function  v  are  not  the  same ;  whereas,  in  the  later  proposition  whereby 
V  is  expressed  in  general  irrationally  in  terms  of  u,  the  arguments  are  the 
same.  The  transition  from  the  first  (which  is  the  less  useful  form)  to  the 
second  is  made  by  expressing  the  functions  of  those  different  arguments  in 
terms  of  functions  of  the  same  argument  when  (as  will  appear  subsequently, 
in  I  121,  in  proving  the  so-called  addition-theorem)  the  irrational  function  of 
u,  represented  by  the  derivative  u',  is  introduced. 

Note  3.  The  theorem  of  this  section,  usually  called  Liouville's  theorem, 
is  valid  only  when  there  .are  no  essential  singularities  in  the  finite  part  of  the 
plane.  The  limitation  arises  in  that  part  of  the  proof,  where  the  irreducible 
zeros  and  the  irreducible  poles  are  considered :  it  is  there  assumed  that  their 
number  is  finite,  which  cannot  be  the  case  when  essential  singularities  exist 
in  the  finite  part  of  the  plane  and  when  therefore  there  are  irreducible 
essential  singularities.  Hence  Liouville's  theorem  is  true  only  for  those 
uniform  doubly-periodic  functions  which  have  their  essential  singularities  at 
infinity. 

In  illustration  of  this  remark,  it  may  be  noted  that  e^^'^\  though  a  uniform 
doubly -periodic  function  of  u,  is  not  expressible  rationally  in  terms  of  sn  u 
and  sn'  u. 

Ex.  If /(%)  be  a  doubly-periodic  function  of  the  third  order  with  poles  at  Cj,  Cg,  c^; 
and  if  (p  {^l)  be  a  doubly-periodic  function  of  the  second  order,  with  the  same  periods,  and 
with  poles  at  a,  ^,  its  value  in  the  neighbourhood  of  u=a  being 

(j)  {u)  =  -—-  +  \i{u-a)  +  X2(u-a)-  +  ...; 

prove 

iX2  { /"  (a)  -/"  m  -X  {/'  (a)  -  /'  m  i  <l>  {c^)  +  {f{a)-fm  {SXXi-t-l  (^  (c,)  0  (c3)}  =  0. 

1  1 

Corollary  1.  Let  fl  denote  the  sum  of  the  irreducible  infinities  or  of 
the  irreducible  zeros  of  the  function  u  of  the  second  order,  so  that  il  =  2y  for 
functions  of  the  first  class,  and  n  =  a  +  /3  for  functions  of  the  second  class. 


282  RELATIONS   BETWEEN  [119. 

Let  u  be  represented  by  ^  (z)  and  v  hj  yjr  (z),  when  the  argument  must  be 
put  in  evidence.     Then 

so  that  x|r  (a  -  ^)  = ^ =Y-Y^  (^)- 

M 
Hence  f  (z)  +  ^|r  (n  -  z)  =  2  j- =  2R, 

N 

yjr  (z)  -  yjr  (n  -  z)  =  2  Y  (f>'  (z)  =  2Scb'  (z). 

First,  if  ■xjr  (z)  = -xlr  (£1  -  z),  then  S=0  and  ■^{z)  =  R:  that  is,  a  function  \|r  (2'), 

which  satisfies  the  equation 

ylr{z)^ir{D.-z), 

can  be  expressed  as  a  rational  meromorphic  function  of  (f)  (z)  of  the  second 
order,  doubly -periodic  in  the  same  periods  and  having  the  sum  of  its  irreducible 
infinities  congruent  with  O. 

Second,  if  f{z)^-f(n-z),  then  R  =  0  and  yjr{z)  =  Scb' (z);  that  is,  a 
function  yjr  (z),  which  satisfies  the  equation 

^|r  (z)  =  -  f  {^  -  z), 

can  be  expressed  as  a  rational  meromorphic  function  of  (j>  (z),  multiplied  by 
(f>'(z),  luhere  <^{z)  is  doubly -per  iodic  in  the  same  periods,  is  of  the  second  order, 
and  has  the  sum  of  its  irreducible  infinities  congruent  with  il. 

Third,  ii  -^{z)  have  no  infinities  except  those  of  u,  it  cannot  become 
infinite  for  finite  values  of  u ;  hence  Z  =  0  has  no  roots,  that  is,  Z  is  a  constant 
which  may  be  taken  to  be  unity.     Then  •x/r  {z)  a  function  of  order  m  can  be 

expressed  in  the  form 

M  +  N<^'{z),   ' 

where,  if  the  function  0  {z)  be  of  the  second  class,  the  degree  of  M  is  not 
higher  than  m ;  but,  if  it  be  of  the  first  class,  the  degree  of  M  is  not  higher 
than  m  —  1 ;  and  in  each  case  the  degree  of  N  is  not  higher  than  m  —  2. 

It  will  be  found  in  practice,  with  functions  of  the  first  class,  that  these 

upper  limits  for  degrees  can  be  considerably  reduced  by  counting  the  degrees 

of  the  infinities  in 

M  +  N4>'  {z). 

Thus,  if  the  degree  of  ilf  in  w  be  /x  and  of  N  be  X,  the  highest  degree  of  an 
infinity  is  either  2/Lt  or  2X  +  3 ;  so  that,  if  the  order  of  y\r  {z)  be  m,  we  should 
have 

m=  2fji,  or  m  =  2\  +•  3, 

according  as  m  is  even  or  odd. 


119.]  HOMOPERIODIC   FUNCTIONS  283 

When  functions  of  the  second  class  are  used  to  represent  a  function  yjr  (z), 
which  has  two  infinities  a  and  /8  each  of  degree  n,  then  it  is  easy  to  see  that 
M  is  of  degi'ee  n  and  iV"  of  degree  n  —  2;  and  so  for  other  cases. 

Corollary  2.  Any  doubly-periodic  function  can  he  expressed  rationally 
in  terms  of  any  other  function  a  of  any  order  n,  doubly -periodic  in  the  same 
periods,  and  of  its  derivative;  and  this  rational  expression  can  always  he  taken 
in  the  form 

Uo  +  U^  U    +  U^U'^  +  . . .  +  Vn-i  U"'~\ 

where Uq,  ...,  Un-i  are  rational  meromorphic  functions  of  u. 

Corollary  3.  If  <^  he  a  doubly -periodic  function,  then  (f)(u-\-v)  can  he 
expressed  in  the  form 

A  4-  B^lr'  {a)  +  C^^'  jv)  +  i)^/r'  {u)  -f'  {v) 
E  ' 

luhere  yjr  is  a  doubly -periodic  function  in  the  same  periods  and  of  the  second 
order:  each  of  the  functions  A,D,  E  is  a  symmetric  function  of-^^  (u)  and  yjr  {v), 
and  B  is  the  same  function  of  yjr  (v)  and  ■xjr  (u)  as  C  is  of  -v|r  (m)  and  -yj/  (v). 

The  degrees  of  A  and  E  are  not  greater  than  m  in  -v/^(w.)  and  than  m 
in  -^(v),  where  m  is  the  order  of  ^;  the  degree  of  D  is  not  greater  than 
m  —  2  in  ■^{u)  and  than  m— 2  in  '\lr{v);  the  degree  of  B  is  not  greater 
than  m  —  2  in  yjr  (u)  and  than  m  in  yjr  (v),  and  the  degree  of  G  is  not 
greater  than  m  —  2  in  yjr  (v)  and  than  m  in  \jr  (u). 

Note  on  Differential  Equations  of  the  First  Order 
HAVING  Uniform  Integrals. 

The  relation  given  in  Proposition  XIV.,  §  119,  immediately  suggests  a  converse 
qufestion  as  follows:  — 

Under  what  conditions  does  an  equation 

possess  integrals  expressing  w  as  a  uniform  function  of  s  ?  Further,  we  should  expect, 
after  the  proposition  which  has  just  been  mentioned,  that  under  fitting  conditions  the 
uniform  function  could  be  doubly-periodic  ;  and  we  have  already  seen  (in  §  104)  that  the 
integral  of  the  equation 

is  a  uniform  doubly-periodic  function  of  z.  But  it  might 'happen  (and  it  does  happen) 
that  other  classes  of  uniform  functions  are  integrals  of  differential  equations  of  the  same 
form. 

The  full  investigation  belongs  to  the  theory  of  differential  equations ;  an  account 
is  given  in  Chapter  X.,  Part  II.  (vol.  ii.)  of  my  Theory  of  Differential  Equations.  The 
following  statement  of  results,  which  are  established  there,  may  be  useful  for  reference. 

The  differential  equation  is  to  be  regarded  as  irreducible.  We  shall  need  the  equation 
satisfied  by  Iju;  so  we  shall  take  v  =  l/u  and  denote  its  derivative  by  v'. 


284  NOTE    ON    DIFFERENTIAL   EQUATIONS   OF  [119. 

I.  In  order  that  the  equation 

F  (u',  u)  =  m'™  -f  m''»  - 1  /i  (m)  + . . .  +/,„  (u)  =  0 
may  have  a  uniform  function  for  its  integral,  the  coefficients  fi{u),  ...,  fmiti)  must  be 
polynomials  in  u  of  degrees  not  higher  than  2,  ...,  2-rti  respectively;  and  the  condition  is 
then  satisfied  for  the  equation 

G  {v',  V)  =  V'"}  -  2/» - 1  /i  (^)  +  ...+■(-  1 )™ /,„  {v)  =  0. 

II.  If  any  finite  value  of  ti  is  a  branch-point  of  u'  determined  as  a  function  of  u  by 
the  equation  F  {%',  ?<)  =  0,  all  the  affected  values  of  u'  must  be  zero  for  that  value  of  u ; 
and  likewise  for  the  value  i'  =  0  in  connection  with  the  equation  O  {v',  v)  —  0.  (The  latter 
condition  covers  an  infinite  value  of  m  as  a  branch-point  of  u'.) 

III.  If  there  is  a  multiple  root  n'  of  F{u',  u)  —  0,  which  is  zero  for  n  branches  for  the 
branch-place  u,  then  the  term  of  lowest  degree  in  the  expansion  of  each  of  those  n  branches 

in  the  vicinity  of  that  branch-place  u  is  of  degree  either  1  — ,  1,  or  1  -I-  - ;  and  likewise 

for  the  value  v  =  0  for  the  equation  G  iv'.  v)  =  0.     The  number  1  — ,  1,  or  1-f--,  is  called 
the  index-degree. 

IV.  The  genus  of  the  equation  F  {u',  m)  =  0,  regarded  as  an  equation  in  «',  is  either 
zero  or  unity — as  is  therefore  also  the  genus  of  the  associated  Eiemann  surface  (see 
Chapter  XV.,  post). 

V.  When  the  index-degree  of  m',  for  any  finite  value  or  for  an  infinite  value  of  u  as 

a  branch  value  for  u',  is  less  than  unity,  being  then  necessarily  of  the  form  1  —  for  each 

branch  value,  though  n  need  not  be  the  same  for  all  the  different  branch-places,  w  is 
a  uniform  doubly -periodic  function  of  z. 

VI.  If  for  some  one  value  of  u  there  is  a  single  set  of  multiple  zero  roots  v!  of  index- 
degree  equal  to  unity,  and  if  for  other  values  of  u  (finite  or  infinite)  all  the  multiple  zero 

roots  xi!  ^re  of  index-degree  less  than  unity  and  therefore  necessarily  of  the  form  1  — , 

then  u  is  a  uniform  singly -periodic  function  of  z. 

VII.  If  for  some  one  value  of  u  there  is  a  single  set  of  multiple  zero  roots  v!  of  index- 
degree  greater  than  unity  and  therefore  necessarily  of  the  form  1  -t-- ,  and  for  other  values 
of  u  (finite  or  infinite)  all  the  multiple  zero  roots  %i!  are  of  index-degree  less  than  unity 
and  therefore  necessarily  of  the  form  1  — ,  then  u  is  a  rational  function  of  z. 

VIII.  When  these  conditions  are  applied  to  the  binomial  equation 


(S)'-/^)- 


where  f{u)  is  a  polynomial  in  u  of  degree  not  greater  than  2s,  so  as  to  obtain  integrals  u 
which  are  uniform  functions  of  z,  the  results  are  as  follows : — 

(A)    Equations,  having  uniform  integrals  which  are  rational  functions  of  z. 


du 
dz 
where  /i  is  a  constant  in  each  case  ; 


Y  =  ^  (j,  _«,).-!  (W-  6)3-1, 


119.]  THE    FIRST   ORDER   HAVING   UNIFORM   INTEGRALS  285 

(B)  Equations,  having  uniform  integrals  which  are  simply-periodic  functions  of  z, 

du        ,         . 
dz     '^  " 

az 

\£)  =  /^  (^* -  «)^  (^* -b){u-  c), 
where  /x  is  a  constant  in  each  case ; 

(C)  Equations,  having  uniform  integrals  which  are  doubly-periodic  functions  of  z, 

-£)  =fi{i(.-af(u-bf, 
{^^^"  =  ^^n-af{u-b)\ 
(^^J  =  f,(^u-anu-b)Hu-cy\ 

{^^'=y.{u-af{u-bf{u-cf, 

/dv\^ 

\dz)  "=  /^  ^^*  ~  ^)^  '^'^  ~  ^)^  "^^  ~  '^)^' 

^j  =fx.{u-a){u-b){u-c), 
[-rj  =fM{u  —  a){u-b){u-c){u-d). 


CHAPTER  XL 

Doubly-Periodic  Functions  of  the  Second  Order. 

The  present  chapter  will  be  devoted,  in  illustration  of  the  preceding 
theorems,  to  the  establishment  of  some  of  the  fundamental  formulae  relating 
to  doubly-periodic  functions  of  the  second  order  which,  as  has  already  (in 
§  119,  Cor.  to  Prop.  XIV.)  been  indicated,  are  substantially  elliptic  functions  : 
but  for  any  development  of  their  properties,  recourse  must  be  had  to  treatises 
on  elliptic  functions. 

It  may  be  remarked  that,  in  dealing  with  doubly-periodic  functions,  we 
may  restrict  ourselves  to  a  discussion  of  even  functions  and  of  odd  functions. 
For,  if  (fi  (z)  be  any  function,  then  ^  {(j)(z)  +  ^(—  z)]  is  an  even  function,  and 
^  {(f)  (z)  —  (f)  {— z)]  is  an  odd  function,  both  of  them  being  doubly-periodic  in 
the  periods  of  (j)(z);  and  the  new  functions  would,  in  general,  be  of  order 
double  that  of  0  (z).  We  shall  practically  limit  the  discussion  to  even 
functions  and  odd  functions  of  the  second  order. 

120.  Consider  a  function  (f)(z),  doubly-periodic  in  2fo  and  2w';  and  let 
it  be  an  odd  function  of  the  second  class,  with  a  and  ^  as  its  irreducible 
infinities,  and  a  and  b  as  its  irreducible  zeros*. 

Then  we  have  cf)  (z)  =  cf)  (ol  +  ^  —  z), 

which  always  holds ;  and  (f)(—  z)  =  -  ^  (z), 

which  holds  because  (f>  (z)  is  an  odd  function.     Hence 

<f)  {a  +  13  +  z)  =  cf)  (- z) 

so  that  a  4-  /3  is  not  a  period ;  and 

(f){a  +  l3  +  a  +  ^  +  z)^-(f)(:i  +  ^  +  z) 

*  To  fix  the  ideas,  it  will  be  convenient  to  compare  it  with  sn  z,  for  which  2w  =  4A',  2u'  —  2iK', 
a  =  iK',  ^^iK'  +  2K,  a  =  0,  and  b  =  2K. 


120.]  DOUBLY-PERIODIC    FUNCTIONS   OF   THE   SECOND    CLASS  287 

whence  2  (a  +  /S)  is  a  period.     Since  oc  +  /3  is  not  a  period,  we  take  o  +  ^  =  &>, 

or  =  ft)',  or  =  ft)  +  &)' ;  the  first  two  alternatives  merely  interchange  a>  and  &>', 

so  that  we  have  either 

a  +  /3  =  &>, 

or  a  -I-  ^  =  ft)  +  ft)'. 

And  we  know  that,  in  general, 

a  +  6  =  a  +  /9. 
First,  for  the  zeros :  we  have 

c/,(0)  =  -(/)(-0)  =  -c^(0), 
so  that  0  (0)  is  either  zero  or  infinite.     The  choice  is  at  our  disposal ;  for 
satisfies  all  the  equations  which  have  been  satisfied  by  0  {z)  and  an 


infinity  of  either  is  a  zero  of  the  other.     We  therefore  take 

<^(0)  =  0, 
so  that  we  have  a  =  0, 

b  =  (o    or    (0  +  ft)'. 

Next,  for  the  infinities :  we  have 

cf>(z)  =  -<f>{-2) 

and  therefore  </>  ( —  a)  =  —  0  (a)  =  oo  , 

The  only  infinities  of  (p  are  a  and  /3,  so  that  either 

—  a  =  a, 

or  -a  =  /3. 

The  latter  cannot  hold,  because  it  would  give  a  +  /3  =  0  whereas  a  -I-  /9  =  &) 

or  =  ft)  +  ft)' ;  hence 

•2oi=  0, 

which  must  be  associated  with  u  +  /3  =  o)  or  with  a  +  0  =  cl>+  co'. 

Hence  a,  being  a  point  inside  the  fundamental  parallelogram,  is  either  0, 
ft),  ft) ,  or  ft)  +  ft)'.  ♦ 

It  cannot  be  0  in  any  case,  for  that  is  a  zero. 

If  a  + 13=  CO,  then  a  cannot  be  co,  because  that  value  would  give  yS  =  0, 
which  is  a  zero,  not  an  infinity.  Hence  either  «  =  &)',  and  then  /3  =  &)'  -I-  ft); 
or  a  =  ft)'  +  ft),  and  then  /3  =  co'.     These  are  effectively  one  solution ;  so  that,  if 

a  +  /S  =  ft),  we  have 

a,  iS  =  ft)' ,  ft)' 4- ft)  1 

and  '        a,  b  =  0,  CO  j 

If  a  + /3  =  ft)  +  ft)',  then  a  cannot  be  co  +&>',  because  that  value  would  give 

/8  =  0,  which  is  a  zero,  not  an  infinity.     Hence  either  a=co  and  then  .d  =  ft)', 

or  a  =  ft)'  and  then  ^  =  co.     These  again  are  effectively  one  solution  ;  so  that, 

if  a  +  /3  =  ft)  +  ft)',  we  have 

(X,  j3  =■  (o,  co'  ] 

and  a,  6  =  0,  ft)  +  ft)'  J 


288  DOUBLY-PERIODIC   FUNCTIONS  [120. 

This  combination  can,  by  a  change  of  fundamental  parallelogram,  be  made 
the  same  as  the  former ;  for,  taking  as  new  periods 

2&)'  =  2&)',         211  =  2w  +  2ft)', 

which  give  a  new  fundamental  parallelogram,  we  have  a  +  ^  =  fl,  and 

w,  /3  =  ft)',  11  —  ft)',  that  is,  ft)',  n  -  ft)'  +  2ft)', 

so  that  a,  |8  =  ft)',  fl  +  ft)'  ] 

and  a,  b  =  0,  Q  J  ' 

being  the  same  as  the  former  with  H  instead  of  ft).  Hence  it  is  sufficient  to 
retain  the  first  solution  alone :  and  therefore 

a  =  ft)',         /S  =  ft)'  +  ft), 

a  =  0,  b  =  CO. 

Hence,  by  |  116,  I.,  we  have 

{Z  —  CO){Z  —  (0  —  CO) 

where  F  {z)  is  finite  everywhere  within  the  parallelogram. 

Again,  (f>(z  +  co')  has  z  =  0  and  z  =  co  SiS  its  irreducible  infinities,  and 
it  has  ^  =  ft)'  and  z  =  co  +  o)'  as  its  irreducible  zeros,  within  the  parallelogram 
of  0  (z) ;  hence 

/  /  '\       (^  —  ^')  (Z—  (O—  ft)')    „    ,    . 

where  F^^  {z)  is  finite  everywhere  within  the  parallelogram.     Thus 

^{z)c^{z  +  co')=^F{z)F,{z), 

a  function  which  is  finite  everywhere  within  the  parallelogram  ;  since  it  is 
doubly-periodic,  it  is  finite  everywhere  in  the  plane  and  it  is  therefore  a 
constant  and  equal  to  the  value  at  any  point.  Taking  —  ^co'  as  the  point 
(which  is  neither  a  zero  nor  an  infinity)  and  remembering  that  0  is  an  odd 
function,  we  have 

k  being  a  constant  used  to  represent  the  value  of  —  [</)(|ft)')}~\ 
Also  ^{z  +  (o)  =  ^{z-\-a  +  li-  2ft)') 

=  </)(^-fa  +  /3)  =  -<^  {z), 
and  therefore  also  ^{co  —  z)  =  (^  {z). 

The  irreducible  zeros  of  4>' {z)  were  obtained  in  §  117,  X.  In  the 
present  example,  those  points  are  ft)'  +  -|&),  co'  +  ^co,  ^co,  fo);  so  that,  as 
there,  we  have 

K[c\^{z)Y-\<^{z)-^{\co)][<l>{z)-c^{%co)]{<l>{z)-^{co'  +  \c^^^^^ 

where  ^  is  a  constant.     But 

</)  (f  ft))  =  (^  (2ft)  - -^ft))  =  (^  ( -  i&))  =  -  (/)  ( J  ft)); 


120.]  OF   THE   SECOND    CLASS  289 

and  (j)  {§ CO  +  0)')  =  (f)  (2ft)  -f  2&)'  —  ^co  —  o)') 

=  (f}{-^(0-(0') 

=  —  (f>{^Q}  +  a)'); 


so  that  {(/)'  (z)}-  =  A 


1  [<t>(^)r 


where  J.  is  a  new  constant,  evidently  equal  to  {^'  (0)]"^.  Now,  as  we  know 
the  periods,  the  irreducible  zeros  and  the  irreducible  infinities  of  the  function 
(f)  (z),  it  is  completely  determinate  save  as  to  a  constant  factor.  To  determine 
this  factor  we  need  only  know  the  value  of  <^  (z)  for  any  particular  finite 
value  of  z.     Let  the  factor  be  determined  by  the  condition 

then,  since  ^(^o))^(|^ft> +  &)')  =  y 

K 

by  a  preceding  equation,  we  have 
and  then 

.[</)'  (z)Y-  =  .[f  (0)}^  [1  -  {<!>  (z)]^]  [1  -  k^  {(^  (z)Y] 

=  f^^[l-{cf>{z)Y][l-k^{cf^(z)Y].     . 

Hence,  since  ^  (z)  is  an  odd  function,  we  have 

(f)  (z)  =  sn  (fjiz). 

Evidently  2/x(o,  2/ji(o'  =  4tK,  2iK',  where  K  and  K'  have  the  ordinary  signifi- 
cations.    The  simplest  case  arises  when  /i  =  1. 

121.  Before  proceeding  to  the  deduction  of  the  properties  of  even 
functions  of  z  which  are  doubly-periodic,  it  is  desirable  to  obtain  the 
addition-theorem  for  (f),  that  is,  the  expression  of  <f)  {y  +  z)  in  terms  of 
functions  of  y  alone  and  z  alone. 

When  (f)(y  +  z)  is  regarded  as  a  function  of  z,  which  is  necessarily  of  the 
second  order,  it  is  (§  119,  XV.)  of  the  form 

M  +  iV(/)'  (z) 
Z  ' 

where  M  and  L  are  of  degree  in  </>  (z)  not  higher  than  2  and  iVis  independent 
of  z.  Moreover  y  +  z=  a  and  y  +  z  =  /3  are  the  irreducible  simple  infinities 
of  (f>{y  +  z);  so  that  L,  as  a  function  of  z,  may  be  expressed  in  the  form 

{cf,(z)-cf>{a-y)]{cf>(z)-cf>(^-y)], 
and  therefore 

P+Qcf>{z)  +  R{<p{z)Y  +  S4>'(z) 
f'^y  +  '^-  jc/,  (,)  -  c^  («  _  y)]{f(-z)  -<^{^-y)Y 

F.   F.  19 


290  DOUBLY-PERIODIC   FUNCTIONS  [121. 

where  P,  Q,  R,  S  are  independent  of  2  but  they  may  be  functions  of  y. 
Now 

and  <^(/3-2/)=^K+^-2/)=^^7^7^  =  X:^^); 

so  that  the  denominator  of  the  expression  for  (f){y+z)  is 

Since  (f>  (z)  is  an  odd  function,  ^'  (z)  is  even ;  hence 

P-Qcf,(z)  +  R{cf>{z)Y  +  S<l,'(z) 
<f>{y-^)  = 1 ' 

2Qcf>(z) 


and  therefore  4>  (y  +  z)  -  cf>{y  -  z)  =  - 

Differentiating  with  regard  to  z  and  then  making  2-  =  0,  we  have 

2<^  (2/)  = T > 


so  that,  substituting  for  Q  we  have 

Interchanging  y  and  2^  and  noting  that  cp  (y  —  z)  =  -  <f)  {z  —  y),  we  have 

1  2</)  (2/)  (/)' (^) 

c^  (2/  +  ^)  +  </>  (2/  -  ^)  =  ^^  ^0)  r^A-  10  (2/)p{cf>  {z)Y ' 

and  therefore  </>  (2/  +  ^)  «/>  (0)  -     i  _  kn4>  {y)Y  {cf>  {z)f~ ' 

which  is  the  addition-theorem  required. 

Ex.  If  f{u)  be  a  doubly -periodiu  function  of  the  second  order  with  infinities  61,  b.^, 
and  (})  (u)  a  doubly- periodic  function  of  the  second  order  with  infinities  Oj,  a.^  such  that, 
in  the  vicinity  of  a^  (for  {=1,  2),  we  have 

(f,(^u)  =  i:L^+p^  +  qi{u-ai)  + -1- Z'i  (w  -  a^)"  + , 


u  —  ai 


then  ^i^^±^=-^{(^(^)  +  c^(62)-^i-p2}, 

the  periods  being  the  same  for  both  functions.     Verify  the  theorem  when  the  functions 
are  sn  «  and  sn(M+i;). 

Prove  also  that  ki  =  {-l)^k2.  (Math.  Trip.,  Part  II.,  1891.) 


122.]  OF   THE    SECOND    CLASS  291 

122.  The  preceding  discussion  of  uneven  doubly-periodic  functions 
having  two  simple  irreducible  infinities  is  a  sufficient  illustration  of  the 
method  of  procedure.  That,  which  now  follows,  relates  to  doubly-periodic 
functions  with  one  irreducible  infinity  of  the  second  degree  ;  and  it  will  be 
used  to  deduce  some  of  the  leading  properties  of  Weierstrass's  o-- function 
(of  §  57)  and  of  functions  which  arise  from  it. 

The  definition  of  the  o--function  is 

where  12  —  2nia)  +  2m' co',  the  ratio  of  w'  :  «  not  being  purely  real ;  and  the 
infinite  product  is  extended  over  all  terms  that  are  given  by  assigning  to 
m  and  to  m'  all  positive  and  negative  integral  values  from  -|-  oo  to  —  oo  , 
excepting  only  simultaneous  zero  values.  It  has  been  proved  (and  it  is 
easy  to  verify  quite  independently)  that,  when  a  (z)  is  regarded  as  the 
product  of  the  primary  factors 


the  doubly-infinite  product  converges  uniformly  and  unconditionally  for  all 
values  of  z  in  the  finite  part  of  the  plane  ;  therefore  the  function  which  it 
represents  can,  in  the  vicinity  of  any  point  c  in  the  plane,  be  expanded  in  a 
converging  series  of  positive  powers  of  ^  —  c,  but  the  series  will  only  express 
the  function  in  the  domain  of  c.  The  series,  however,  can  be  continued  over 
the  whole  plane. 

It  is  at  once  evident  that  cr  (z)  is  not  a  doubly-periodic  function,  for  it  has 
no  infinity  in  any  finite  part  of  the  plane. 

It  is  also  evident  that  cr  (z)  is  an  odd  function.  For  a  change  of  sign  in  z 
in  a  primary  factor  only  interchanges  that  factor  with  the  one  which  has 
equal  and  opposite  values  of  m  and  of  m',  so  that  the  product  of  the  two  factors 
is  unaltered.  Hence  the  product  of  all  the  primary  factors,  being  independent 
of  the  nature  of  the  infinite  limits,  is  an  even  function ;  when  z  is  associated 
as  a  factor,  the  function  becomes  uneven  and  it  is  or  {z). 

The  first  derivative,  a'  (z),  is  therefore  an  even  function ;  and  it  is  not 
infinite  for  any  point  in  the  finite  part  of  the  plane. 

It  will  appear  that,  though  a  (z)  is  not  periodic,  it  is  connected  with 
functions  that  have  2(o  and  2g)'  for  periods ;  and  therefore  the  plane  will  be 
divided  up  into  parallelograms.  When  the  whole  plane  is  divided  up,  as  in 
§  105,  into  parallelograms,  the  adjacent  sides  of  which  are  vectorial  repre- 
sentations of  2&)  and  2co',  the  function  a  (z)  has  one,  and  only  one,  zero  in 
each  parallelogram ;  each  such  zero  is  simple,  and  their  aggregate  is  given 
by  ^  =  ft.     The  parallelogram  of  reference  can  be  chosen  so  that  a  zero  of 

19—2 


292  DOUBLY-PERIODIC   FUNCTIONS  [122. 

o-  (z)  does  not  lie  upon  its  boundary ;  and,  except  where  explicit  account  is 
taken  of  the  alternative,  we  shall  assume  that  the  argument  of  co'  is  greater 
than  the  argument  of  co,  so  that  the  real  part*  of  co'jio)  is  positive. 

Before  proceeding  further,  it  is  convenient  to  establish  some  proi^ositions  relating  to 
series  which  will  be  used  almost  immediately. 

We  have  seen  that  the  series  Sfi"^,,^,,  where  Q„,,  ^'  denotes  2mco  +  2,m' a  for  all  positive 
and  negative  integers  m  and  m'  ranging  independently  from  —  oo  to  +  co  (only  the 
simultaneous  zero  values  being  excepted),  converges  absolutely.     Now  consider  the  series 

1 

V"»i,  in'      '') 

for  the  same  range  of  summation;  and  assume  that  z  can  have  any  value  except  a 
quantity  Q„j,  „',  when  there  is  obviously  an  infinite  term  of  order  three.  From  the  series, 
we  temporarily  exclude  all  the  terms  for  which 

as  I  2  I  is  finite,  these  terms  are  finite  in  number  and  their  sum  does  not  affect  the 
convergence  of  the  series. 

For  all  the  remaining  terms,  we  have 

|0|<1|Q|. 

Now  |12-2|>|G|-|5|, 

i  O-zl        ,       Izl 
so  that  I      I     >  I  -  j-— -| 

consequently  the  series 


-{St-zf 

converges  absolutely  for  all  finite  values  of  z  except  the  isolated  values  given  by  z  =  Q; 
and,  by  Weierstrass's  J/-test+,  the  same  inequality  shews  that  the  series  converges 
uniformly. 

It   is  a  known   property  (p.  22)   of  uniformly  converging   series  that  they  can  be 
integrated  term  by  term  within  a  finite  range  and  the  resulting  series  will  also  converge 

uniformly.     Now 

dz 


0  (0-2)3      2  t(Q- 

choosing  the  path  of  integration  merely  to  avoid  a  possible  infinity  of  the  subject  of 
integration — a  choice  that  does  not  affect  the  result  in  this  case.  Hence  dropping  the 
factor  -^j  we  see  that  the  series 

f      1  '     _  V\ 

^\a-zf      Q^j 

is  a  series  that  converges  uniformly  for  all  finite  values  of  2,  except  the  isolated  values 
given  by  z  =  Q.. 

*  This  quantity  is  often  denoted  by  9{  (  —  j . 

t  Bromwicb,  Theory  of  Infinite  Series,  §  81. 


122.]  OF   THE    FIRST    CLASS  293 

As  this  series  converges  uniformly,  it  also  can  be  integrated  term  by  term  within 
a  finite  range  and  the  resulting  series  will  also  converge  uniformly.     Now 


(I        z  I 

hence  2  i  —  +  tt^  H 

[a      fl2   '  2-12 

is  a  series  which  converges  uniformly  for  all  finite  values  of  z,  except  the  isolated  values 
given  by  z  =  O. 

Again  integrating  within  the  finite  range  from  0  to  s,  we  have 


+  .-^}^^  =  l  +  l5  +  ^°H'-i2 


^^^^^  ^{l  +  ^l  +  ^^s^-^yj 

is  a  series  which  converges  uniformly  for  all  finite  values  of  s,  except  the  isolated  values 
given  hj  z  =  Q. 

123.  We  now  proceed  to  obtain  other  expressions  for  a  (z),  and  particu- 
larly, in  the  knowledge  that  it  can  be  represented  by  a  converging  series  in 
the  vicinity  of  any  point,  to  obtain  a  useful  expression  in  the  form  of  a  series, 
converging  in  the  vicinity  of  the  origin. 

Since  a  (z)  is  represented  by  an  infinite  product  that  converges  uniformly 
and  unconditionally  for  all  finite  values  of  z,  its  logarithm  is  equal  to  the  sum 
of  the  logarithms  of  its  factors,  so  that 

log.(.)=log.+_Si(^  +  l^,+log(l-^)j. 

where  the  series  on  the  right-hand  side  extends  to  the  same  combinations  of 
m  and  m   as  the  infinite  product  for  z.     When  it  is  regarded  as  a  sum  of 

functions  o  +  s  7v  +  ^°S"  P-  "  n    '  ^^^  series  converges  uniformly  and  uncon- 


n    2  a-     ^  V     ^/ 

ditionally,  except  for  points  ^  =  H.     This  expression  is  valid  for  log  cr  {z)  over 
the  whole  plane. 

Now  let  these  additive  functions  be  expanded,  as  in  §  82.     In  the  imme- 
diate vicinity  of  the  origin,  we  have 

z       \  z"'       ^      ( ^       z 
+  on-2  +  log    l- 


Xl  '  2  n^       ^  V        ^ 

\  z^  _\^_\^_ 

""siP'So^    sn-'    ■■■' 

a  series  which  by  itself  converges  uniformly  and  unconditionally  in  that 
vicinity.  When  this  expression  is  substituted  in  the  right-hand  side  of  the 
foregoing  expression  for  log  a  {z),  we  have  a  triple  series 

00      00       f     CO      1       ^r 


294  WEIERSTRASS'S  [123. 

It  is  easy  to  see  that  this  triple  series  converges  uniformly  for  the  values  of  z 
considered.  As  in  the  lemma  at  the  end  of  §  122,  we  omit  temporarily  all 
the  terms  for  which  |  ^  |  ^  -^  |  fi  1  ;  they  are  finite  in  number  for  finite  values 
of  z,  and  their  omission  does  not  affect  the  convergence  of  the  series.  Now 
the  modulus  of  the  remainder 

00         00         00  1         \     ^    [>" 

<  2  2  2  i       '  ■ 


»  00         CO 


<i2  2   S 

—  00  —00  r=S 


3 Ifl 


3  —  I  n  p 

'    '  1- 


But  for  each  of  these  terms  1 5;  |  <  ^  |  fl  | ;  and  therefore 

1 


1 

so  that  the  modulus  is  less  than 


z 

n 


<% 


00        00 


3 

—  00    —00 


a  finite  quantity.     Hence  the  whole  triple  series  converges  uniformly  and 
absolutely  for  the  values  of  z  considered  ;  and  so  we  may  take  it  in  the  form 


00       -,»•    f      00       00 

-  s  -   X  S  n-' 


r=3  r  (_c 
In  §  56,  it  was  proved  that  each  of  the  coefficients 

—  00    —00 

for  r  =  3,  4,  . . . ,  is  finite,  and  has  a  value  independent  of  the  nature  of  the 
infinite  limits  in  the  summation.  When  we  make  the  positive  infinite  limit 
for  m  numerically  equal  to  the  negative  infinite  limit  for  m,  and  likewise  for 
m,  then  each  of  these  coefficients  determined  by  an  odd  index  r  vanishes, 
and  therefore  it  vanishes  in  general.     We  then  have 

log  a-  {z)  =  log  ^  -  iz'ii.n-'  -  ^z'tsn-'  -  ^z'tm-'  - . . . , 

a  series  which  converges  uniformly  and  unconditionally  in  the  vicinity  of  the 
origin. 

The  coefficients,  which  occur,  involve  &>  and  w',  two  independent  constants. 
It  is  convenient  to  introduce  two  other  magnitudes,  g^  and  g^,  defined  by  the 
equations 

g2  =  6022^l-^    g,  =  uottn-', 


123.]  ELLIPTIC    FUNCTIONS  295 

SO  that  gz  and  y^  are  evidently  independent  of  one  another  ;  then  all  the 
remaining  coefficients  are  functions*  of  g^  and  g^.     We  thus  have 

l0gc.(.)  =  l0g.-2^^,^-gL^3,e_...__L,..X2a--..., 

and  therefore  a{z)  =  ze  ^-^'^^"'    s^o^^^    -^ 

where  the  series  in  the  index,  containing  only  even  powers  of  z,  converges 
uniformly  and  unconditionally  in  the  vicinity  of  the  origin. 

It  is  sufficiently  evident  that  this  expression  for  a  {z)  is  an  effective 
representation  only  in  the  vicinity  of  the  origin ;  for  points  in  the  vicinity  of 
any  other  zero  of  a  {z),  say  c,  a  similar  expression  in  powers  of  ^  —  c  instead 
of  in  powers  of  z  would  be  obtained.     . 

124.     From  the  first  form  of  the  expression  for  log  a  {z),  we  have 

a  {z)  z^  n  -^  \^^  ^^  z-  n 
where  the  quantity  in  the  bracket  on  the  right-hand  side  is  to  be  regarded 
as  an  element  of  summation,  being  derived  from  the  primary  factor  in  the 
product-expression  for  a  (z).  We  have  seen  (p.  293)  that  this  double  series 
converges  uniformly  for  the  values  of  z  concerned,  except  of  course  the 
isolated  values  CI. 

We  write  ^(^)  =  ^\ 

so  that  ^(z)  is,  by  §  122,  an  odd  function,  a  result  also  easily  derived  from  the 
foregoing  equation ;  and  so 

This  expression  for  ^(z)  is  valid  over  the  whole  plane. 
Evidently  ^(z)  has  simple  infinities  given  by 

z  =  n, 

for  all  values  of  m  and  of  ni  between  +  x  and  —  oo ,  including  simultaneous 
zeros.  There  is  only  one  infinity  in  each  parallelogram,  and  it  is  simple  ;  for 
the  function  is  the  logarithmic  derivative  of  a  (z),  which  has  no  infinity  and 
only  one  zero  (a  simple  zero)  in  the  parallelogram.  Hence  ^(z)  is  not  a 
doubly -periodic  function. 

For  points,  which  are  in  the  immediate  vicinity  of  the  origin,  we  have 


d 

dz 

['°g^    L^'^'-L^'^'    -    al^-S^"-'"--] 

1 

z 

eV^^'    140^'^=     ...-.-22n-»_...; 

*  See  Quart.  Journ.,  vol.  xxii,  pp.  4,  5.     The  magnitudes  g.2  and  g^  are  often  called  the 
invariants. 


296  WEiERSTR  ass's  [124. 

but,  as  in  the  case  of  cr  (z),  this  is  an  effective  representation  of  ^  (z)  only 
in  the  vicinity  of  the  origin ;  and  a  different  expression  would  be  used  for 
points  in  the  vicinity  of  any  other  pole. 

We  again  introduce  a  new  function  ^j  (z)  defined  by  the  equation 

Because  ^  is  an  odd  function,  g)  {z)  is  an  even  function ;  and 

,  ,  _  1      ^   ^  j  1  1      ]  _  1      V   V  1      1  11 

i,  {Z)  -  ^.^  -    2^    _Z      ^^,  ^-^-^1   -  ^,3  +    -,      2^  1^^  _  ^^^  -  ^,|    , 

where  the  quantity  in  the  bracket  is  to  be  regarded  as  an  element  of 
summation.  We  have  seen  (p.  293)  that  this  double  series  converges  uni- 
formly for  the  values  of  z  concerned,  except  of  course  the  isolated  values  H. 
Thus  the  expression  for  ^  (z)  is  valid  over  the  whole  plane.  Evidently  ^J  (z) 
has  infinities,  each  of  the  second  degree,  given  by  ^^  =  0,  for  all  values  of  m 
and  of  m'  between  +  oo  and  —  oo  ,  including  simultaneous  zeros ;  and  there 
is  one,  and  only  one,  of  these  infinities  in  each  parallelogram.  One  of  these 
infinities  is  the  origin  ;  using  the  expression  which  represents  log  a  (z)  in 
the  immediate  vicinity  of  the  origin,  we  have 


^^'^  =  -d^^ 


1  1  4  1  6 

°        240^  840^ 


for  points  z  in  the  immediate  vicinity  of  the  origin. 

A  corresponding  expression  exists  for  ^J  (z)  in  the  vicinity  of  any  other 
pole. 

125.  The  importance  of  this  function  ^o  (z)  lies  in  its  periodic  character ; 
and  the  importance  of  the  functions  a-{z)  and  ^{z)  partly  lies  in  their 
pseudo-periodic  character.  To  establish  the  necessary  properties,  we  use  the 
derivative  of  ^  (z) ;  we  differentiate  term  by  term  the  series  in  the  expression 

^'^'^  =  z^  +  ll\{^^ny^-m' 


and  we  have 


9  .  00  00 


=  -2  S    S 


(z-ny 
1 


(z  -  ny ' 

where  the  double  summation  no  longer  excludes  the  simultaneous  zero 
values  of  m  and  m'  in  the  expression  of  fl.  The  series  on  the  right-hand 
side  converges  uniformly  and  absolutely  (p.  292)  for  all  values  of  z  except  the 
isolated  places  given  by  ^^  =  ft  ;  and  so  this  expression  for  ^J  (z)  is  valid  over 
the  whole  plane. 


125.]  ELLIPTIC   FUNCTIONS  297 

Evidently  ^j'  (2)  has  infinities,  each  of  the  third  degree,  given  by  2  =  H 

for  all  values  of  m  and  m'  within  the  range  from  +00  to  —  00  ,  including  the 

simultaneous  zero  values ;  and  there  is  one,  and  only  one,  of  these  infinities 

within  each  parallelogram.     Using  the  expression  for  ^j{z)  in  the  vicinity 

of  2  =  0,  we  have 

2       1  1 

Clearly  g?'  is  an  odd  function  of  z. 

The  periodicity  of  ^y  (z)  can  be  deduced  at  once.     We  have 

^'(^)  =  -  2SS -^7—  =  -  21X  , ^ — ^^^-7-^x3; 

''    ^  ^  {z  —  ^Y  {z  —  2m&)  —  2m  &>  y 

and  therefore 

*    ^  '  {z-2  (vi  -  1)  ft)  —  2?7z.  CO  Y 

Now  the  series  S2  {z  —  fl)"^  converges  absolutely;  and  so  (p.  21)  its  sum 
does  not  depend  upon  the  order  in  which  the  terms  are  taken.  The  series 
in  §)'  (z  +  2&))  differs  fi'om  the  series  in  ^'  (z)  merely  in  taking  the  terms  in 
the  order  of  values  of  m  —  1  from  -  00  to  +  00  instead  of  the  terms  in  the 
order  of  values  of  m  from  —  -jd  to  +  00  ;  this  negative  unit  derangement  in 
the  summation  for  m  is  permissible  under  the  convergence;  and  so  we  have 

^y  (z  +  2&))  =  p'  (z). 
Similarly  we  have 

^'  (z  +  2a)')  =  g>  (z), 

equations  which  shew  the  double  periodicity  of  ^j'  (z).     Further,  we  have 

g>'  (z  +2a)+  2&)0  =  ^0  (z  +  2ft)')  =  ^'  (z); 
or  writing 

ft)"  =  ft)  +  ft)', 
we  have 

p'  (z  +  2ft)")  =  p'  (z). 

Integrating  these  equations  respectively,  we  have 

p{z  +  2o})  =  p{z)  + A,         p{z+  2(i)')=p{z)  +  B, 

where  A  and  B  are  constants.  To  determine  these  constants,  take  z=  —  co 
in  the  former  equation  and  z  =  —  co'  in  the  latter ;  we  have 

^(co)  =  p{-(o)+A,         p(co')  =  iJ(-co')+B. 

Neither  ft)  nor  &)'  is  a  pole  of  p  (z),  for  the  isolated  poles  of  ^  (z)  are  given  by 
z  =  2mo)  +  2m  co',  for  integer  values  of  m  and  m' ;  and  ^  (z)  is  an  even 
function.     Thus  ^4.  =  0,  B  =  0;   and  so  we  have 

p  (z  +  2co)  =  p  (z),         p(z  +  2co')  =  p(z), 

and  therefore  also 

p(z+2co")  =  i0(z), 

equations  which  shew  the  double  periodicity  of  p  (z). 


298  WEIERSTRASS'S  [125. 

The  poles  of  ^j  (z)  are  given  hy  z  =  H,  and  each  is  of  order  2.  Thus  in 
any  parallelogram  whose  adjacent  sides  are  2a)  and  2(o',  there  is  one  (and 
there  is  only  one)  pole,  and  it  is  of  order  2.  Hence  by  §  116,  Prop.  III., 
Cor.  3,  2&)  and  2co'  determine  a  primitive  parallelogram  for  ^j(z).  Conse- 
quently our  function  ^  (z)  is  of  the  first  class  and  the  second  order. 

We  shall  assume  that  the  parallelogram  of  reference  is  so  chosen  as  to 
include  the  origin  in  its  interior. 

126.  In  the  preceding  chapter,  we  have  seen  (§119)  that  there  exists  an 
algebraical  relation  between  ^j  (z)  and  g)'  (z).  Owing  to  the  order  of  ^j  (z), 
this  must  have  the  form 

^'2  (z)  =  A^j'  (z)  +  Bf  {z)  +  Gf  (z)  +  D, 
where  A,  B,  C,  D  are  constants. 

The  only  irreducible  infinity  of  (^'  {z)  is  of  the  third  order,  being  the 
origin ;   and  the  function  ^'  {z)  is  odd.     As 

<^j'  (z  +  2(o)  =  gy  (z)  =  p'(z  +  2co')  =  ^y  (z  -f  2(o"), 
we  have 

^'(&))  =  -^'(ft)),         p' (o)')  =  -  g>' («'),         p'(«")  =  -^j'(«"), 

so  that  the  irreducible  zeros  of  ^'  (z)  are  a,  co',  co".     We  write 

^o(w)  =  e„         ^j{o)")  =  e^,         ^{co')  =  es,         ^j(z)  =  ^,         ^o' (z)  =  ^j' ; 
and  then  the  foregoing  relation  becomes 

^j''  =  A(^-e,)(^-e,)(^j-e.,), 
where  A  is  some  constant.     To  determine  the  equation  more  exactly,  we 
substitute  the  expression  of  ^J  in  the  vicinity  of  the  origin.     Then 

^'=     ^^  +  ^^^"''  +  ^^^"'+-" 

2       1  1 

so  that  ^  =--3  +  iq92^  +  Y^'s^'  +  •  •  •  • 

When  substitution  is  made,  it  is  necessary  to  retain  in  the  expansion  all 
terms  up  to  2"  inclusive.     We  then  have,  for  jp'^  the  expression 

?~5z'-~7^'^"-' 
and  for  A(f  —  e^)  (^  —  e.^)  (^  —  63),  the  expression 


A 


z'^20  z''^28^''^ 


-  {e,  +  So  +  63)  (-  +  ^ 5^2  +  •••  j  +  (ei^o  +  e.e.i  +  e,e,)  ^-  +  ...  j  -  e^e^e^ 

When  we  equate  coefficients  in  these  two  expressions,  we  find 

^  =  4, 
6^  +  62  +  63  =  0,        6162  +  6.263  +  6361  =  - i^'-^.        616,63  =  ^5^3; 


126.]  ELLIPTIC   FUNCTIONS  299 

therefore  the  differential  equation  satisfied  by  p  is 

Evidently  ^"  =  6^J^  —  ^g2, 

and  so  on ;  it  is  easy  to  verify  that  the  2?ith  derivative  of  p  is  a  rational 
integral  function  of  ^  of  degree  n  + 1,  and  that  the  (2n  +  l)th  derivative 
of  ^  is  the  product  of  <p'  by  a  rational  integral  function  of  ip  of  degree  n. 

The  differential  equation  can  be  otherwise  obtained,  by  dependence  on 
Cor.  2,  Prop.  V.  of  §  116.     We  have,  by  differentiation  of  ^y, 

/,      6       1  3       , 

for  points  in  the  vicinity  of  the  origin;  and  also 

Hence  p"  and  ^-  have  the  same  irreducible  infinities  in  the  same  degree  and 
their  fractional  parts  are  essentially  the  same :  they  are  homoperiodic  and 
therefore  they  are  equivalent  to  one  another.  It  is  easy  to  see  that  ^"  —  6^- 
is  equal  to  a  function  which,  being  finite  in  the  vicinity  of  the  origin,  is  finite 
in  the  parallelogram  of  reference  and  therefore,  as  it  is  doubly-periodic,  is 
finite  over  the  whole  plane.  It  therefore  has  a  constant  value,  which  can  be 
obtained  by  taking  the  value  at  any  point ;  the  value  of  the  function  for 
2  =  0  is  —^g2  and  therefore 

so  that  ^y  =  6^^  —  ^g., 

the  integration  of  which,  with  determination  of  the  constant  of  integration, 
leads  to  the  former  equation. 

This  form,  involving  the  second  derivative,  is  a  convenient  one  by  which 
to  determine  a  few  more  terms  of  the  expansion  in  the  vicinity  of  the  origin : 
and  it  is  easy  to  shew  that 

fi:om  which  some  theorems  relating  to  the  sums  220~^'''  can  be  deduced*. 

^x.     If  c„  be  the  coefficient  of  ^2)1-2  [-^  ^}jg  expansion  of  ^  (z)  in  the  vicinity  of  the 
origin,  then 

3  r=n-2 

^n=7^ — TTw ^     ^     CrCn_r-  (Weierstrass.) 

(2%  +  l)(?i-3)    ,.=2 

*  See  a  paper  by  the  author,  Quart.  Jmirn.,  vol.  xxii,  (1887),  pp.  1 — 43,  where  other  references 
are  given  and  other  applications  of  the  general  theorems  are  made. 


300  PERIODICITY  [126. 

We  have  ^'^  =  4<p' -  g^^  - g^; 

the  function  p'  is  odd,  and  in  the  vicinity  of  the  origin  we  have 

2 

hence,  representing  by  —  (4jj)^— ^fg^  — ^'s)*  tliat  branch  of  the  function  which 
is  negative  for  large  real  values,  we  have 

dip 
and  therefore  2  = 


^j  {4>^^-g,^-g,f' 
The  upper  limit  is  determined  by  the  fact  that  when  z  =  0,  ^=  co;  so  that 

dp 


dp 


This  is,  as  it  should  be,  an  integral  with  a  doubly-infinite  series  of  values. 
Wehave,  by  Ex.  7  of§104, 

dp 


0)1  =  0)     = 


0)9  =  O)     = 


^.  (4>f -92^-93)'' 
=°  dp 

e.{'^p'-g,p-gzf' 

e^i^p^-g^p-gzf 
with  the  relation  o)"  =  o)  +  o)'. 

127.     We  have  seen  (§  125)  that  p  {z)  is  doubly-periodic,  so  that 

p  (z  +  2o))  =  p  {z), 

and  therefore  , =  — -. —  ; 

dz  dz 

hence  integrating  ^ (^  +  2o))  =  ^  (z)  +  A. 

Now  ^  is  an  odd  function  ;  hence,  taking  z  =  —  q)  which  is  not  an  infinity  of  ^, 

we  have 

say,  where  77  denotes  ^  (o)) ;  and  therefore 

^{z  +  2(o)-^(z)  =  2'n, 
which  is  a  constant. 

Similarly  l;{z  +  2co')-^  (z)  =  2r]', 

where  ^'  =  ^  (o)')  and  is  constant. 


127.]  OF   WEIERSTRASS'S   FUNCTIONS  301 

Similarly 

r(^+2a)'o-r(^)  =  2V', 

where  r]"  =  ^{co")  and  is  constant.     Moreover, 

l;{2  +  2&)")  =^(z+2co  +  2(o') 
=  l;  {z  +  2co)  +  27]' 

=  ^(2)+2l]  +  27]', 

and  therefore 

v    =  v  +  V  > 
a  relation  which  merely  expresses  ^  (co  +  co')  as  the  sum  of  ^(eo)  and  ^  (&)')• 
Combining  the  results,  we  have 

^(z  +  2mco  +  2m'(o')  —  ^(^)  =  2m7}  +  2m  tj', 
where  m  and  m'  are  any  integers. 

It  is  evident  that  rj  and  rj'  cannot  be  absorbed  into  ^;  so  that  ^is  not  a 
periodic  function,  a  result  confirmatory  of  the  statement  in  §  124. 

There  is,  ho^yever,  a,  pseudo-periodicity  of  the  function  ^:  its  characteristic 
is  the  reproduction  of  the  function  with  an  added  constant  for  an  added 
period.  This  form  is  only  one  of  several  simple  forms  of  pseudo-periodicity 
which  will  be  considered  in  the  next  chapter. 

128.  But,  though  ^  {z)  is  not  periodic,  functions  which  are  periodic  can 
be  constructed  by  its  means. 

Thus,  if         cj) {z)  =  A  ^(z- a)  +  B ^(z -b)  +  C ^(2 - c)  ^  ... , 

then  4>{z+2(o)-cf>(z)  =  tA  {^{z- a  +  2oo)- ^(z -a)} 

=  2r]{A+B  +  C-{-...), 

and  (f)(z+2oo')-(f>{z)^2v{A  +  B  +  C+...), 

so  that,  subject  to  the  condition 

A+B  +  C+...  =  0, 

(f)(z)  is  a  doubly-periodic  function. 

Again,  we  know  that,  within  the  fundamental  parallelogram,  ^  has  a 
single  irreducible  infinity  and  that  the  infinity  is  simple ;  hence  the  irre- 
ducible infinities  of  the  function  (f>  (z)  are  z  =  a,  h,  c,  ...,  and  each  is  a  simple 
infinity.  The  condition  A  -\-  B  +  G  +  ...  =  Q  is  merely  the  condition  of 
Prop.  III.,  §  116,  that  the  'integral  residue'  of  the  function  is  zero. 

Conversely,  a  doubly-periodic  function  with  m  assigned  infinities  can  be 
expressed  in  terms  of  ^  and  its  derivatives.  Let  aj  be  an  irreducible  infinity 
of  $  of  degree  n,  and  suppose  that  the  fractional  part  of  $  for  expansion  in 
the  immediate  vicinity  of  a-^  is 

A^  B,  K, 

— 1 — '+      -\ — . 

z  —  a-i^     {z  —  a^Y     '"     (z  —  a-^Y 


302  PSEUDO-PERIODIC  [128. 

Then 


A,U^-ci^)-B,^'(z-a,)  +  ^^r(z-a,)-... 


is  not  infinite  for  z  =  o.i. 

Proceeding  similarly  for  each  of  the  irreducible   infinities,  we  have  a 
function 


^  (^)  -  S 


Ar^{z-  ar)  -  5,r  {Z  -  «r)  +  ^  K"  (^  "  «-•)  " 


which  is  not  infinite  for  any  of  the  points  z  —  a-^,  a^,  ....  But  because  <I>  {z) 
is  doubly-periodic,  we  have 

^1  +  ^0+ ... +^,,  =  0, 
and  therefore  the  function 

m 

S    Ar^{z  —  ar) 
r=l 

is  doubly-periodic.  Moreover,  all  the  derivatives  of  any  order  of  each  of  the 
functions  ^  are  doubly-periodic ;  hence  the  foregoing  function  is  doubly- 
periodic. 

The  function  has  been  shewn  to  be  not  infinite  at  the  points  ai ,  a^,  ... , 
and  therefore  it  has  no  infinities  in  the  fundamental  parallelogram ;  con- 
sequently, being  doubly-periodic,  it  has  no  infinities  in  the  plane  and  it  is 
a  constant,  say  G.     Hence  we  have 

m 

with  the  condition   ^   Ar  =  0,  which  is  satisfied  because  4>  (z)  is  doubly- 

r=l 

periodic. 

This  is  the  required  expression*  for  <i>  (z)  in  terms  of  the  function  ^and 
its  derivatives ;  it  is  evidently  of  especial  importance  when  the  indefinite 
integral  of  a  doubly-periodic  function  is  required. 

129.  Constants  v  and  tj',  connected  with  co  and  &)',  have  been  introduced 
by  the  pseudo-periodicity  of  ^(z);  the  relation,  contained  in  the  following 
proposition,  is  necessary  and  useful : — 

The  constants  rj,  rj',  co,  w  are  connected  hy  the  relation 

'r}(ii'  —  7]'(0  =  +  ^TTl, 

the  +  or  —  sign  being  taken  according  as  the  real  pa7-t  of  w' /coi  is  positive  or 
negative. 

*  See  Hermite,  Ann.  de  Toulouse,  t.  ii,  (1888),  C,  pp.  1 — 12. 


129.] 


FUNCTIONS 


303 


A  fundamental  parallelogram  having  an  angular  point  at  z^  is  either  of 

f(o'\ 


the  form  (i)  in  fig.  34,  in  which  case  9t  (  — .  1  is 


positive ;  or  of  the  form  (ii),  in  which  case  9{ 


03lj 


2n  +  2£o', 


is  negative.  Evidently  a  description  of  the  paral- 
lelogram ABGD  in  (i)  will  give  for  an  integral  the 
same  result  (but  with  an  opposite  sign)  as  a  de- 
scription of  the  parallelogram  in  (ii)  for  the  same 
integral  in  the  direction  ABGD  in  that  figure. 

We  choose  the  fundamental  parallelogram,  so 
that  it  may  contain  the  origin  in  the  included 
area.  The  origin  is  the  only  infinity  of  ^  which 
can  lie  within  the  area :  along  the  boundary  ^  is 
always  finite. 

Now  since 

^(z  +  2co')-^{z)  =  2v', 
the  integral  of  ^(z)  round  ABGD  in  (i),  fig.  34,  is  (§  116,  Prop.  II.,  Cor.) 


B/?o  +  2co 


Fig.   34, 


27]dz 


2'n'dz, 


the  integrals  being  along  the  lines  AD  and  AB  respectively,  that  is,  the 
integral  is 

4  {^TjO)'  -  1]  w). 

But  as  the  origin  is  the  only  infinity  within  the  parallelogram,  the  path  of 
integration  ABGD  A  can  be  deformed  so  as  to  be  merely  a  small  curve  round 
the  origin.     In  the  vicinity  of  the  origin,  we  have 


r(^)  = 


\92Z' 


1 


^gz^  - 


and  therefore,  as  the  integrals  of  all  terms  except  the  first  vanish  when  taken 
round  this  curve,  we  have 

=  27ri. 


Hence 

and  therefore 


4  (?;&)'  —  •?/'&))  =  2Tri, 
7]w'  —  r]'(o    =  ^iri. 


This  is  the  result  as  derived  from  (i),  fig.  34,  that  is,  when  Ot  ( —  )  is  positive. 

When  (ii),  fig.  34,  is  taken  account  of,  the  result  is  the  same  except 
that,    when   the   circuit   passes   from   z^   to   ^o  +  2&),  then  to   ^(,  +  2&) -h  2&)', 


304  PSEUDO-PERIODICITY   OF   WEIERSTRASS'S  [129. 

then  to  Zq  +  Iw  and  then  to  z^,  it  passes  in  the  negative  direction  round  the 
parallelogram.  The  value  of  the  integral  along  the  path  ABCDA  is  the 
same  as  before,  viz.,  4  (770)'  —  //'&>) ;  when  the  path  is  deformed  into  a  small 

[dz 
curve  round  the  origin,  the  value  of  the  integral  is  I  —  taken  negatively,  and 

therefore  it  is  —  l-iri :  hence 

7]ai'  —  rj'o)  =  —  ^Tri. 

Combining  the  results,  we  have 

?;&)'  —  77'ft)  =  +  ^TTl, 

according  as  9i  (— .)  is  positive  or  negative. 

Corollary.     If  there  be  a  change  to  any  other  fundamental  parallelo- 
gram, determined  by  211  and  2Q',  where 

n  =pQ)  +  qco',         D'  =p'(o  -f  qo)', 

p,  q,  p,  q'  being  integers  such  that  pq'  —p'q  =  ±  1,  and  if  H,  H'  denote  ^(fi), 
^(n'),  then 

H=pr]  +  qr]',         H'  =  prj  +  q'7)' ; 

therefore  HO,'  -  H'O  =  ±  ^iri, 

M'  . 
according  as  the  real  part  of  -7^  is  positive  or  negative. 

130.     It  has  been   seen   that   ^{z)  is  pseudo-periodic;    there  is  also  a 
pseudo-periodicity  for  a{z),  but  of  a  different  kind.     We  have 

^(^-h2ft))=^(^)+277,   ^ 

that  is,  — -, — ,  o    X  —  — T\  +  ^'7' 

'  a  {z  -\-  2&))      <j  {z) 

and  therefore  a  {z  +  2fo)  =  ^e*''^  a  {z), 

where  J.  is  a  constant.     To  determine  A,  we  make  z  =  -  w,  which  is  not  a 

zero  or  an  infinity  of  a  {z) ;  then,  since  cr  {z)  is  an  odd  function,  we  have 

—  Ae-"''^'^  =  l, 
so  that  a{z  +  2w)  =  -  e'-''  '^+'"'  a  {z). 

Hence  a{z  +  ^co)  =  -  e--^  (^+^'-»  a  (^  +  2to) 

=  e--^  <22+'""'  o-  {z) ; 
and  similarly  a  {z  +  2mco)  =  (-  1)»*  e'^  imz+m^<.)  ^  (^^l 

Proceeding  in  the  same  way  from 

^{z  +  2(o')=^^(z)+2^', 
we  find  a{z  +  2m'«')  =  (-  I)'"'  e""^'  "«'^+™'="')  a  (z). 

Then      a  {z  +  2m(o  +  2m  co')  =  (-  1  )"^ e'^  (»«z+m^'<o+2,«m'.o)  ^.  (^  ^  2)}ico') 

_.  / T\mH-?«,'  g2Z  (mT)+?n.'r|') +2T)m2(o+4r)mm'a)'+27)m'2o)'  q- ( 2;\ 

_  /         -J^Nm+w'  g2()?ir)  +  m'T)')(Z+7«w+m'w')  +  2m7n'(7;aj'— 7)'(o)   g- Z^") 


130.]  PRODUCT-FUNCTION  305 

But  77ft)'  —  7]  (ji  =  +  -^TTl, 

SO  that  g2mrn'(i)<o'-,,'<o)  =  g±m7n'«  ^  ^_  l^mm-  . 

and  therefore 

O-  (^  +  2mft)  +  2??i'&)')  =  (-  lynm'+m+m'  g2(m,,+m'V)  (z+mco+m'o)')  q.  ^^-^^ 

which  is  the  law  of  change  of  a-  {z)  for  increase  of  z  by  integral  multiples  of 
the  periods. 

Evidently  a  {z)  is  not  a  periodic  function,  a  result  confirmatory  of  the 
statement  in  §  122.  But  there  is  a  pseudo-periodicity  the  characteristic  of 
which  is  the  reproduction,  for  an  added  period,  of  the  function  with  an 
exponential  factor,  the  index  being  linear  in  the  variable.  This  is  another 
of  the  forms  of  pseudo-periodicity  which  will  be  considered  in  the  next 
chapter. 

131.  But  though  a  {z)  is  not  periodic,  we  can  by  its  means  construct 
functions  which  are  periodic  in  the  pseudo-periods  of  a  {z). 

By  the  result  in  the  last  section,  we  have 

^^ L  =    i I   /,2(»M7)+m'j)')0— a)   . 

a(z-^  +  2ma)  +  2m'co')      a{z-^) 
and  therefore,  if  ^  (z)  denote 

(7  {Z  -  a^)  or  {Z  -  a^ (T{z-OLn) 

a{z-^,)a{z-^.^ a-(z-0n)' 

then  (t>(^  +  ^mco  +  2,mco')  =  e^-(mr,+mW)  (s^^-sa,.)  ^  ^^-^^ 

so  that  <f)  (z)  is  doubly-periodic  in  2&)  and  2&)'  provided 

t/3r  -  tfXr  =  0. 

Now  the  zeros  of  (p  (z),  regarded  as  a  product  of  cr-functions,  are  cfj,  a^,  ...,  a^ 
and  the  points  homologous  with  them;  and  the  infinities  are  ySj,  /Sa,  ...,  ^^ 
and  the  points  homologous  with  them.  It  may  happen  that  not  all  the  points 
a  and  /3  are  in  the  parallelogram  of  reference ;  if  the  irreducible  points 
homologous  with  them  be  a^,  ...,  an  and  61,  ...,  6„,  then 

2a,.  =  Ibr  (mod.  2(o,  2&>'), 

and  the  new  points  are  the  irreducible  zeros  and  the  irreducible  infinities  of 
(f)  (z).     This  result,  w^e  know  from  Prop.  III.,  §  116,  must  be  satisfied. 

It  is  naturally  assumed  that  no  one  of  the  points  a  is  the  same  as,  or  is 
homologous  with,  any  one  of  the  points  l3 :  the  order  of  the  doubly-periodic 
function  would  otherwise  be  diminished  by  1. 

If  any  a  be  repeated,  then  that  point  is  a  repeated  zero  of  (p  (z);  similarly 
if  any  /3  be  repeated,  then  that  point  is  a  repeated  infinity  of  cf>(z).  In 
every  case,  the  sum  of  the  irreducible  zeros  must  be  congruent  with  the  sura 
of  the  irreducible  infinities  in  order  that  the  above  expression  for  (f)  (2)  may 
be  doubly-periodic. 

F.  F.  20 


306  DOUBLY-PERIODIC    FUNCTIONS  [131. 

Conversely,  if  a  doubly-periodic  function  (b(z)  be  required  with  m  assigned 

irreducible  zeros  a  and  m  assigned  irreducible  infinities  b,  which  are  subject 

to  the  congruence 

2a  =  26  (mod.  2(w,  2&)'), 

we  first  find  points  a  and  l3  homologous  with  a  and  with  b  respectively  such 

that 

la  =  1/3. 

(t{z—  ai) a  (z  —  a„i) 

Then  the  function  — t- — w\ —jz — ^-^ 

o-{z  —  (3i) a{z-  /dm) 

has  the  same  zeros  and  the  same  infinities  as  ^  (z),  and  is  homoperiodic  with 
it;  and  therefore,  by  §  116,  IV., 

"^^^"^(^-A) ^(^-^j: 

where  A  is  &  quantity  independent  of  z. 

Ex.  1.  Consider  g>'  {£).  It  has  the  origin  for  an  infinity  of  the  third  degree  and  all  the 
remaining  infinities  are  reducible  to  the  origin ;  and  its  three  irreducible  zeros  are  w,  co',  m". 
Moreover,  since  a"  =  (•>'  + a,  we  have  co  +  co'  +  w"  congruent  with  but  not  equal  to  zero. 
We  therefore  choose  other  points  so  that  the  sum  of  the  zeros  may  be  actually  the  same 
as  the  sum  of  the  infinities,  which  is  zero  ;   the  simplest  choice  is  to  take  m,  a',  —  co". 

Hence 

(T  {z-  co)  (x  (z  -  co')  a-  (z  +  o)") 
^('^  =  '^ ^3-(i) ' 

where  4  is  a  constant.     To  determine  A,  consider  the  expansions  in  the   immediate 
vicinity  of  the  origin  ;   then 

2  a- {-co)  cr{-co')cT{(o") 

Z^  Z^ 

cr  {z  —  co)  (7  (z  —  co')  (T  {z  +  co") 


SO  that  ^'(2)=~2  /    \     /    '\     t    "\    ■6/~\      • 

°  cr  {co)  cr  {(o  )  CT  {co  )  a"^  {zj 

Another  method  of  arranging  zeros,  so  that  their  sum  is  equal  to  that  of  the  infinities, 
is  to  take  -co,  -  co',  co"  ;  and  then  we  should  find 

^a{z  +  co)cr{z  +  co')cr{z-co") 
r^')-^      a{co)a{co')cT{co")a^{z)      ' 

This  result  can,  however,  be  deduced  from  the  preceding  form  merely  by  changing  the 
sign  of  z. 

Ex.  2.     Consider  the  function 

cT{u  +  v)a{u-v) 
^  ^Hu)  ' 

where  v  is  any  quantity  and  A  is  independent  of  ?«.  It  is,  qua  function  of  u,  doubly- 
periodic  ;  and  it  has  u=0  as  an  infinity  of  the  second  degree,  all  the  infinities  being 
homologous  with  the  origin.  Hence  the  function  is  homoperiodic  with  f  {u)  and  it  has 
the  same  infinities  as  ^  {u) :   thus  the  two  are  equivalent,  so  that 

^<ri^^+l)^(^^-')^B^{u)-C, 
a-{u) 


131.]  EXAMPLES  307 

where  B  and  C  are   independent  of  u.     The   left-hand   side  vanishes  if  u  =  v;    hence 
C=B^{v),  and  therefore 

where   A'   is   a   new   quantity   independent   of  u.      To   determine    4'  we   consider  the 
expansions  in  the  vicinity  of  u=0  ;   we  have 


A' a-  iv)a{- 

-^), 

_1 

+ 

u^ 

so  that 

-  A'<T^ 

(iO=i, 

and  therefore 

(t{u  +  v)(t 
0-2  (u)  0- 

(■w-v) 

=  ^iv)- 

-g>w> 

a  formula  of 

very 

great 

importance. 

£x.  3.     Taking  logarithmic  derivatives  with  regard  to  «  of  the  two  sides  of  the  last 
equation,  we  have 

f("+''>-^f(''-'')-^f<«'->w-^(»)' 

and,  similarly,  taking  them  with  regard  to  v,  we  have 


^'hence  C(»  +  i^)  -  t(»)-C(^0  =  i  ^!S~  ^' ^ , 

giving  the  special  value  of  the  left-hand  side  as  (§  128)  a  doubly-periodic  function.  It  is 
also  the  addition-theorem,  so  far  as  there  is  an  addition-theorem,  for  the  {"-function. 

Ex.  4.     We  can,  by  diflferentiation,  at  once  deduce  the  addition-theorem  for  ^{u  +  v). 
Evidently 

which  is  only  one  of  many  forms :  one  of  the  most  useful  is 

which  can  be  deduced  from  the  preceding  form. 

The  result  can  be  used  to  modify  the  expression  for  a  general  doubly -periodic  function 
^(z)  obtained  in  §  128.     We  have 

2ArC  {z  -  a,)  =^2^^,  |f  (.)  -  C  («.)  -  2  ^  («,,,)  _^(,)  I 

Each  derivative  of  f  can  be  expressed  either  as  a  polynomial  function  of  p  {z  —  a,.)  or  as 
the  product  of  ^'{z-aj.)  by  such  a  function;  and  by  the  use  of  the  addition-theorem, 
these  can  be  expressed  in  the  form 

M+JVjJ'iz) 
Z         ' 

20—2 


308  EXAMPLES  [131. 

where  Z,  J/,  iV  are  rational  integral  functions  of.  ^  (2).  Hence  the  function  $  {z)  can  be 
expressed  in  the  same  form.  The  simplest  case  arises  when  all  its  infinities  are  simple, 
and  then 


$(2)=C'+  2   ArCiz-dr) 


=  C-  2  J,.f(a,.)  +  i  2  A 


r(^)+rK) 


_„,  1  ™  .  r(^)+rK) 

~^  +  ^=i    '■g>(^)-P(a.)' 

m 

with  the  condition  2  ^4^  =  0. 

r=l 

Ex.  5.  The  function  ^  {z)  -  e^  is  an  even  function,  doubly-periodic  in  2<b  and  2o)'  and 
having  2=0  for  an  infinity  of  the  second  degree  ;  it  has  only  a  single  infinity  of  the  second 
degree  in  a  fundamental  parallelogram. 

Again,  2  =  0)  is  a  zero  of  the  function;  and,  since  ^'(a))  =  0  but  ^"  (w)  is  not  zero, 
2=0)  is  a  double  zero  of  ^{z)  —  e-^.  All  the  zeros  are  therefore  reducible  to2  =  (B  ;  and  the 
function  has  only  a  single  zero  of  the  second  degree  in  a  fundamental  parallelogram. 

Taking  then  the  parallelogram  of  reference  so  as  to  include  the  points  2=0  and  z  —  a>, 
we  have 

where  Q  (2)  has  no  zero  and  no  infinity  for  points  within  the  parallelogram. 

Again,  for  ^(2  +  0))  — ei,  the  irreducible  zero  of  the  second  degree  within  the  parallelo- 
gram is  given  by  2  +  co  =  &),  that  is,  it  is  2=0;  and  the  irreducible  infinity  of  the  second 
degree  within  the  parallelogram  is  given  by  2-|-(»  =  0,  that  is,  it  is  2  =  0).     Hence  we  have 

^(2  +  co)-e,  =  ^-^§i(2), 
{Z-ca) 

where  Qi  (2)  has  no  zero  and  no  infinity  for  points  within  the  parallelogram. 
Hence  {^  (2)  -  e,}  {^  (^  +  «)  -  e^}  =  $  (2)  Q,  (2)  ; 

that  is,  the  function  on  the  left-hand  side  has  no  zero  and  no  infinity  for  points  within  the 
parallelogram  of  reference.  Being  doubly-periodic,  it  therefore  has  no  zero  and  no  infinity 
anywhere  in  the  plane ;  it  consequently  is  a  constant,  which  is  the  value  for  any  point. 
Taking  the  special  value  2  =  a>',  we  have  ^(co')  =  e3,  and  g>(co'-|-a))  =  e2 ;   ^ncl  therefore 

'         {^(^)-ei}{^^(^  +  a>  )-ei}  =  (63-61)  (62-^1). 

Similarly  [<^o  {z)-e<i}  {f  {z  +  a>")- e.^  =  {e^-e^){e^-e.\ 

and  /^  (s)  _  eg}  |g>  (2  +  ^' )  _  ^3}  =  (e^  _  e^)  (e^  -  e^). 

It  is  possible  to  derive  at  once  from  these  equations  the  values  of  the  ^i>-function  for 
the  quarter-periods. 

Note.  In  the  preceding  chapter  some  theorems  were  given  which  indicated  that 
functions,  which  are  doubly-periodic  in  the  same  periods,  can  be  expressed  in  terms  of 
one  another :  in  particular  cases,  care  has  occasionally  to  be  exercised  to  be  certain  that 
the  periods  of  the  functions  are  the  same,  especially  when  transformations  of  the  variables 
are  eflfected.  For  instance,  since  g>  (2)  has  the  origin  for  an  infinity  and  snu  has  it  for  a 
zero,  it  is  natural  to  express  the  one  in  terms  of  the  other.  Now  g)  (2)  is  an  even  function, 
and  sn  u  is  an  odd  function ;  hence  the  relation  to  be  obtained  will  be  expected  to  be 
one  between  g>  (2)  and  sn^  u.     But  one  of  the  periods  of  sn^ «  is  only  one-half  of  the 


131.]  OF   DOUBLY-PERIODIC    FUNCTIONS  .  309 

corresponding  period  of  anu;  and  so  the  period-parallelogram  is  changed.     The  actual 
relation*  is 

where  7i  =  (ei  —  espz  and  k^={e2  — 63)1(61  — es). 

Again,  with  the  ordinary  notation  of  Jacobian  elliptic  functions,  the  periods  of  snz 
are  45"  and  2iK',  those  of  dnz  are  2K  and  4iA'',  and  those  of  en  2  are  4:K  and  2K+2iK'. 
The  squares  of  these  three  functions  are  homoperiodic  in  2K  and  2iK' ;  they  are  each 
of  the  second  order,  and  they  have  the  same  infinities.  Hence  sn^  z,  cn^  z,  dn^  z  are 
equivalent  to  one  another  (§  116,  V.). 

But  such  cases  belong  to  the  detailed  development  of  the  theory  of  particular  classes 
of  functions,  rather  than  to  what  are  merely  illustrations  of  the  general  propositions. 

Ex.  6.     Prove  that 

,   .__(r{u  +  Ui)(T(w  +  U2)  (r(u  +  Us)(T{u  +  Ui) 
^  a-[2u+^  (%  +  U2  +  U3  +  %4)] 

is  a  doubly-periodic  function  of  u,  such  that,  with  the  ordinary  notation, 

g(u)+g{7i  +  co)+ff  {u  +  co')+g  {u  +  o)  +  a') 

=  -2a(^ 2 j"V ^ hi ^ 

Prove  further  that,  if  S  denote  the  substitution 

i(-i   111), 

1-111 

11-11 

111-1 

and  {Ui,   U2,   Us,   U^  =  S(ui,  u^,  M3,  M4)  and  0{ii)  denote  what  g  (u)  becomes  when, 
therein,   U^,  U^,   Uz,   Ui  are  written  for  Ui,  %,  u^,  u^  respectively,  then  also 

(-6^(w),   -G{n  +  (o\   -G{u  +  ci'),   -G{u  +  o)  +  o}')) 

=S(g{zi),  giu  +  m),  g(ii  +  J),  g  {u-ira>+co')). 

(Math.  Trip.,  Part  II.,  1893.) 

Ex.  7.  All  the  zeros  of  a  function,  doubly -periodic  in  the  periods  of  U>(s),  are  simple 
and  are  given  by  pec  +  qa,  where  p  and  q  are  integers  such  that  ^  -f  g'  is  odd ;  all  its 
infinities  are  simple  and  are  given  by  pa  +  qa,  where  p  and  q  are  integers  such  that 
p  +  q  is,  even.     Shew  that  the  function  is  a  constant  multiple  of 

f'/^^    ;  (Trinity  Fellowship,  1896.) 

iHz)-e2  "■  ^ 

Ex.  8.     Construct  the  diflferential  equation  of  the  first  order,  satisfied  by 

az-a)-C{z-h). 

(Trinity  Fellowship,  1899.) 

132.  As  a  last  illustration  giving  properties  of  the  functions  just  con- 
sidered, the  derivatives  of  an  elliptic  function  with  regard  to  the  periods 
will  be  obtained. 

Let  (^{z)  be  any  function,  doubly-periodic  in  2ft)  and  2ft)'  so  that 
(f)(2;  +  2mft)  +  27110)')  =  (f>  (z). 
*  Halphen,  Fonctions  Elliptiques,  t.  i,  pp.  23—25. 


310  .  PERIOD-DERIVATIVES  [132. 

The  coefficients   in    </>    implicitly  involve    eo    and    w .     Let    ^i,   (^2,   and   ^ 
respectively  denote  dcp/do),  d(f)jdw',  dcfi/dz;  then 

(^1  {2  +  %nw  +  2riiw')  +  2m<l>  (z  +  2mco  +  2m  co')  =  (f)^  (2), 
(f)o  {2  +  2moi  +  27h'g)')  +  277i'(^'  (^  +  ^mw  +  2?>i'q)')  =  <^2  {z), 

(j)'  (2  +  2m(o  +  2m  Qi')  =  4/  {2). 
Multiplying  by  w,  on',  2  respectively  and  adding,  we  have 
ftx^i  {2  +  2mw  +  2m  w)  +  w'^s  (-^  +  2m&)  +  2m  co') 

+  (2^  +  2??-i&)  +  2??i'&)')  0'  (2  +  27?i&)  +  2m' 0)') 
=  co^i  (^)  +  &)'^2  {z)  +  z(\i'  {2). 
Hence,  if  f  {z)  =  wgbi  {2)  +  w'^a  (^)  +  ^s^'  {z), 

then  f  (2)  is  a  function  doubly-periodic  in  the  periods  of  cf). 

Again,  multiplying  by  77,  tj',  ^{2),  adding,  and  remembering  that 
^(2  +  2mco  +  2ni(o')  =  ?(^)  +  2mr)  +  2m  7]', 
we  have 

77^1  (2  +  2mo)  +  2m' w)  +  7?'<^2  {z  +  2/7i&)  +  2m' a>') 

+  ^(^  +  277ZG)  +  2m  w)  (f)'  {2  +  2mco  +  2m V) 

=  r,cf,,(2)  +  v'cl>,(2)  +  U2)cf>'(2). 

Hence,  i/  g{2)  =  v4>,  (z)  +  v'^2  (z)  +  ?  (^)  0'  (z), 

then  g(2)  is  a  function  doubly -periodic  in  the  periods  of  cj). 

In  what  precedes,  the  function  ^  (2)  is  any  function,  doubly-periodic  in 
2(o,  2(1)' ;  one  simple  and  useful  case  occurs  when  0(^)  is  taken  to  be  the 
function  i^j{z).     Now 

and  ^(')=]-io^"-''-m^-^''-MOO^'''''-' 

hence,  in  the  vicinity  of  the  origin,  we  have 

ft)  -^  -t-  ft)'  ^,  +  2^  tJ-  = ;  +  even  integral  powers  of  2^ 

do)  9ft)         02         2' 

since  both  functions  are  doubly-periodic  and  the  terms  independent  of  z 
vanish  for  both  functions.  It  is  easy  to  see  that  this  equation  merely 
expresses  the  fact  that  <^,  which  is  equal  to 

is  homogeneous  of  degree  —  2  in  2^,  «,  w  . 


132.]  OF   AVEIERSTRASS'S   FUNCTION  311 

Similarly 

V  ^  +  v'  J^'  +  ^  i^)  4-  "^  ~  —^  +  T^  9^  +  ^^'^^  integral  powers  of  z. 
'  dco         dw  oz         z*      lo'^ 

But,  in  the  vicinity  of  the  origin, 

92/^      6       1 

-e^  =  — f-  T7^  f/o  +  even  integral  powers  of  z, 

dz-      z'     10^'  6       ^ 

so  that 

V^  +  V  ^,  +  ^(^)^  +  o^2  =  a9'^  +  even  integral  powers  of  z. 
dco         d(o  dz      S  dz^      b  "^ 

The  function  on  the  left-hand  side  is  doubly-periodic :  it  has  no  infinity 
at  the  origin  and  therefore  none  in  the  fundamental  parallelogram ;  it  there- 
fore has  no  infinities  in  the  plane.  It  is  thus  constant  and  equal  to  its  value 
anywhere,  say  at  the  origin.     This  value  is  Igz,  and  therefore 

"^dco^^  d<o'^^^  hz      sdz'^a^' 

This  equation,  when  combined  luith 

dco  00)  dz 

,         ,         .  did        ,    9|» 
qives  the  vatue  of  ^  and  ^~-, , 
^  ''003  da> 

The  equations  are  identically  satisfied.  Equating  the  coefficients  of  z'^ 
in  the  expansions,  which  are  valid  in  the  vicinity  of  the  origin,  we  have 

OO)  00) 

and  equating  the  coefficients  of  z*  in  the  same  expansions,  we  have 

dft)  00) 

da.        ,  8f/3         1 
00)  dco  o 

Hence  for  any  function  u,  which  involves  w  and  w  and  therefore  implicitly 
involves  g<2,  and  g^-,  we  have 

9it        ,  9u         /,     'bii  .  ^     9tt^ 


di 


ft)  9co  V       95^2  ogj 

du        ,  du         1  /,  „     dii      2       du 


312  EVEN  [132. 

Since  ^  is  such  a  function,  we  have 


being  the  equations  which  determine  the  derivatives  of  p  with  regard  to  the 
invariants  g^  and  g^. 

The  latter  equation,  integrated  twice,  leads  to 

^"(^     -I «     9o"      2     „  9o-       1        „        „ 

a  differential  equation*  satisfied  by  cr{z). 

133.  The  foregoing  investigations  give  some  of  the  properties  of  doubly- 
periodic  functions  of  the  second  order,  whether  they  be  uneven  and  have  two 
simple  irreducible  infinities,  or  even  and  have  one  double  irreducible  infinity, 

If  a  function  TJ  of  the  second  order  have  a  repeated  infinity  at  2^  =  7,  then 
it  is  determined  by  an  equation  of  the  form 

V'"'  =  ^a''[{U  -\){U  -  iJi){V  -v)], 
or,  taking  U—^{\-\-^-\-v)  =  Q,  the  equation  is 

Q'-'  =  4a^  [(Q  -  e,)  (Q  -  e,)  {Q  -  e,)], 
where  ej  +  62  +  ^3  =  0-     Taking  account  of  the  infinities,  we  have 

Q  =  iO  {az  -  ay) ; 
and  therefore      U'—^(\  +  /jb-^v)  =  ^ {az  —  ay) 

by  Ex.  4,  p.  308.  The  right-hand  side  cannot  be  an  odd  function;  hence 
an  odd  function  of  the  second  order  cannot  have  a  repeated  infinity.  Similarly, 
by  taking  reciprocals  of  the  functions,  it  follows  that  an  odd  function  of  the 
second  order  cannot  have  a  repeated  zero. 

It  thus  appears  that  the  investigations  in  §§  120, 121  are  sufficient  for  the 
included  range  of  properties  of  odd  functions.  We  now  proceed  to  obtain 
the  general  equations  of  even  functions.  Every  such  function  can  (by  §  118, 
XIII.,  Cor.  1)  be  expressed  in  the  form  \ai^{z)  +  h]^{cip{z)-{-d],  and  its 
equations  could  thence  be  deduced  from  those  of  ^  {z) ;  but,  partly  for 
uniformity,  we  shall  adopt  the  same  method  as  in  §  120  for  odd  functions. 
And,  as  already  stated  (p.  286),  the  separate  class  of  functions  of  the  second 
order  that  are  neither  even  nor  odd,  will  not  be  discussed. 

*  For  this  and  other  deductions  from  these  equations,  see  Frobenius  und  Stiekelberger, 
Grelle,  t.  xcii,  (1882),  pp.  311—327 ;  Halphen,  Traite  des  fonctions  elliptiques,  t.  i,  (1886), 
chap.  IX. ;   and  a  memoir  by  the  author,  quoted  on  p.  299,  note. 


134.]  DOUBLY-PERIODIC   FUNCTIONS  313 

134.  Let,  then,  (f)  (2)  denote  an  even  doubly-periodic  function  of  the 
second  order  (it  may  be  either  of  the  first  class  or  of  the  second  class)  and  let 
2&),  2(o'  be  its  periods ;  and  denote  2&)  +  2co'  by  2&)".     Then 

c}>iz)  =  cf>(-z), 

since  the  function  is  even ;  and  since 

(f)  (co  +  z)  =  (f)  (—  (o  —  z) 

=  (^  (2&)  —  0}  —  z) 

=  (f)((0-z), 

it   follows   that   cf)  (o)  +  z)    is    an    even    function.     Similarly,    (f)  (co'  +  z)   and 
<j>  (ft)"  +  z)  are  even  functions. 

Now  ^  (o)  4-  z),  an  even  function,  has  two  irreducible  infinities,  and  is 
periodic  in  2ft),  2&)' ;  also  <^  {z),  an  even  function,  has  two  irreducible  infinities 
and  is  periodic  in  2ft>,  2ft)'.  There  is  therefore  a  relation  between  <^  {z)  and 
<\>{(>)  +  z),  which,  by  §  118,  Prop.  XIII.,  Cor.  1,  is  of  the  first  degree  in  4>{z) 
and  of  the  first  degree  in  ^  (ft)  -f-  2^) ;  thus  it  must  be  included  in 

5(/)  {z)  4>{co  +  z)-  C(t>  (z)  -C'(f)(co+z)  +  A=  0. 

But  <f)  (z)  is  periodic  in  2ft) ;  hence,  on  writing  z  +  co  for  z  in  the  equation,  it 
becomes 

Bcji  {a}  +  z)cf>  (z)  -  Ccf)  (ft)  +  z)-C'^{z)  +  A=0; 

thus  C=C'. 

If  B  be  zero,  then  C  may  not  be  zero,  for  the  relation  cannot  become 
evanescent :  it  is  of  the  form 

<j>{z)  +  ct>{co+z)  =  A' (1). 

If  B  be  not  zero,  then  the  relation  is 

'f'^''  +  '^  =  B4>(z)-C ^^^- 

Treating  (f)  (co'  +  z)  in  the  same  way,  we  find  that  the  relation  between  it 
and  <f)  (z)  is 

F(j)  (z)  (j)  (co'  +  z)-Dcf)  (z)  -  D(f>  (co'  +  z)  +  E  =  0, 

so  that,  if  F  be  zero,  the  relation  is  of  the  form 

(f>(z)  +  (i>{co'  +  z)  =  E' (ly, 

and,  if  F  be  not  zero,  the  relation  is  of  the  form 

D<^{z)-E 
^^''^'^=F^{z)-D ^^>- 

Four  cases  thus  arise,  viz.,  the  coexistence  of  (1)  with  (1)',  of  (1)  with  (2)', 
of  (2)  with  (1)',  and  of  (2)  with  (2)'.     These  will  be  taken  in  order. 


314  EVEN  [134. 

I. :  the  coexistence  of  (1)  with  (1)'.     From  (1)  we  have 

(f)((0'  +  Z)  +  (J3{0)"  +  Z)=A', 

so  that  (])  (z)  +  (f)  {(o  +  z)  +  (}>  (co' +  z)  +  (f)  (&)"  +  z)  =  2A'. 

Similarly,  from  (1)',  ■ 

ct)(z)  +  (t)((o'  +  z)  +  (f)(w  +  z)+  cf)  (a)"  +z)  =  2E'; 

so  that  A'  =  E',  and  then 

^(&)  +  ^)  =  ^(&)'  +  2r), 

whence  to  ~  w'  is  a  period,  contrary  to  the  initial  hypothesis  that  2w  and  2(u 
determine  a  fundamental  parallelogram.     Hence  equations  (1)  and  (1)'  cannot 
coexist. 

II. :  the  coexistence  of  (1)  with  (2)'.     From  (1)  we  have 

<^{(o" -\-z)  =  A' -<^{oi' +  z) 

_  {A'F  -B)cf)  (z)  -(A'D-  E) 
F<f>(z)-I) 

on  substitution  from  (2)'.     From  (2)'  we  have 

(bid)    +  Z)=  r^  .   . r R 

_  (A'D  -E)-D(\>  {z) 
~  A'F-D-F<\>{z)  ' 

on  substitution  from  (1).     The  two  values  of  cj)  (m"  +  z)  must  be  the  same, 

whence 

A'F-D  =  D, 

which  relation  establishes  the  periodicity  of  (f>  (z)  in  2&)",  when  it  is  considered 

as  given  by  either  of  the  two  expressions  which  have  been  obtained.     We 

thus  have 

A'F=2D; 

and  then,  by  (1),  we  have 

(i>{z)-j  +  (f>((o  +  z)-j  =  0; 
and,  by  (2)',  we  have 

If  a  new  even  function  be  introduced,  doubly-periodic  in  the  same  periods 
having  the  same  infinities  and  defined  by  the  equation 

<^i  (^)  =  0  (^)  -  J  . 

the  equations  satisfied  by  0i  (z)  are 

01  (a)  +  z)  +  </>!  {z)  =  0  I 

01  (&)'  +  z)  01  (z)  =  constant]  ' 


134.]  DOUBLY-PERIODIC    FUNCTIONS  315 

To  the  detailed  properties  of  such  functions  we  shall  return  later ;  meanwhile 
it  may  be  noticed  that  these  equations  are,  in  form,  the  same  as  those  satisfied 
by  an  odd  function  of  the  second  order. 

III. :  the  coexistence  of  (2)  with  (1/.  This  case  is  similar  to  II.,  with  the 
result  that,  if  an  even  function  be  introduced,  doubly-periodic  in  the  same 
periods  having  the  same  infinities  and  defined  by  the  equation 

G 

the  equations  satisfied  by  ^2  (z)  are 

fji,(a>'  +  2)  +  <}>,(z)  =  0  [_ 

(^2  («  +  ^)  ^2  (^)  =  constant)  ' 
It  is.  in  fact,  merely  the  previous  case  with  the  periods  interchanged. 

IV. :  the  coexistence  of  (2)  with  (2)'.     From  (2)  we  have 
C(f,{co'  +  z)-A 

_(CD  -  AF)  6  (z)  -  (CE  -  AD) 
~~  (BD  -  CF)  (t>  (z)  -  (BE -CD) ' 

on  substitution  from  (2)'.     Similarly  from  (2)',  after  substitution  fi:om  (2),  we 

have 

„  (GD-BE)<t>iz)  +  {CE-AD) 

^^"   +  ^'^  -  (CF-BD)  4>  {z)  +  {CD  -  AF)' 
The  two  values  must  be  the  same ;  hence 

CD-AF  =  -(CD-BE), 
which  indeed  is  the  condition  that  each  of  the  expressions  for  (f)(o)"  +  z) 
should  give  a  function  periodic  in  2(o".     Thus 

AF  +  BE=2CD. 
One  sub-case  may  be  at  once  considered  and  removed,  viz.  if  C  and  D 
vanish  together.     Then  since,  by  the  hypothesis  of  the  existence  of  (2)  and 
of  (2)',  neither  B  nor  F  vanishes,  we  have 

A__F 
B~     F' 

so  that  ^(-  +  ^>  =  -^^)  =  F^)=-^^"'  +  ^^' 

and  then  the  relations  are     <^  {to  -\-  z)  -\-  <^  {w  +  z)  =  0, 

or,  what  is  the  same  thing,  (f>(z)  +  cf)  {(o"  -f  ^)  =  0) 

and  </)  (z)  (f)((o  +  z)  =  constant]  " 

The  sub-case  is  substantially  the  same  as  that  of  II.  and  III.,  arising  merely 
from  a  modification  (§  109)  of  the  fundamental  parallelogram,  into  one  whose 
sides  are  determined  by  2&)  and  2a)". 


316  EVEN  [134. 

Hence  we  may  have  (2)  coexistent  with  (2)',  provided 

AF  +  BE=2CD; 
C  and  D  do  not  both  vanish,  and  neither  B  nor  F  vanishes. 
IV.  (1).     Let  neither  C  nor  D  vanish  ;  and  for  brevity  write 

^(&)+^)=^l,         <^  (ft)"   +   ^)  =  (^2,         <^  (ft)'  +  ^)  =  (^3,         (j)(z)  =  (f). 

Then  the  equations  in  IV.  are 

B<f^(i>,-C((f>  +  4>,)  +  A  =  0, 

Now  a  doubly-periodic  function,  with  given  zeros  and  given  infinities,  is 
determinate  save  as  to  an  arbitrary  constant  factor.  We  therefore  introduce 
an  arbitrary  factor  \,  so  that 

CD 
and  then  taking  5\  =  ^^ '       FX^^^' 

we  have  (i/r  -  d)  {-^^  -  c^)  =  Cj^  -  ^, , 

W 

(yjf  -  c)  (yjrs  -  C3)  =  C,^  -  j^,  • 

The  arbitrary  quantity  X  is  at  our  disposal:  we  introduce  a  new  quantity  Ca, 
defined  by  the  equation 

A    _ 

and  therefore  at  our  disposal.     But  since 

AF  +  BE  =  2CD, 

,  A        E       ^   G    D      ^ 

^^^"^^  BX^  +  FX^  =  ^BXFX  =  ^'^'- 

and  therefore  -f^—  =  Cg  (c,  +  c)  —  c;iC.2. 

Hence  the  foregoing  equations  are 

(■f  -  Ci)  if  I  -  Ci)  =  (Ci  -  Ca)  (Ci  -  C3), 
if  -  C3)  (-^/^s  -  C3)  =  (Cs  -  Ci)  (Cs  -  C2). 

The  equation  for  ^2,  which  is  ^{ay"  +  z),  is 

where       L  =  CD  -  BE  =  AF  -  GB,     M  =  AD-GE,     N=GF-BD, 
so  that  AN  +  BM^2GL. 


134.]  DOUBLY-PERIODIC   FUNCTIONS  317 

As  before,  one  particular  sub-case  may  be  considered  and  removed.     If  N 

be  zero,  so  that 

C  _D_ 

A     E     ^GD     ^  , 
say,  and  B'^  W^     BF^ 

then  we  find  (^  -h  ^2  =  ^i  +  <^3  =  ^a, 

or  taking  a  function  •)(  =  4>  —  a, 

the  equation  becomes  X  (^)  +  X  (•"    -\-  z)  =  0. 

The  other  equations  then  become 

X  (^)  X  {''"'  +  ^)  =  «'  -  J 

and  therefore  they  are  similar  to  those  in  Cases  II.  and  III. 
If  N  be  not  zero,  then  it  is  easy  to  shew  that 

N=BF\{c,-c,), 

L=  BF\"(c,-c3)c.2, 

M=BF\^(ci-Cs)(c2Ci  + Co^c-i-c^Ca); 
and  then  the  equation  connecting  <^  and  ^2  changes  to 

(^|r  -  C2)  (-^o  -  Co)  =  (Co  -  Ci)  (Cn  -  C3)  \ 
which,  with  (-yjr  —  Ci)  (-v//^:  —  Ci)  =  (Cj  —  c,)  (Cj  —  Cg) 

("^  -  C3)  (-»/^3  -  C3)  =  (Cs  -  Ci)  (C3  -  Co) 

are  relations  between  -v/r,  yfr^,  -v/r.,,  -v|r3,  where  the  quantity  Cg  is  at  our  disposal. 

IV.  (2).  These  equations  have  been  obtained  on  the  supposition  that 
neither  C  nor  JD  is  zero.  If  either  vanish,  let  it  be  C :  then  D  does  not 
vanish ;    and  the  equations  can  be  expressed  in  the  form 

F 

D\/  .       I)\     D'-EF 


E\f         E\  E{D^-EF) 

We  therefore  obtain  the  following  theorem  : — 

If  ^  he  an  even  function  doubly -periodic  in  2&)  and  2w'  and  of  the  second 
order,  and  if  all  functions  equivalent  to  (j)  in  the  form  R^  +  S  {where  R  and 


318  EVEN  [134. 

S  are  constants)  he  regarded  as  the  same  as  </>,  then  either  the  function  satisfies 
the  system  of  equations 

(f)(z)     </)(«'  +z)  =  H     )- (!)*> 

(f,{z)     (ji{(o"  +  z)  =  -H\ 
where  H  is  a  constant ;  or  it  satisfies  the  system  of  equations 

[(j)  (2)  -  Ci}  [(f)  (oy    +Z)-  Ci}  =  (Ci  -  Ca)  {C^  -  C3)  j 

{(f>  (z)  -  C^  \(f>  {(o'  +Z)-C.^  =  (C3  -  Ci)  (C3  -  C2)  [  (11), 

{(f)  {2)  -  Co}  {(f)  (ft)"  +2)-  C2I  =  (Ca  -  Ci)  (C2  -  C3)  ) 

where  of  the  three  constants  c^,  c^,  c,  one  can  be  arbitrarily  assigned. 
We  shall  now  very  briefly  consider  these  in  turn. 

135.  So  far  as  concerns  the  former  class  of  equations  satisfied  by  an  even 
doubly-periodic  function,  viz., 

(f)  (2)  +  (f)  {cO    +2)  =  i)    ] 

(f>{z)    (})  {(o'  +  2)  =  H  y 

we  proceed  initially  as  in  (§  120)  the  case  of  an  odd  function.     We  have  the 

further  equations 

(f)(z)  =  (f)(-z), 

^(o)  +  z)  =  (f){o)  —  z),        (f)  (o)'  +z)  =  <f)(a)'  —  z). 

Taking  2;  =  -  ^6),  the  first  gives 

<^(ia>)  +  (/,(icu)  =  0, 

so  that  ^ft)  is  either  a  zero  or  an  infinity. 

If  ^  ft)  be  a  zero,  then 

<^  (|a))  =  ^  (ft)  +  ^co)  =  -  </)  {\w)  by  the  first  equation 

=  0, 

so  that  \(o  and  f «  are  zeros.     And  then,  by  the  second  equation, 

W    -\-\w,  ft)'  +  f  ft) 

are  infinities. 

If  ^ft)  be  an  infinity,  then  in  the  same  way  fo)  is  also  an  infinity;  and 
then  ft)'  +  ^  ft),  ft)'  + 1  ft)  are  zeros.  Since  these  amount  merely  to  interchanging 
zeros  and  infinities,  which  is  the  same  functionally  as  taking  the  reciprocal  of 
the  function,  we  may  choose  either  arrangement.  We  shall  take  that  which 
gives  ^  ft),  f  ft)  as  the  zeros ;  and  ft)'  +  ift),  ft)'  +  f  &>  as  the  infinities. 

The  function  ^  is  evidently  of  the  second  class,  in  that  it  has  two  distinct 
simple  irreducible  infinities. 

*  The  systems  obtained  by  the  interchange  of  tu,  w',  w"  among  one  another  in  the  equations 
are  not  substantially  distinct  from  the  form  adopted  for  the  system  I. ;  the  apparent  difference 
can  be  removed  by  an  appropriate  corresponding  interchange  of  the  periods. 


135.]  DOUBLY-PERIODIC    FUNCTIONS  319 

Because  co'  +  ^(o,  w  +  f  w  are  the  irreducible  infinities  of  ^iz),  the  four 
zeros  of  <^  {z)  are,  by  §  117,  the  irreducible  points  homologous  with  to", 
o)"  +  ft),  ft)"  -!-  ft)',  ft)"  +  ft)",  that  is,  the  irreducible  zeros  of  ^'  {£)  are  0,  &),  ft)',  w" . 
Moreover 

(^  (0)  +  </)  (ft))  =  0, 

(^  (ft)')  +  0  (ft)")  =  0, 

by  the  first  of  the  equations  of  the  system ;  hence  the  relation  between  ^  {z) 
and  0'  {z)  is 

f  ^  (5)  =  ^  {</,  (^)  -  (/,  (0)1  {(^  {£)  -  </>  (o))}  {c^  {z)  -  0  (ft,')}  {(/>  {z)  -  (^  (ft)")} 

=  ^  [(/)H0)  -  0^^)}  {0^^')  -  </>H^)}- 

Since  the  origin  is  neither  a  zero  nor  an  infinity  of  ^  {z),  let 

(^(^)  =  <^(0)<^,(^), 

so  that  ^i(O)  is  unity  and  ^/(O)  is  zero;  then 

<^,"-{z)  =  ^-{\-<^^\z)]{^^-^^-{z)] 

the  differential  equation  determining  </)i  {£). 

The  character  of  the  function  depends  upon  the  value  of  jx  and  the 
constant  of  integration.    The  function  may  be  compared  with  en  ii,  by  taking 

2ft),  2ft)' =  4^,  2K-\-^iK'\    and  with  -^,  by   taking   2ft),    2ft)' =  2A",  4tX', 

which  (§  131,  note)  are  the  periods  of  these  (even)  Jacobian  elliptic  functions. 

We  may  deal  -even  more  briefly  with  the  even  function  characterised  by 
the  second  class  of  equations  in  §  134.  One  of  the  quantities  Ci,  Cg,  Cg  being 
at  our  disposal,  we  choose  it  so  that 

Ci  +  Co  +  Ca  =  0  ; 

and    then   the    analogy   with    the  equations    of  Weierstrass's  ^-function  is 
complete  (see  §  133). 


CHAPTER   XII. 

Pseudo-Periodic  Functions. 

136.  Most  of  the  functions  in  the  last  two  chapters  are  of  the  type 
called  doubly-periodic,  that  is,  they  are  reproduced  when  their  arguments  are 
increased  by  integral  multiples  of  two  distinct  periods.  But,  in  §§  127,  130, 
functions  of  only  a  pseudo-periodic  type  have  arisen :  thus  the  ^-function 
satisfies  the  equation 

^  (^  -t-  m2ci)  +  'm'2(o')  =  ^(^)  +  m2'r)  +  mlt] , 

and  the  <r-function  the  equation 

O-  (^  -f  m2tO  +  77i'2a)')  =  (-  l)»"«'+"'+™'  e2(m7,+»i'V)  I^+wco+tti'o,')  ^  (^^^_ 

These  are  instances  of  the  most  important  classes :  and  the  distinction 
between  the  two  can  be  made  even  less  by  considering  the  function 
e  ^'^'  =  ^  {£),  when  we  have 

In  the  case  of  the  f-function,  an  increase  of  the  argument  by  a  period  leads 
to  the  reproduction  of  the  function  multiplied  by  an  exponential  factor 
that  is  constant.  In  the  case  of  the  cr-function,  a  similar  change  of  the 
argument  leads  to  the  reproduction  of  the  function  multiplied  by  an 
exponential  factor  having  its  index  of  the  form  az  -f-  h. 

Hence,  when  an  argument  is  subject  to  periodic  increase,  there  are  three 
simple  classes  of  functions  of  that  argument. 

First,  if  a  function  f{z)  satisfy  the  equations 

/(^  +  2to)=/(^),         /(^+2a>')=/(^), 

it  is  strictly  periodic :  it  is  sometimes  called  a  douhly -periodic  function  of  the 
first  kind.  The  general  properties  of  such  functions  have  already  been 
considered. 

Secondly,  if  a  function  F{z)  satisfy  the  equations 

F {z -v  2ai)  =  fjiF {z),        F(z  +  2co')  =  fju'F{z), 


136.]  PSEUDO-PERIODIC   FUNCTIONS  321 

where  jm  and  /i'  are  constants,  it  is  pseudo-periodic :  it  is  called  a  doubly- 
periodic  function  of  the  second  kind.  The  first  derivative  of  the  logarithm 
of  such  a  function  is  a  doubly-periodic  function  of  the  first  kind. 

Thirdly,  if  a  function  </>  {z)  satisfy  the  equations 

0  (^  +  2o))  =  e«^+&  (^  {z),         (f>{z+  2a)')  =  e'»'^+*'  (f>  (z), 

where  a,  b,  a,  b'  are  constants,  it  is  pseudo-periodic  :  it  is  called  a  doubly- 
periodic  function  of  the  third  kind.  The  second  derivative  of  the  logarithm 
of  such  a  function  is  a  doubly-periodic  function  of  the  first  kind. 

The   equations  of  definition  for   functions   of  the    third   kind   can   be 
modified.     We  have 

<f>{z+  2(0  +  2ft)')  =  ea(2+2'o')+6+«'^+&'  (f)  (^) 

whence  a'co  —  aco'  —  —  nfiiri, 

where  m  is  an  integer.  Let  a  new  function  E  {z)  be  introduced,  defined  by 
the  equation 

E{z)  =  e^^-+''^<^{z); 

then  X  and  /z  can  be  chosen  so  that  E{z)  satisfies  the  equations 
E{z-\-2(o)  =  E  {z),         E{z  +  2&)')  =  e^^^  E  (z). 
From  the  last  equations,  we  have 

E(z-i-2a)+  2ft)')  =  e^^'+"^>+^E(z) 
=  e^'+^E{z), 
so  that  2 J. ft)  is  an  integral  multiple  of  27ri. 

Also  we  have         E  (z  -\-  2ft))  =  g  A(2+2<")=+M(2+2a>)  cf){z-{-  2(o) 

so  that  4Xft)  -I-  a  =  0, 

and  4A,ft)2 -\-  2/j,co -\-b  =  0  (mod.  2^). 

Similarly,  E(z  +  2co')  =  e^(2+2a.') 2+^.(2+20.')  0  (^  +  2&)') 

_  g  4A2(o'+4Xio'2+2|a(o'+«'2+6'  E  (z^ 

so  that  4Xft)'  -{-  a  =  A, 

and  4Xft)'2  -f-  2/Aft)'  +  b'  =  B  (mod.  2^). 

From  the  two  equations,  which  involve  X  and  not  /*,  we  have 

u4ft)  =  a' ft)  —  aft)' 
=  —  miri, 
agreeing  with  the  result  that  2 Aw  is  an  integral  multiple  of  27ri. 

And  from  the  two  equations,  which  involve  [ju,  we  have,  on  the  elimination 
of  /i  and  on  substitution  for  \ 

b'oo  —  boo'  -  aw  (&)'  —  &))  =  Bw  (mod.  27ri). 
F.  F.  21 


322  DOUBLY-PERIODIC   FUNCTIONS  [136. 

If  A  be  zero,  then  E  (z)  is  a  doubly-periodic  function  of  the  first  kind 
when  e^  is  unity,  and  it  is  a  doubly-periodic  function  of  the  second  kind 
when  e^  is  not  unity.  Hence  A,  and  therefore  m,  may  be  assumed  to  be 
different  from  zero  for  functions  of  the  third  kind.  Take  a  new  function 
^(2),  such  that 

then  ^  (z)  satisfies  the  equations 

_nnTi 

.     (P  (2  +  2(o)  =  ^  (z),         ^(z-\-2(o')  =  e~^~^  ^{z), 
which  will  be  taken  as  the  canonical   equations   defining  a  doubly -periodic 
function  of  the  third  kind. 

Ex.     Obtain  the  values  of  X,  /x,  A,  B  for  the  Weierstrassian  function  o-  (2). 
We  proceed  to  obtain  some  properties  of  these  two  classes  of  functions 
which,  for  brevity,  will  be  called  secondary -periodic  functions  and  tertiary- 
'periodic  functions  respectively. 

Doubly-Periodic  Futfctions  of  the  Second  Kind. 

For  the  secondary-periodic  functions  the  chief  sources  of  information  are  : — 

Hermite,  Comptes  Rendus,  t.  liii,  (1861),  pp.  214—228,  ib.,  t.  Iv,  (1862),  pp.  11—18, 
85 — 91;  Si0'  quelques  applications  des  fonctions  elliptiqices,  §§  i — iii,  separate 
reprint  (1885)  from  Comptes  Re7idus ;  "  Note  sur  la  theorie  des  fonctions  ellip- 
tiques''  in  Lacroix,  vol.  ii,  (6th  edition,  1885),  jjJ).  484 — 491;  Cours  d^ Analyse, 
(4'^«  ed.),  pp.  227—234. 

Mittag-LefEer,  Comptes  Rendus,  t.  xc,  (1880),  pp.  177—180. 

Frobenius,  Crelle,  t.  xciii,  (1882),  pp.  53—68. 

Brioschi,  Comptes  Rendihs,  t.  xcii,  (1881),  pp.  325 — 328. 

Halphen,  Traite  des  fonctions  elliftiques,  t.  i,  pp.  225—238,  411—426,  438—442,  463. 

137.     In  the  case  of  the  periodic  functions  of  the  first  kind  it  was  proved 

that  they  can  be  expressed  by  means  of  functions  of  the  second  order  in  the 

same  period — these  being  the  simplest  of  such  functions.     It  will  now  be 

proved  that  a  similar  result  holds  for  secondary-periodic  functions,  defined  by 

the  equations 

F{2^2(o)  =  fj^F{2),         F(z  +  2co')  =  fM'F{z). 

Take  a  function  G  (2)  =  "^ /\     ^^\  e^^; 

then  we  have  G{2  +  2oy)  =  /^(^+;^  +  ^^)  ^x^+^x. 

=  e2^^+2^"  G  (2), 
and  G(2-^  2a)')  =  e^Va+sxa,'  q  (^). 

The  quantities  a  and  \  being  unrestricted,  we  choose  them  so  that 

and  then  G{z),  a  known  function,  satisfies  the  same  equation  as  F(2). 


137.]  OF   THE   SECOND    KIND  323 

Let  u  denote  a  quantity  independent  of  z,  and  consider  the  function 
f^z)  =  F{z)G{u-z). 
We  have  f{z  -v2oi)  =  F{z+  2&))  G{u-z-  2a)) 

=  tiF{z)'^G{u-z) 

and  similarly  f{z  +  2a>')  =f(^z), 

so  that  f{z)  is  a  doubly-periodic  function  of  the  first  kind  with  2a)  and  2aj' 
for  its  periods. 

The  sum  of  the  residues  of  f{z)  is  therefore  zero.  To  express  this  sum, 
we  must  obtain  the  fractional  part  of  the  function  for  expansion  in  the 
vicinity  of  each  of  the  (accidental)  singularities  of  f{z),  that  lie  within  the 
parallelogram  of  periods.  The  singularities  oif{z)  are  those  of  G  {u  —  z)  and 
those  oi  F{z). 

Choosing  the  parallelogram  of  reference  so  that  it  may  contain  u,  we  have 
^  =  li  as  the  only  singularity  of  G  (u  —  z)  and  it  is  of  the  first  order,  so  that, 
since 

G(^)  =  -r  +  positive  integral  powers  of  ^ 
in  the  vicinity  of  ^=0,  we  have,  in  the  vicinity  of  u, 

f{z)  =  [F  (u)  +  positive  integral  powers  of  u  —  z]  \ h  positive  powers [ 

Fhi)  ...  , 

= +  positive  integral  powers  oi  z  —  u; 

hence  the  residue  off(z)  for  ii  is  —  F  (u). 

Let  z  =  c  be  a  pole  of  F{z)  in  the  parallelogram  of  order  n+1;  and,  in 
the  vicinity  of  c,  let 

^  (^)  -  ^  +  <^^  ^  (^c)  +  •  •  •  +  ^'^+^  &  (^c)  +  P°'^^^^'  '""^^^^^  P^^^^'- 
Then  in  that  vicinity 

G{u-z)  =  G(u-c)-(z-c)^^G(u-c)  +  ^-^^^^G(u-c)-..., 

and  therefore  the  coefficient  of in  the  expansion  of  f(z)  for  points  in  the 

^  —  c 

vicinity  of  c  is 

C,G{u-c)  +  a,gG(u-c)+G,I^^G(u-c)  +  ...  +  Cn^,^,G{u-c), 
which  is  therefore  the  residue  off(z)  for  c. 

This  being  the  form  of  the  residue  of  /(z)  for  each  of  the  poles  of  F  (z), 
then,  since  the  sum  of  the  residues  is  zero,  we  have 


-F(u)  +  1. 


G,G(a-c)  +  C,-g^G(u-c)  +  ...  +  Cn+r-^G(u-c) 


=  0, 
21—2 


\ 


324 

or,  changing  the  variable 

F{z)  =  t 


HERMITE  S   THEOREM 


[137. 


C,G{z-c)  +  C,^^G{z^c)+...+C,,^,^^G{z-o) 


where  the  summation  extends  over  all  the  poles  of  F{z)  within  that  parallelo- 
gram of  periods  in  which  z  lies.     This  result  is  due  to  Hermite. 

138.     It  has  been  assumed  that  a  and  X,  parameters  in  G,  are  determinate, 

an  assumption  that  requires  [x  and  yu,'  to  be  general  constants :  their  values 

are  given  by 

r]a-\-  (o\  =  \  log  /i,     7]' a  +  w'X  =  -|  log  yu.', 

and,  therefore,  since  ??«'  —  77'a)  =  ±  ^iir,  we  have 

+  iira  =      co'  log  IX  — w  log  jjl 

±  iifk  =  —  r]'  log  /i  +  77  log  jjb 

Now  X.  may  vanish  without  rendering  G  {z)  a  null  function.     If  a  vanish  (or, 

what  is  the  same  thing,  be  an  integral  combination  of  the  periods),  then  G  {z) 

is  an  exponential  function  multiplied  by  an  infinite  constant  when  \  does  not 

vanish,  and  it  ceases  to  be  a  function  when  X  does  vanish.     These  cases  must 

be  taken  separately. 

First,  let  a  and  X,  vanish*;  then  both  [x  and  [x  are  unity,  the  function  F 

is  doubly-periodic  of  the  first  kind ;  but  the  expression  for  F  is  not  determinate, 

owing  to  the  form  of  G.     To  render  it  determinate,  consider  \  as  zero  and  a 

as  infinitesimal,  to  be  made  zero  ultimately.     Then 

0-  {z)  -\-  acr  {z)-\- ... 


G{z)  = 


aa  (z) 


(1  +  powers  of  a  higher  than  the  first) 


=  -  ^-  ^{z)  +  positive  powers  of  a. 

Since  a  is  infinitesimal,  /x  and  /x'  are  very  nearly  unity.  When  the 
function  J^is  given,  the  coefficients  Cj,  C^, ...  may  be  affected  by  a,  so  that 
for  any  one  we  have 

Gjs  =  bk  +  (^Jk  +  higher  powers  of  a, 
where  7^  is  finite ;  and  bjc  is  the  actual  value  for  the  function  which  is  strictly 
of  the  first  kind,  so  that 

26i  =  0, 

the  summation  being  extended  over,  the  poles  of  the  function.    Then  retaining 
only  a~^  and  a°,  we  have 


2 


G,G{ii-c)  +  G2-^^G{u-c)  + +  Cn+, -^^  G  (u  -  c) 


br 


^f^  +  Xj.  +  X 


a 


d  cf 

6it  (m  -  c)  +  62  ^^  ?  (m  -  c)  +  . . .  +  hn+,  ;^  ^(m  -  C) 


diC' 


d"' 


bi^{u  -c)+  ...+  bn+i -^,  ^{u -  c) 


This  case  is  discussed  by  Hermite,  (I.e.,  p.  322). 


138.]  mittag-leffler's  theorem  325 

where   C^,  equal  to  S71,  is  a  constant  and   the  term  in  -  vanishes.     This 

a 

expression,  with  the  condition  26i  =  0,  is  the  value  of  F{u);  changing  the 
variables,  we  have 

with  the  condition  S61  =  0,  a  result  agreeing  with  the  one  formerly  (§  128) 
obtained. 

When  F  is  not  given,  but  only  its  infinities  are  assigned  arbitrarily,  then 
20=0  because  jP  is  to  be  a  doubly-periodic  function  of  the  first  kind;  the 

term  -  2(7  vanishes,  and  we  have  the  same  expression  for  F(z)  as  before. 
Secondly,  let  a  vanish*  but  not  \,  so  that  /x  and  /x'  have  the  forms 

We  take  a  function  g  (z)  =  e^^  t(^) ; 

then  g{z-2o})  =  fi-^e'^^(z-2o)) 

=  /x-'e'^\^{z)-2v} 
=  fji-'{g{z)-2ve^], 
and  g(z—2(a')  =  jji'-'^{g(z)  —  2r}'e'^}. 

Introducing  a  new  function  H(z)  defined  by  the  equation 

H{z}  =  F{z)g(u-z), 
we  have  H{z  +  2w)  =  H {z)  -  277e^(«-^'  F{z), 

and  H{z  +  2(o')  =  H{z)-  27;'e^<«-^'  F{z). 

Consider  a  parallelogram  of  periods  which  contains  the  point  u;  then,  if  @  be 
the  sum  of  the  residues  of  H  {z)  for  poles  in  this  parallelogram,  we  have 

2'jri®=fH{z)dz, 
the  integral  being  taken  positively  round  the  parallelogram.     But,  by  §  116, 
Prop.  II.  Cor.,  this  integral  is 

4e^«  I  corj'  r  e-Mi^+2a.f,  ^(^  ^  2(ot)  dt  -  w't)  [  g-^'^+^-'f)  F  {p  +  2(o't)  dt  \  , 

where  p  is  the  corner  of  the  parallelogram  and  each  integral  is  taken  for 
real  values  of  t  from  0  to  1.  Each  of  the  integrals  is  a  constant,  so  far  as 
concerns  u ;  and  therefore  we  may  take 

@  =  -  Ae^'', 
the  quantity  inside  the  above  bracket  being  denoted  by  —  ^iirA. 

The  residue  of  H(z)  for  z=  u,  arising  from  the  simple  pole  of  g(u  —  z),  is 
-F{u)  as  in  §  137. 

If  2;  =  c  be  an  accidental  singularity  of  F{z)  of  order  n+1,  so  that,  in  the 
vicinity  of  ^  =  c, 

*  This  is  discussed  by  Mittag-Leffler,  (I.e.,  p.  322). 


326  SECONDARY  "  [138. 

then  the  residue  of  H  (z)  for  ^  =  c  is  , 

d  d"" 

and  similarly  for  all  the  other  accidental  singularities  of  F{z).     Hence 

F(z)  =  Ae>^^  +  %i^C,  +  C,^^+...+C^^,^^g{z-c), 

where  the  summation  extends  over  all  the  accidental  singularities  of  F{z)  in  a 

parallelogram  of  periods  which  contains  2,  and  g{z)  is  the  function  e^t,{z). 

This  result  is  due  to  Mittag-LefBer. 

Since  /i  =  e'^^'"  and 

^r  (^^  -  c  +  2&))  = /z^  (^r  -  c)  +  2?7/ie'^<2-<'>, 
we  have 

ixF{z)^F{z  +  1ai) 

id  fJ'>^  ^ 

=  f,Ae^^  +  %\^C,  +  G,^^  +  ...  +  C^^,^J^f.g{z-c) 

and  therefore  2  (C^  +  CgX  +  . . .  +  Gn+jV)  e-^"  =  0, 

the  summation  extending  over  all  the  accidental  singularities  of  F  (z).     The 

same  equation  can  be  derived  through  fji!F{z)  =  F{z  +  2(o'). 

Again  2(7i  is  the  sum  of  the  residues  in  a  parallelogram  of  periods,  and 

therefore 

2'jriXG,=jF(z)dz, 

the  integral  being  taken  positively  round  it.    If  p  be  one  corner,  the  integral  is 

2a)  (1  -  fi')  rF(p  +  2cot)  dt  -  2co'  (1  -/x)['  F(p+  2o)'t)  dt, 

Jo  Jo 

each  integral  being  for  real  variables  of  t. 

Hermite's  special  form  can  be  derived  from  Mittag-Leffler's  by  making 
X  vanish. 

Note.  Both  Hermite  and  Mittag-Leffler,  in  their  investigations,  have 
used  the  notation  of  the  Jacobian  theory  of  elliptic  functions,  instead  of 
dealing  with  general  periodic  functions.  The  forms  of  their  results  are  as 
follows,  using  as  far  as  possible  the  notation  of  the  preceding  articles. 

I.     When  the  function  is  defined  by  the  equations 

F{z  +  2K)  =  fiF  (z),       F(z  +  2iK')  =  ix'F  {z), 

then  i?'(^)  =  slc,  +  a^  +  ...  +  a.+i^}G^i^-c), 

where  a^,^-H' {^)  H{z^  .) 

where  (^{z)-      ^^^-^^^^^      ^   , 


138.]  ^  PEEIODIC   FUNCTIONS  327 

(the  symbol  H  denoting  the  Jacobian  fi'-function),  and  the  constants  co  and  \ 
are  determined  by  the  equations 


^  =  e^^^,     iJb  =e    ^ 


II.     If  both  A,  and  a  be  zero,  so  that  F  {z)  is  a  doubly-periodic  function 
of  the  first  kind,  then 

d   ^  ,        d^}  H'iz-c) 


with  the  condition  26i  =  0. 


H{z-c)' 


III.     If  a  be  zero,  but  not  \,  then 


F(z)  =  Ae^^  +  t\G,  +  G.^^^-^...  +  G,^,^J^g(z-c), 


Xz 


where     ,  ^^^^^~WU)^' 

the  constants  being  subject  to  the  condition 

2  (Ci  +  ax  + . . .  +  a+A'O  e-^'' = 0, 

and  the  summations  extending  to  all  the  accidental  singularities  of  F  {£)  in  a 
parallelogram  of  periods  containing  the  variable  z. 

139.     Reverting  now  to  the  function  F  {£),  we  have  G  {z)^  defined  as 

o-  iz)  a  (a)       ' 
when  a  and  \  are  properly  determined,  satisfying  the  equations 

G{z  +  2co)  =  ^lG{z),    G(z  +  2co')  =  /u,'G(z). 

Hence  fl  (z)  =  F  {z)/G  (z)  is  a  doubly-periodic  function  of  the  first  kind ;  and 
therefore  the  number  of  its  irreducible  zeros  is  equal  to  the  number  of  its 
irreducible  infinities,  and  their  sums  (proper  account  being  taken  of  multi- 
plicity) are  congruent  to  one  another  with  moduli  2co  and  2co\ 

Let  Ci,  Ca,  ...,  Cm  be  the  set  of  infinities  of  F (z)  in  the  parallelogram  of 
periods  containing  the  point  z;  and  let  71,  ...,  7^  be  the  set  of  zeros  of  F{z) 
in  the  same  parallelogram,  an  infinity  of  order  n  or  a,  zero  of  order  n  occurring 
n  times  in  the  respective  sets.  The  only  zero  of  G{z)  in  the  parallelogram  is 
congruent  with  —  a,  and  its  only  infinity  is  congruent  with  0,  each  being 
simple.     Hence  the  m  +  1  irreducible  infinities  of  H  (z)  are  congruent  with 

(I,  Cj ,  C2 ,  . . . ,  Cin , 

and  its  fi  +  1  irreducible  zeros  are  congruent  with 

0,  7i,  72,  •••,  7/*; 
and  therefore  i^^^K     m  -t- 1  =  //.  -f- 1 , 

a  -h  2c  =  27. 


328  SECONDARY   FUNCTIONS  ^  [139. 

From  the  first  it  follows*  that  the  numher  of  infinities  of  a  doubly -periodic 
function  of  the  second  kind  in  a  parallelogram  of  periods  is  equal  to  the 
numher  of  its  zeros,  and  that  the  excess  of  the  sum  of  the  former  over  the  sum 
of  the  latter  is  congruent  with 

^.log^--.log^j, 


< 


CO  \ 


the  sign  being  the  same  as  that  o/  9t  ( —  j . 

The  result  just  obtained  renders  it  possible  to  derive  another  expression 
for  F  {z),  substantially  due  to  Hermite.     Consider  a  function 

(t{z-c^)(t{z-c.)  ...  cr{z-  c„i)- 
where  p  is  a  constant.     Evidently  F^  (z)  has  the  same  zeros  and  the  same 
infinities,  each  in  the  same  degree,  as  F  (z).     Moreover 

i^i  (^  +  2g)  )  =  J^i  (^)  e2')(2c-2y)+2p.^ 
F^  (Z  +  2&)')  =  F,  (Z)  e2V(2c-2v)+2pa.'^ 

If,  then,  we  choose  points  c  and  7,  such  that 

2c  —  27  =  a, 
and  we  take  p  =  \  where  a  and  \  are  the  constants  of  G  (z),  then 

F,  {z  +  2(o)  =  fxF,  {z),     F,  (z  +  2co')  =  fi'F,  {z). 
The  function  F^  {z)/F  (z)  is  a  doubly-periodic  function  of  the  first  kind,  and 
by  the  construction  of  F-l(z)  it  has  no  zeros  and  no  infinities  in  the  finite 
part  of  the  plane :  it  is  therefore  a  constant.     Hence 

F{z)  =  A  ^(;-7:)^(^-7.)-.^(^-7>n)  ^,,^ 
a{z-Cj,)a{z  -C2)  ...  (t{z-  c^) 

where  2c  —  27  =  a,  and  a  and  X,  are  determined  as  for  the  function  G  (z). 

140.  One  of  the  most  important  applications  of  secondary  doubly-periodic 
functions  is  that  which  leads  to  the  solution  of  Lame's  equation  in  the  cases 
when  it  can  be  integrated  by  means  of  uniform  functions.  This  equation 
is  subsidiary  to  the  solution  of  the  general  equation,  which  is  characteristic 
of  the  potential  of  an  attracting  mass  at  a  point  in  free  space ;  and  it  can  be 
expressed  "I"  either  in  the  form 

-r—  =  (Ak'  sn^z  +  B)  w, 
dz' 

or  in  the  form  -^Y  —  {-^^  (^)  +  ^1  '^^' 

*  Frobenius,  Crelle,  xciii,  pp.  55 — 68,  a  memoir  which  contains  developments  of  the  properties 
of  the  function  G  (2).  The  result  appears  to  have  been  noticed  first  by  Brioschi,  (Gomptes  Rendits, 
t.  xcii,  p.  325),  in  discussing  a  more  limited  form. 

t  The  equation  arises  when  the  coordinates  of  any  point  in  space  are  taken  to  be  the 
parameters  of  the  three  confocal  quadries  through  the  point.  For  the  actual  derivation  of 
the  equation  in  either  of  the  forms  stated,  see  my  Theory  of  Differential  Equations,  vol.  iv,  §  148. 


140.]  lamia's  equation  329 

according  to  the  class  of  elliptic  functions  used.  In  order  that  the  integral 
may  be  uniform,  the  constant  A  must  be  n  (n  +  1),  where  n  is  a  positive 
integer;  this  value  of  A,  moreover,  is  the  value  that  occurs  most  naturally  in 
the  derivation  of  the  equation.     The  constant  B  can  be  taken  arbitrarily. 

The  foregoing  equation  is  one  of  a  class,  the  properties  of  which  have 
been  established*  by  Picard,  Floquet,  and  others.  Without  entering  into 
their  discussion,  the  following  will  suffice  to  connect  them  with  the  secondary 
period^  function. 

Let  two  independent  special  solutions  be  g  (z)  and  h  (z),  uniform  functions 
of  z ;  every  solution  is  of  the  form  ag  (z)  +  j3h  (z),  where  a.  and  /3  are  constants. 
The  equation  is  unaltered  when  z  +  2(o  is  substituted  for  z ;  hence  g(z  +  2&)) 
and  h(z  +  2&))  are  solutions,  so  that  we  must  have 

g{z  +  2co)  =  Ag  (z)  +  Bh  {z),     h  (z  +  2co)  =  Cg  (z)  +  Dh  (z), 

where,  as  the  functions  are  determinate.  A,  B,  C,  D  are  determinate  constants, 
such  that  AD  —  BC  is  different  from  zero. 

Similarly,  we  obtain  equations  of  the  form 

g(z  +  2«')  =  A'g  (z)  +  B'h  {z),     A  (5  +  2a)' )  =  G'g  {z)  +  D'h  (z). 

Using  both  equations  to  obtain  g{z  ^-^.a  +  2co')  in  the  same  form,  we  have 

BC  =  B'C,    AB'-hBD'  =  A'B  +  B'D ; 

and  similarly,  for  h  {z  -\-2(o  +  2&)'),  we  have 

CA'  +  DC  =  CA  +  D'C,    BC  =  B'C ; 

.       „  G     C     .      A-D     A'-D' 

therefore  B^W~         — W  ^  — R —  ^  ^' 

Let  a  solution  F{z)  =  ag  (z)  +  bh  (z) 

be  chosen,  so  as  to  give 

F{z  +  2co)  =  fiF(z),     F(z  +  2co')=/xT(z), 

if  possible.     The  conditions  for  the  first  are 

aA  +hC     aB  +  bD 
a  b 

so  that  ajb  (=  |)  must  satisfy  the  equation 

A-D  =  ^B-^^; 

and  the  conditions  for  the  second  are 

aA'  +  bC     aB'+bD'       , 

= T =  /^> 

a  0 

*  Picard,  Comptes  Rendus,  t.  xc,  (1880),  pp.  128—131,  293—295 ;  Grelle,  t.  xc,  (1880), 
pp.   281—302. 

Floquet,  Comptes  Rendus,  t.  xcviii,  (1884),  pp.  82—85;  Ann.  de  VEc.  Norm.  Sup.,  3™  Ser., 
t.  i,  (1884),  pp.  181—238. 


330  lamp's  [140. 

so  that  I  must  satisfy  the  equation 

A'-D'  =  ^B'-^. 

These  two  equations  are  the  same,  being 

p  _  e^  _  g  =  0. 

Let  |i  and  ^o  be  the  roots  of  this  equation  which,  in  general,  are  unequal;  and 
let  yLti,  fu,i  and  //-a,  H^z  he  the  corresponding  values  of  //-,  /jf.  Then  two  functions, 
say  ^1  (z)  and  F2  (^),  are  determined :  they  are  independent  of  one  another,  so 
therefore  are  g  (z)  and  h  (z) ;  and  therefore  every  solution  can  be  expressed  in 
terms  of  them.  Hence  a  linear  differential  equation  of  the  second  order,  having 
coefficients  that  are  doubly -periodic  functions  of  the  first  kind,  can  generally  he 
integrated  by  means  of  doubly -periodic  functions  of  the  second  kind. 

It  therefore  follows  that  Lame  s  equation,  which  will  be  taken  in  the  form 

can  be  integrated  by  means  of  secondary  doubly-periodic  functions. 

141.     Let  z  =  c  be  an  accidental  singularity  of  w  of  order  m ;  then,  for 
points  z  in  the  immediate  vicinity  of  c,  we  have 


^  "  {z  -  cT  (1  +  i^  (^  -  c)  +  ?  (^  -  c)^  +  . . 


and  therefore 


1  d^w     m-\-m^     2mp  ,        ...  p 

— - —  =  -; ; ~  +  positive  powers  01  z  —  c. 

w  dz^      {z-cf     z-c     ^  ^ 

Since  this  is  equal  to  n  {n -{- 1)  (^  {z)  +  B 

it  follows  that  c  must  be  congruent  to  zero  and  that  m,  a  positive  integer, 

must  be  n.     Moreover,  ^  =  0.     Hence  the  accidental  singularities  of  tv  are 

congruent  to  zero,  and  each  is  of  order  n. 

The  secondary  periodic  function,  which  has  no  accidental  singularities 

except  those  of  order  n  congruent  to  z  =  0,  has  n  irreducible  zeros.    Let  them 

be  —  Oi,  -a-i,  ■■■>  —cin',  then  the  form  of  the  function  is 

cr(z  +  ai)  (r(z-{-a2)  ...  cr(z+  an)    „^ 
w  =  — ^^ -rrr eP". 

(7^  {z) 

1  dw  "" 

Hence  ^  =  p-n^(z)+  2  l;(z  +  a,.), 

tu  dz      ^  r=\ 

or,  taking  p  —  —  2^(a^),  we  have 

W     dz  y.=  l 

and  therefore  ^ f-^ )  =  ni^  (z)  -  1  iJ{z  +  a,.). 

w  dz^      w^  \dz.l  r=l 


141.]  DIFFERENTIAL   EQUATION  331 

But,  by  Ex.  3,  §  131,  we  have 


lu-  \dz  J       4  (^=1  gj {ar)  —  f  {z) 

by  Ex.  4,  §  131.     Thus 

Now 

^y(a,)-^'(^)  (^'{as)-<p'{z) 

io{ar)-i^{z)'  io{a,)-<^{z) 

^  4,^^  {z)  -  g^^  (z)  -g,  +  g/  (g^)  g)^  (g^)  -  {p'  (a,)  +  jp'  (a,)}  g>^  (^) 

where  ^  ^  ^^X;^.)  +  P>.)  ^  _  ^ 

^j  (a,.)  -  ^J  (a,) 

Let  the  constants  a  be  such  that 

^y(ai)  +  ^y(a2)  ^  gy  (ai)  +  g)^  (go)  _^       ^  ^ , 

^j  (gO  -  §)  (gg)       g.)  (gO  -  ^J  W 

gy(g,)  +  gy(gi)  _^ ^y(g2)  +  |P^(a3)  ^      _  (^i  , 
^J  (ga)  -  ^i)  (g-i)       ^J>  (go)  -  ^J  (gg) 

?i  equations  of  which  only  n—1  are  independent,  because  the  sum  of  the  n 
left-hand  sides  vanishes.     Then  in  the  double  summation  the  coefficient  of 

each  01  the  fractions  " — y-^ — —, — {  is  zero ;  and  so 
gj  (ir)  -  gj  (g,.) 

.    2    2  i^-H^ — S^T  —7-^^ — ^-^  r  =  27i  (71  - 1)  ^  (^)  +  4  (n  -  1)  2  g? (g^), 

1    (i^W  ^ 

and  therefore         —  -7^  =  n  (w  +  1)  p  (^r)  +  (2n  —  1)2^  (g^). 

Hence  it  follows  that 

_  q-  (^  +  g,)  o-  (^  +  g^)  ...  cr(^  +  ^n)    "^f/K) 
J^Kz)-tu,-  ^^^ 

satisfies  Lame's  equation,  provided  the  n  constants  a  he  determined   by  the 
preceding  equations  and  by  the  relation 

n 

B  =  (2n-1)  X  iJ  (ar). 

r=l 


332  lamp's  equation  [141. 

Evidently  the  equation  is  unaltered  when  —  z  is  substituted  for  z ;  and 
therefore 

n 

is  another  solution.     Every  solution  is  of  the  form 

MF{z)  +  NF{-z), 
where  M  and  N  are  arbitrary  constants. 

Corollary.     The  simplest  cases  are  when  n  =  l  and  n  =  2. 

When  n  =  1,  the  equation  is 

lu  dz-        ^  ^  ■^ 
there  is  only  a  single  constant  a  determined  by  the  single  equation 

and  the  general  solution  is 

-.^criz  +  a)        .,  .      ^^  a  (z  —  a)      .,  . 
(t{z)  a{z) 

When  n=  2,  the  equation  is 

The  general  solution  is 

0-2  (^)  <T^(z) 

where  a  and  &  are  determined  by  the  conditions 

p{a)-^j{b)  ^  ^  -^     ^  ^  ^      ^ 

Rejecting  the  solution  a  +  6  =  0,  we  have  a  and  b  determined  by  the  equations 
io  (a)  +  iJ  (6)  =  ^B,         p  (a)  ^  (b)  =  ^B^  -  Ig,. 

For  a  full  discussion  of  Lame's  equation  and  for  references  to  the  original  sources  of 
information,  see  Halphen,  Traite  des  fonctions  elliptiques,  t.  ii,  chap,  xii.,  in  particular, 
pp.  495  et  seq. 

Ex.    When  Lame's  equation  has  the  form 

=--r  =  ?2:  (n  +  1)  Fsn^  3-A, 

obtain  the  solution  for  n  =  l,  in  terms  of  the  Jacobian  Theta-Functions, 

e(z)  e{z) 

where  a>  is  determined   by  the   equation   dn"a  =  h-k^;   and  discuss  in  particular  the 
solution  when  h  has  the  values  1+F,  1,  F. 
Obtain  the  solution  for  n  =  2  in  the  form 


dz\_    e  (z) 


E(i±^  >-|g]}  ']+B±  [^-(i^>  e-^-ir^  ' 

dz      e(z) 


141.]  TERTIARY   PERIODIC    FUNCTIONS  333 

where  X  and  co  are  given  by  the  equations 

(2F  sn2  a  - 1  -F)  (2F  sn2  a  - 1)  (2  sn2  a- 1) 


sn2  0)  = 


3F  sn*  a-2  (l  +  F)  sn^  a  +  1 
sn*a(2Fsn2a-l-y(;2) 


"3Fsn*a-2(l+F)sn2a  +  l' 
and  d  is  derived  from  h  by  the  relation 

A=4(l+X-2)-6/l-2sn2a. 
Deduce  the  three  solutions  that  occur  when  X  is  zero,  and  the  two  solutions  that  occur 
when  X  is  infinite.  (Hermite.) 

Doubly -Periodic  Functions  of  the  Third  Kind. 

142.  The  equations  characteristic  of  a  doubly-periodic  function  <f>  {z)  of 
the  third  kind  are 

_  m-rri  ^ 

<l>  (^r  +  2&))  =  $  {z),     (^{z  +  2(o')  =  e~^^  ^  <l>  (z), 
where  m  is  an  integer  different  from  zero. 

Obviously  the  number  of  zeros  in  each  parallelogram  is  invariable,  as  well 
as  the  number  of  infinities.  Let  a  parallelogram,  chosen  so  that  its  sides 
contain  no  zero  and  no  infinity  of  <l>  (z).  have  p,  p+  2co,  p  +  2&)'  for  three 
of  its  angular  points;  and  let  a^,  a^,  ...,ai  be  the  zeros  and  Ci,  ...,  Cm  be  the 
infinities,  multiplicity  of  order  being  represented  by  repetitions.    Then  using 

■^  {z)  to  denote  -j-  {log  <^  {z)],  we  have,  as  the  equations  characteristic  of 
^  (5  +  2co)  =  ^  {z),     ^{z  +  2co')  =  ^(z)-''-^^; 

CO 

and  for  points  in  the  parallelogram 

II  ^»        1 

r  =  l  ^        ^r       s  =  l  ^        Gg 

where  H  {z)  has  no  infinity  within  the  parallelogram.     Hence 

2'7Ti{l-n)  =  j^r{z)dz, 
the  integral  being  taken  round  the  parallelogram  :  by  using  the  Corollary  to 
Prop.  II.  in  §  116,  we  have 

ZTTi  {i  —  n)  =  —  \  —  ( 1  dz  =  zmiri, 


\    CO    J 

so  that    '  l  =  n  +  m: 

or  the  algebraical  excess  of  the  number  of  irreducible  zeros  over  the  number  of 

irreducible  infiiiities  is  equal  to  m. 

Z  IM 

A  gam,  since  =  1  + 


Z  —  fX  Z—  [Jb 

we  have  2  -^  -S^  -^  +1  -  n  =  z^  {z)  -  zH  {z), 

z  —  a         z  —  c 

and  therefore  ^-ni  (%a  —  2c)  =  /^^  {z)  dz, 


334  PSEUDO-PERIODIC    FUNCTIONS  [142. 

the  integral  being  taken  round  the  parallelogram.     As  before,  this  gives 

27ri  (Sa  -  2c)  -  2g)^  {z)  dz-  2co'^  (z)  -  -—  (z  +  2a)') \  dz. 

The  former  integral  is 


'25  +  2-'  (^'  /^\ 

i<a  I  ^  ■  /  dz 


P         ^  (^) 
=  2ft)  ( i^ )  =  ~"  ^inTTip, 

for  the  side  of  the  parallelogram  contains  *  no  zero  and  no  infinity  of  <I>  (z). 
The  latter  integral,  with  its  own  sign,  is 

-  2ft)'  ^iTT^/  (^^  +  (Z  +  2ft)  )  (^0 

jp  ^(Z)  ft)     j^        ^  ^ 

=  0  +  ^'  {(^  +  2ft)  +  20,')^  -{p  +  2<oJ} 

Zft) 

=  2m7ri  (p  +  CO  +  2ft)'). 
Hence  Sa  —  2c  =  m  (&)  +  2ft)'), 

giving  ^/ie  excess  of  the  sum  of  the  zeros  over  the  sum  of  the  infinities  in  any 
parallelogram  chosen  so  as  to  contain  the  variable  z  and  to  have  no  one  of  its 
sides  passing  through  a  zero  or  an  infinity  of  the  function. 

These  will  be  taken  as  the  irreducible  zeros  and  the  irreducible  infinities  : 
all  others  are  congruent  with  them. 

All  these  results  are  obtained  through  the  theorem  II.  of  §  116,  which 
assumes  that  the  argument  of  &)'  is  greater  than  the  argument  of  o)  or,  what 
is  the  equivalent  assumption  (§  129),  that 

7](o'  —  77'ft)  =  ^  7^^. 
143.     Taking  the  function,  naturally  suggested  for  the  present  class  by 
the  corresponding  function  for  the  former  class,  we  introduce  a  function 
^  ^,,,^^^  a(.-a,U(.-a,)...a(.-ad 
^  ^  a{z  —  c-i)a(z—  Co)  ...  (T  {z  -  Cn) 

where  the  a's  and  the  c's  are  connected  by  the  relations 
2a  —  2c  =  w^  (ft)  +  2ft)'),     I  —  n  =  m. 
Then  </>  (z)  satisfies  the  equations  characteristic  of  doubly-periodic  functions 
of  the  third  kind,  if 

j  0  =  4X,ft)  +  2mr}, 

[k  .  27^^  =  4X,ft)-  -|-  2m7]oi  +  2fxco  +  miri  —  2m7)  {w  +  2a) ; 

=  4Xft)  +  2mr) , 

ft) 

k'.  2Tri  =  4\ft)'-  +  2mriw  -f  2yu-ft)'  +  mivi  —  2mi)  (ft)  -f  2ft)'), 

*  Both  in  this  integral  and  in  the  next,  which  contain  parts  of  the  form  /  — ,  there  is,  as 

in  Prop.  VII.,  §  116,  properly  an  additive  term  of  the  form  Ik-kI,  where  k  is  an  integer.  But, 
as  there,  both  terms  can  be  removed  by  modification  of  the  position  of  the  parallelogram ;  and 
this  modification  is  supposed,  in  the  proof,  to  have  been  made. 


143.]  OF   THE   THIRD    KIND  335 

k  and  k'  being  disposable  integers.     These  are  uniquely  satisfied  by  taking 

_         1   1117] 

with  k  =  0,     k'  =m. 

Assuming  the  last  two,  the  values  of  \  and  jx  are  thus  obtained  so  as  to  make 
^  {z)  a  doubly-periodic  function  of  the  third  kind. 

Now  let  «!,...,  a;  be  chosen  as  the  irreducible  zeros  of  ^  {z)  and  Cj,  ...,Cn 
as  the  irreducible  infinities  of  ^  {z),  which  is  possible  owing  to  the  conditions 
to  which  they  were  subjected.  Then  (^{z)l(^{z)  is  a  doubly-periodic  function 
of  the  first  kind;  it  has  no  zeros  and  no  infinities  in  the  parallelogram  of 
periods  and  therefore  none  in  the  whole  plane ;  it  is  therefore  a  constant,  so 
that 

$ (z)  =  Ae~^^ " '"'"'^^^ "  +(r,+2r,')}mz o-(z  -  a,)  (t(z  -  a,)  ..: ct(z  -  ai) 

(t{z-  Ci)  a{z-  C.2)  ...  cr(z-  Cn)  ' 
a  representation  of  <t>  (z)  in  terms  of  known  quantities. 

Ex.  Had  the  representation  been  effected  by  means  of  the  Jacobian  Theta-Functions 
which  would  replace  <t{z)  by  JI{z),  then  the  term  in  z^  in  the  exponential  would  be  absent. 

144.  No  limitation  on  the  integral  value  of  ni,  except  that  it  must  not 
vanish,  has  been  made :  and  the  form  just  obtained  holds  for  all  values. 
Equivalent  expressions  in  the  form  of  sums  of  functions  can  be  constructed : 
but  there  is  then  a  difference  between  the  cases  of  m  positive  and  m 
negative. 

If  m  be  positive,  being  the  excess  of  the  number  of  iiTeducible  zeros  over 
the  number  of  irreducible  infinities,  the  function  is  said  to  be  of  positive 
class  m ;  it  is  evident  that  there  are  suitable  functions  without  any 
irreducible  infinities — they  are  integral  functions. 

When  771  is  negative  (=  —  01),  the  function  is  said  to  be  of  negative  class  7i ; 
but  there  are  no  corresponding  integral  functions. 

145.  First,  let  m  be  positive. 

i.  If  the  function  have  no  accidental  singularities,  it  can  be  expressed  in 
the  form 

A e^s^'+f'^  a  {z  -  tt:^)  a  (z  -  a^) ...  (T  (z  —  a^), 

with  appropriate  values  of  \  and  yu,. 

ii.  If  the  function  have  n  irreducible  accidental  singularities,  then  it  has 
m-\-n  irreducible  zeros.  We  proceed  to  shew  that  the  function  can  be 
expressed  by  means  of  similar  functions  of  positive  class  m,  with  a  single 
accidental  singularity. 


336  TERTIAKY   FUNCTIONS  [145. 

Using  X,  and  /a  to  denote 

1  mr)         ,    1  niTri  ,         _    ,, 

which  are  the  constants  in  the  exponential  factor  common  to  all  functions  of 
the  same  class,  consider  a  function,  of  positive  class  m  with  a  single  accidental 
singularity,  in  the  form 

o-  (m  —  6i)  cr  (tt  —  62)  . . .  cr  (w  —  &,«+i)  (7  {z  —  u)  ' 
where  61,  h^,  ...,  h^n  ^-^e  arbitrary  constants,  of  sum  s,  and 

m  (ft)  +  2ft)')  =  hm+i  +  h-i_-\-ho+  ...  +  h,n-u 

The  function  -v/r,^  satisfies  the  equations  , 

mvzi 
^m  {Z  +  2ft),  U)  =  l/r,„  (^,  U),       l/r,„  (5  +  2ft)',  U)  =  e       "     -v/r^  (^,  1*)  ; 

regarded  as  a  function  of  z,  it  has  m  for  its  sole  accidental  singularity, 
evidently  simple. 

The  function  -, — ; .   can  be  expressed  in  the  form 

^K(u^-z^)+a<u-z)  0-(u-2)a-(u-b,)  ...O-JLl-  b,rd  (T  {g  -  m  (ft)  +  2ft)')} 

(t(z  —  bi) a(z  —  bm)     cr  {u  —  z  —  s  +  m{(i)  +  2ft)')} ' 

Regarded  as  a  function  of  u,  it  has  z,bi,  ...,b^n  for  zeros  and  z  +  s  —  m{(o  +  2ft)') 
for  its  sole  accidental  singularity,  evidently  simple :  also 

z  +  bi+  ...  +bm.—  {z  +  s  —  m(co  +  2ft)')}  =  m{co  +  2ft)'). 

Hence  owing  to  the  values  of  A,  and  fi,  it  follows  that  — — ^ r,   when 

regarded  as  a  function  of  u,  satisfies  all  the  conditions  that  establish  a 
doubly-periodic  function  of  the  third  kind  of  positive  class  m,  so  that 

1  _        1 

■^m  (Z,  U  +  2ft))        ^|r,n  (z,  t()  ' 

"1  irnrui  1 

=  e 


y\r,n  {z,  u  -I-  2ft)')  yjr,^  (z,  «-)  ' 

and  therefore 

mnui 
-v/r^  (^,  W  +  2ft))  =  n/r^j  (^,  w),.      t/t,^  (^,  It  +  2ft)')  =  e    "     ■s\r,n{z,u). 

Evidently  t/t,^  {z,  u)  regarded  as  a  function  of  u  is  of  negative  class  m :  its 
infinities  and  its  sole  zero  can  at  once  be  seen  from  the  form 

^lr,n  iz,  li)  =  eM--«^)+M(.-«)  <ri^-h)...c7{z-b^)a[u-z-s  +  m{co-\-2ay')] 

a-{ii  — z)a{u  —  b-^)...a-{u  —  bm)o'[s  —  m{a)  +  2(a')]' 

Each  of  the  infinities  is  simple.     In  the  vicinity  of  u  =  z,  the  expansion  of 
the  function  is 

h  positive  integral  powers  of  u  —  z  : 


145.]  OF   POSITIVE   CLASS  337 

and,  in  the  vicinity  of  u  =  hy,  it  is 

-— y-  +  positive  integral  powers  of  u  —  h,-, 
where  Gr(2)  denotes 

(7(b,-bi)...  (T(b,.-br-i)o-(br-br4-i) ...  a-{br-b,n)cr{s-m(o)  +  2o)')][ 
and  is  therefore  an  integral  function  of  z  of  positive  class  m. 

Let  ^(u)  be  a  doubly-periodic  function  of  the  third  kind,  of  positive 
class  m;  and  let  its  irreducible  accidental  singularities,  that  is,  those  which 
occur  in  a  parallelogram  containing  the  point  u,  be  ctj  of  order  1  +/ii,  otg  of 
order  1  +  /^a,  and  so  on.     In  the  immediate  vicinity  of  a  point  or,.,  let 

$  (U)  =(Ar-Br~  +  C,f--...+Mr  P^]  -^—+Pr(u  -  a,). 

Then  proceeding  as  in  the  case*  of  the  secondary  doubly-periodic  functions 
(§  137),  we  construct  a  function 

F{u)  =  ^(u)ylr„,(2,it). 
We  at  once  have  F(u  +  2co)  =  F{u)  =  F(u  +  2co'), 

so  that  F  (u)  is  a  doubly-periodic  function  of  the  first  kind ;  hence  the  sum 
of  its  residues  for  all  the  poles  in  a  parallelogram  of  periods  is  zero. 

For  the  infinities  of  F  (u),  which  arise  through  the  factor  T|r„j  (z,  u),  we 
have  as  the  residue  for  u  =  z 

and  as  the  residue  for  ii  =  b,.,  where  r  =  1,  2,...,vi, 

^(br)Or(z). 

In  the  vicinity  of  a^,  we  have 

■^m (z,  u)  =  -v/r,^ (z,  a.,)  +  {u-  a,.)  yjr.n  {z,  ar)  +     ^  i'"    yfrm"  {z,  a,.)  +  ..., 

where  dashes  imply  differentiation  of  yfr„^  (z,  u)  with  regard  to  u,  after  which 
II  is  made  equal  to  a,.;  so  that  in  ^  (m) -v/r^i,  (2,  it)  the  residue  for  m  =  a,., 
where  r  =  1,2, ...,  is 

Er  (z)  =  Ar^lr^, {z,  cr,.)  -1-  5,i/r„;  {z,  a,)  +  Cr^.n"  (z,  a,)  +  ...+  MryjrJ'^r') (z,  ar).    ■ 

Hence  we  have 

m 
-CI>{Z)+    2    ^(br)Gr(z)+    S    Es{z)  =  0, 
r=l  s=l 

m 

and  therefore  ^  (z)  =  t  Es  (^)  -t-  2  <I>  (6,)  Gr  {z), 

giving  the  expression  of  $  {z)  by  means  of  donbly -periodic  functions  of  the 
third  kind,  which  are  of  positive  class  m  and  either  have  ■  no  accidental 
singularity  or  have  only  one  and  that  a  simple  singularity. 

F.  F.  22 


338  TERTIARY    FUNCTIONS  [145. 

The  m  quantities  h^,  .■■,hm  are  arbitrary;  the  simplest  case  occurs  when 
the  m  zeros  of  ^(z)  are  different  and  are  chosen  as  the  values  of  61,  ...,  6,^. 
The  value  of  <I>  (z)  is  then 

S  =  l 

where  the  summation  extends  to  all  the  irreducible  accidental  singularities ; 
while,  if  there  be  the  further  simplification  that  all  the  accidental  singularities 
are  simple,  then 

^  (2)  =  A.yjr.r, (z,  a,)  +  A^ylr,n  (z,  0^)+  ..., 

the  summation  extending  to  all  the  irreducible  simple  singularities. 
The  quantity  ^m{^,  ^r),  which  is  equal  to 

g;,(^.  _  „,2)+^(,  -  .,)    <T{z-h,)...<T{z-  hm)  CT  {^  +  26  -  m  (a>  +  2oy')  ~  ff,}     ^ 

(7{ar  —  61)...  (7  (a,.  —  hm)  o-[S6  — m(&)  +  2(«')}  cr{z  —  a^)' 

and  is  subsidiary  to  the  construction  of  the  function  E  {z),  is  called  the 
simple  element  of  positive  class  m. 

In  the  general  case,  the  portion 

^<^(br)Gr{z) 

gives  an  integral  function  of  z,  and  the  portion  S  £'«  {z)  gives  a  fractional 

s  =  l 

function  of  z. 

146.  Secondly,  let  m  be  negative  and  equal  to  —  n.  The  equations 
satisfied  by  ^{z)  are 

nwzi 

^  (^  +  2a))  =  4>  {z),         <t>{z+  2&)0  =  e~^  ^  {z), 

and  the  number  of  irreducible  singularities  is  greater  by  n  than  the  number 
of  irreducible  zeros. 

One  expression  for  ^  {z)  is  at  once  obtained  by  forming  its  reciprocal, 
which  satisfies  the  equations 

and  is  therefore  of  the  class  iust  considered :  the  value  of  ^  ,  ,  is  of  the 
form 

lE,{z)  +  ^ArGr(z). 

For  purposes  of  expansion,  however,  this  is  not  a  convenient  form  as  it  gives 
only  the  reciprocal  of  <l>  (z). 

To  represent  the  function,  Appell  constructed  the  element 

%n(^,2/)  =  ^.  2    e  cot — , 


146.]  OF  NEGATIVE  CLASS  339 

which,  since  the  real  part  of  w  jwi  is  positive,  converges  for  all  values  of  z  and 
y,  except  those  for  which 

z.^y  (mod.  2&),  2&)'). 

For  each  of  these  values  one  term  of  the  series,  and  therefore  the  series 
itself,  becomes  infinite  of  the  first  order. 

Evidently  Xn  (^.  i/  +  2a,)  =  ^n  (^>  v), 

_mryi 

Xn  {z,  y  +  2ft)0  =  e    '^  Xn  (z,  y) ; 

therefore  in  the  present  case 

regarded  as  a  function  of  y,  is  a  doubly-periodic  function  of  the  first  kind. 

Hence  the  sum  of  the  residues  of  its  irreducible  accidental  singularities 
is  zero. 

Within  the  parallelogram,  which  includes  z,  these  singularities  are: — 

(i)     y  =  z,  arising  through  %«  {z,  y)  ; 

(ii)     the  singularities  of  <i>  (y),  which  are  at  least  w  in  number,  and  are 
n+  I  in  number  when  4>  has  I  irreducible  zeros. 

The  expansion  of  Xn  {z,  y),  in  powers  oi  y  ~  z,  in  the  vicinity  of  the  point 
z,  is 

h  positive  integral  powers  of  y  —  z\ 

y  ~  ^ 

therefore  the  residue  of  Vl{y)  is 

Let  Or  be  any  irreducible  singularity,  and  in  the  vicinity  of  a^  let  <J>  {y')  denote 

d      ^   d?  ,   T^   dP\      1 


(^'-^'<i.  +  ^'3?  +  -±^' 


dy        ^ dy^      '"         ^ dy^J  y  —  a^ 
+  positive  integral  powers  oi  y  —  a^, 

where    the   series   of  negative  powers  is  finite   because   the  singularity  is 
accidental ;    then  the  residue  of  U,  (y)  is 

^7-Xn  {Z,  dr)  +  BrXn   {z,  O.^)  +  C^%/  {z,  01^)  +  . . .  +  Pr%n'^'  {z,  «,.), 

where  %„<^>  {z,  a^)  is  the  value  of 

d'^Xn  jz,  y) 
dy^ 

when  y  =  ccr  after  differentiation.     Similarly  for  the  residues  of  other  singu- 
larities :  and  so,  as  their  sum  is  zero,  we  have 

a>  (Z)  =  2  {ArXn  (Z,   Or)  +  BrXn  {z,  a,.)  -F  . . .  -f  PrXn'^)  (z,  a,)}, 

the  summation  extending  over  all  the  singularities. 

22—2 


340  TERTIARY  [146. 

The  simplest  case  occurs  when  all  the  N  (>  n)  singularities  a  are  accidental 
and  of  the  first  order ;  the  function  ^  (z)  can  then  be  expressed  in  the  form 

The  quantity  %,i(^,  o),  which  is  equal  to 

is  called  the  simple  element  for  the  expression  of  a  doubly -periodic  function  of 
the  third  kind  of  negative  class  n. 

Hx.     Deduce  the  result 

=     2     (  -  1)«  cot   '     ^ 


TT    sn  u     s=_cc  1         2^ 

147.     The  function  'x^n  (z,  y)  can  be  used  also  as  follows.     Since  %,„  {z,  y), 
qua  function  of  y,  satisfies  the  equations 

%m  (z,  y  +  2co)  =  xm  {z,  y), 

imryi 

Xm.  {z,  y  +  2ft)')  =  e     -    Xm  (^,  y\ 
which  are  the  same  equations  as  are  satisfied  by  a  function  of  y  of  positive 
class  m,  therefore  %,„(«>  z),  which  is  equal  to 

^     Z    e  cot ^ ■-  , 

2ft)s=_a>  ^ft> 

being  a  function  of  z,  satisfies  the  characteristic  equations  of  §  142 ;  and,  in 
the  vicinity  of  z  =  a, 

Xm  («)  z)  = 1-  positive  integral  powers  of  ^  —  of. 

If  then  we  take  the  function  ^  {z)  of  §  14.5,  in  the  case  when  it  has  simple 
singularities  at  Wj,  a,,  ...  and  is  of  positive  class  m,  then 

<|)  {z)  +  A^X^n  («!,  Z)  +  ^2Xm  (O2,  Z)+... 

is  a  function  of  positive  class  m  without  any  singularities:    it  is  therefore 
equal  to  an  integral  function  of  positive  class  m,  say  to  G(z),  where 

G{z)=Ae''^'+''^a-(z-a,)...a(z-a„,), 

so  that  (^(z)  =  G  (2)  -  AjXm  {^1,  z)  -  ^2%7»  (a,,  ^)  -  . . . . 

Ex.  As  a  single  example,  consider  a  function  of  negative  class  2,  and  let  it  have 
no  zero  within  the  parallelogram  of  reference.  Then  for  the  function,  in  the  canonical 
product-form  of  §  14.3,  the  two  irreducible  infinities  are  subject  to  the  relation 

Ci  +  C2  =  2((u  +  2co'), 


and  the  function  is         ^{z)  =  Ke'"        ^^  ' 


(t{z-  Ci)  <t{z  —  C2)  ' 


147.]  FUNCTIONS  341 

The  simple  elements  to  express  4>  {z)  as  a  sum  are 

X2  (2,  ^i)  =  2^  _^  ^  "  ^°*^  2^  (2-  Ci-2sco  ), 


,  TT      1       — '{(«-l)<o'  +  2<0+4<o'-e,}  TT     ,  „  ,      ,        o        'N 

X2  (■^5  ^2)  =  5—  2  e  *"  cot  —  (s  +  Ci  —  2a)  —  4co  -  2sa) ) 

TT     — (Ci-a>')°°      {(r-l)<o'-<:-,}       ^    77    ,  ^      ,, 

=  -—  e  *"  2  e  "  cot  ;r-  (2  +  c,  —  2ra) ) 

2a)  _  00  2a) 

after  an  easy  reduction, 

47rz 

(Ci  -  (0  ) 

=  6"  X2(2,   -Ci). 

The  residue  of  $  (2)  for  Ci ,  which  is  a  simple  singularity,  is 

V.,2-g+2r,+4v)c,  1 


Ji  =  /ire<' 


0-  (Cj  -  C2)  ' 

and  for  C2,  also  a  simple  singularity,  it  is 

o-  (C2  -  Ci) 

,,     ,  ^1  -(Ci-Csi)  — (^1-2(0) 

so  that  — i=-e'»  =_eio 

^2 
Hence  the  expression  for  $  (2)  as  a  sum,  which  is 

^1X2(2.  Cl)  +  ^2X2(^5  C2), 

277i 

becomes  •  A  j  {;t2  (.~,  cj)  -  e  "  '''  X2  (-,  "  ^i)}  ; 

that  is,  it  is  a  constant  multiple  of 

TTJ  -ni 

e    '-'"'x2(3,  Ci)-e<"'"'x2(2,  -Ci). 


a-  (2  — Ci)  o-  (s  +  Ci-  2co-4a)') 
^^  -z2-(— +27,  +  4v)2+2(r,+2r,')(«  +  ei-w-2cu')  1 

=  — Ae"        ^^  /  


0-(s-Ci)o-(2  +  Ci) 


-,    -z- or  (2Ci) 


0-(2-Ci)  0-(2  +  Ci)' 

on  changing  the  constant  factor.     Hence  it  is  possible  to  determine  L  so  that 

■ni  -ni 

$  (2)  =  e    CO  ''^  ;^2  (2,  ci)  -  e'"  '''  X2  (2,  -  Cl)- 
Taking  the  residues  of  the  two  sides  for  2  =  Cj ,  we  have 

-  Ci- Ci  -  —  C\ 

i^Qia  ft)         =  e      <o       J 

and  therefore  finally  we  have 

•         =-    i   /^^'^-^^'^^^'|/^*:^^'?cot,-  (.-O,-2.a)0-.-'"'^^'^%otf  (.  +  C,-2.a)') 
2a)  _oo  (  2a)  iito 

the  right-hand  side  of  which  admits  of  further  modification  if  desired. 


342  PSEUDO-PERIODIC  [147. 

Many  examiDles  of  such  developments  in  trigonometrical  series  are  given  by  Hermite*, 
Biehlert,  Halphen;,  Appell§,  and  Krause||. 

148.  We  shall  not  further  develop  the  theory  of  these  uniform  doubly- 
periodic  functions  of  the  third  kind.  It  will  be  found  in  the  memoirs  of 
Appell|  to  whom  it  is  largely  due;  and  in  the  treatises  of  Halphen**,  and 
of  Rausenbergerff. 

It  need  hardly  be  remarked  that  the  classes  of  uniform  functions  of  a 
single  variable  which  have  been  discussed  form  only  a  small  proportion  of 
functions  reproducing  themselves  save  as  to  a  factor  when  the  variable 
is  subjected  to  homographic  substitutions,  of  which  a  very  special  example 
is  furnished  by  linear  additive  periodicity.  Thus  there  are  the  various 
classes  of  pseudo-automorphic  functions,  (§  305)  called  Thetafuchsian  by 
Poincare,  their  characteristic  equation  being 

for  all  the  substitutions  of  the  group  determining  the  function:  and  other 
classes  are  investigated  in  the  treatises  which  have  just  been  quoted. 

The  following  examples  relate  to  particular  classes  of  pseudo-periodic 
functions. 

Ex.  1.     Shew  that,  if  F{z)  be  a  uniform  function  satisfying  the  equations 

where  6  is  a  primitive  int\\  root  of  unity,  then  F{z)  can  be  expressed  in  the  form 

d  c^" 

dz^ +  ^'^^2^ 

where  /(«)  denotes  the  function 

/       2a)\      ,  -,    /       4a)\  ,       , ,  /       2«to  -  2a)\ 

f(^)+K(--)+'.'f(--)  + +6»-f(= sr-)^ 

and  prove  that  \F  (z)  dz  can  be  expressed   in  the  form  of  a  doubly-periodic   function 
together  with  a  sum  of  logarithms  of  doubly-periodic  functions  with  constant  coefficients. 

(Goursat.) 

*  Comptes  Rendus,  t.  Iv,  (1862),  pp.  11—18. 

+  Sur  les  developpements  en  serie.i  des  fonctions  doublevient  periodiques  de  troisieme  espece, 
(These,  Paris,  Gauthier-Villars,  1879). 

X  Traite  des  fonctions  elliptiques,  t.  i,  chap.  xiii. 

§  Annales  de  VEc.  Norm.  Sup.,  S"'  S^r.,  t.  i,  pp.  135—164,  t.  ii,  pp.  9—36,  t.  iii,  pp.  9—42. 

II   Math.  Ami.,  t.  xxx,  (1887),  pp.  425—436,  516—534. 

**  Traite  des  fonctions  elliptiques,  t.  i,  chap.  xiv. 

ft  Lehrbuch  der  Theorie  der  periodischen  Functionen,  (Leipzig,  Teubuer,  1884),  where  further 
references  are  given. 


2  [A^  +  A^^-  + +  ^»i)/(^-«), 


148.]  "  FUNCTIONS  343 

Ex.  2.     Sliew  that,  if  a  pseudo-periodic  function  be  defined  by  the  equations 

f{z  +  2ay)=fiz)+\, 
f(z  +  2a,')=f{z)  +  \', 
and  if,  in  the  parallelogram  of  periods  containing  the  point  z,  it  have  infinities  c,  ...  such 
that  in  their  immediate  vicinity 


then  f{z)  can  be  expi'essed  in  the  form 

^%^,+.{c,+^,|+ +<^...Sf(-«), 

the  summation  extending  over  all  the  infinities  of  f{z)  in  the  above  parallelogram  of 
periods,  and  the  constants  Cj,  ...  being  subject  to  the  condition 

+  ^7r  2  Cj  =  Xco'  —  X'o). 
Deduce   an  expression   for  a  doubly-periodic   function    ^  (3)    of  the   third   kind,    by 
assuming 

/(.)  =  |1|.  (Halphen.) 

Ex.  3.     If  S{z)  be  a  given  doubly-periodic  function  of  the  first  kind,  then  a  pseudo- 
periodic  function  F  {z),  which  satisfies  the  equations 

F{z  +  2c.)  =  F{z), 

niriz 

F(z  +  2a>')  =  e~  S{z)F{z), 
where  71  is  an  integer,  can  be  expressed  in  the  form 

F{z)  =  Aei^sJzj+^r'^''^'', 
where  A  is  a  constant  and  tt  {3)  denotes 


iTT  \    '         '  dz         ''' dz^ 

the  summation  extending  over  all  points  h^  and  the  constants  B,.  being  subject  to  the 
relation 

25,=  -^. 

Explain  how  the  constants  6,  G  and  B  can  be  determined.  (Picard.) 

Ex.  4.     Shew  that  the  function  F  {£)  defined  by  the  equation 

F{^Z)=    "2"   Z2«+1(1_22")2^ 
n=  — CO 

for  values  of  k|,  which  are  <  1,  satisfies  the  equation 

F{f)  =  F{z)  ; 

and  that  the  function  F^  {x)  =     2         ,  g/   %     > 

where  ^(^)  =  ^3_  1^  and  4>n{x),  for  positive  and  negative  values  of  n,  denotes  ^[(^{(^...(^(.^)}], 
(^  being  repeated  n  times,  and  a  is  the  positive  root  of  a^ -a  —  1  =0  ;  satisfies  the  equation 

F,{x^-\)  =  F^{oo) 
for  real  values  of  the  variable. 

Discuss  the  convergence  of  the  series  which  defines  the  function  F^  {x).     (Appell.) 


CHAPTER  XIII. 

Functions  possessing  an  Algebraical  Addition-Theorem. 

149.  We  may  consider  at  this  stage  an  interesting  set*  of  important 
theorems,  due  to  Weierstrass,  which  are  a  justification,  if  any  be  necessary, 
for  the  attention  ordinarily  (and  naturally)  paid  to  functions  belonging  to 
the  three  simplest  classes  of  algebraic,  simply-periodic,  and  doubly-periodic, 
functions. 

A  function  <fi  (u)  is  said  to  possess  an  algebraical  addition-theorem,  when 
among  the  three  values  of  the  function  for  arguments  u,  v,  and  u  +  v,  where  u 
and  V  are  general  and  not  merely  special  arguments,  an  algebraical  equation 
exists f  having  its  coefficients  independent  of  u  and  v. 

150.  It  is  easy  to  see,  from  one  or  two  examples,  that  the  function  does 
not  need  to  be  a  uniform  function  of  the  argument.  The  possibility  of 
multiformity  is  established  in  the  following  proposition: — 

A  function  defined  hy  an  algebraical  equation,  the  coefiicients  of  which  are 
rational  functions  of  the  argument,  or  are  uniform  simply -periodic  functions 
of  the  argument,  or  are  uniform  doubly-periodic  functions  of  the  argument, 
possesses  an  algebraical  addition-theorem. 

*  They  are  placed  in  the  forefront  of  Schwarz's  account  of  Weierstrass's  theory  of  elliptic 
functions,  as  contained  in  the  Formeln  und  Lehrsatze  zum  Gehrauchc  cler  clliptischen  Functionen ; 
but  they  are  there  stated  (§§  1 — 3)  without  proof.  The  only  proof  that  has  appeared  is  in  a 
memoir  by  Phragmen,  Acta  Math.,  t.  vii,  (1885),  pp.  33—42  ;  and  there  are  some  statements 
(pp.  390 — 393)  in  Biermann's  Theorie  der  analytischen  Functionen  relative  to  the  theorems. 
The  proof  adopted  in  the  text  does  not  coincide  with  that  given  by  Phragmen. 

t  There  are  functions  which  possess  a  kind  of  algebraical  addition-theorem ;  thus,  for 
instance,  the  Jacobian  Theta-f  unctions  are  such  that  Q^  (u  +  v)  Q^{u-v)  can  be  rationally 
expressed  in  terms  of  the  Theta-functions  having  u  and  v  for  their  arguments.  Such  functions 
are,  however,  naturally  excluded  from  the  class  of  functions  indicated  in  the  definition. 

Such  functions,  however,  possess  what  may  be  called  a  multiplication-theorem  for  multipli- 
cation of  the  argument  by  an  integer,  that  is,  the  set  of  functions  0  [mu)  can  be  expressed 
algebraically  in  terms  of  the  set  of  functions  e(u).  This  is  an  extremely  special  case  of  a  set 
of  transcendental  functions  having  a  multiplication-theorem,  which  are  investigated  by  Poincare, 
Liouville,  4"^  Ser.,  t.  iv,  (1890),  pp.  313—365. 


150.]  FUNCTIONS   POSSESSING   AN    ADDITION-THEOREM  345 

First,  let  the  coefficients  be  rational  functions  of  the  argument  u.  If 
the  function  defined  by  the  equation  be    U,  we  have 

W^g,  (u)  +  W^-^g,  {u)  +...+g,n  {u)  =  0, 

where  go{u),  g^{u),  ...,^^(u)  are  rational  integral  functions  of  u  of  degree, 
say,  not  higher  than  n.     The  equation  can  be  transformed  into 

u-fo  ( U)  +  u--^f,  (U)+...+f],(U)  =  0, 

where  fl ( U),  fi ( U),  ...,  fn(U)  are  rational  integral  functions  of  U  of  degree 
not  higher  than  m. 

Let  V  denote  the  function  when  the  argument  is  v,  and  W  denote  it 
when  the  argument  is  u  +  v;  then 

^"/o {V)+  v»-\f\  ( F)  +  . . .  +/;  ( F)  =  0, 

and  ( a  +  vy\f,  {W)  +  {u+  vr-\t\  (  Tf )  +  ...+/;  ( W)  =  0. 

The  algebraical  elimination  of  the  two  quantities  u  and  v  between  these 

three    equations    leads    to    an  algebraical   equation  between   the  quantities 

f{U),  f{V)  and  f{W),  that  is,  to  an  algebraical  equation  between  U,  V,  W, 

say  of  the  form 

G{U,  V,  Tf)  =  0, 

where  G  denotes  a  polynomial  function,  with  coefficients  independent  of 
u  and  V.  It  is  easy  to  prove  that  G  is  symmetrical  in  U  and  V,  and  that 
its  degree  in  each  of  the  three  quantities  U,  V,  W  is  mn'^.  The  equation 
G  =  0  implies  that  the  function  U  possesses  an  algebraical  addition-theorem. 

Secondly,  let  the  coefficients*  be  uniform  simply-periodic  functions  of 
the  argument  u.     Let  (o  denote  the  period :   then,  by  §  113,  each  of  these 

functions    is   a   rational   function   of  tan  —  .      Let   u'  denote  tan  —  ;  then 

CO  0) 

the  equation  is  of  the  form 

U'^g,  (u')  +  U^-'g,  (u')  +  ...+g,n  ("0  =  0, 

where  the  coefficients  g  are  rational  (and  can  be  taken  as  integral)  functions 
of  u.  If  p  be  the  highest  degree  of  ic  in  any  of  them,  then  the  equation 
can  be  transformed  into 

u'pfo ( U)  +  u'P-\t\  (U)  +  ...+fpiU)  =  0, 

where  /„  ( U),  fiiU),  ...,  fp{U)  are  rational  integral  functions  of  U  of  degree 
not  higher  than  m. 

*  The  limitation  to  uniformity  for  the  coefficients  has  been  introduced  merely  to  make  the 
illustration  simpler;  if  in  any  case  they  were  multiform,  the  equation  would  be  replaced  by 
another  which  is  equivalent  to  all  possible  forms  of  the  first  arising  through  the  (finite) 
multiformity  of  the  coefficients :  and  the  new  equation  would  conform  to  the  specified 
conditions. 


846  FUNCTIONS    POSSESSING  [150. 

Let  v  denote  tan  —  ,  and  w'  denote  tan''^^^ ;  then  the  corresponding 

values  of  the  function  are  determined  by  the  equations 

v'^f.{V)  +  v'P-^f,{V)+...+f,{V)  =  Q, 
and  w'i^fo {W)  +  w'P-'f, (  PT)  +  . . .  +/^ ( F)  =  0. 

The  relation  between  v',  v ,  w'  is 

u'v'w'  +  u'  +  v'  —  w  =0. 
The  elimination  of  the  three  quantities  u',  v',  w   among  the  four  equations 
leads  as  before  to  an  algebraical  equation 

G{JJ,  V,  W)  =  0, 
where  G  denotes  a  polynomial  function  (now  of  degree  mp^)  with  coefficients 
independent  of  u  and  v.     The  function  U  therefore  possesses  an  algebraical 
addition-theorem. 

Thirdly,  let  the  coefficients  be  uniform  doubly-periodic  functions  of  the 
argument  u.  Let  w  and  co'  be  the  two  periods ;  and  let  g?  (u),  the  Weier- 
strassian  elliptic  function  in  those  periods,  be  denoted  by  |.  Then  every 
coefficient  can  be  expressed  in  the  form 

L 

where  L,  M,  N  are  rational  integral  functions  of  f  of  finite  degree.  Unless 
each  of  the  quantities  N  is  zero,  the  form  of  the  equation  when  these  values 
are  substituted  for  the  coefficients  is 

A  +  B^y  {u)  =  0, 
so  that  ^2^52(4^-^0.1^-^3); 

and  this  is  of  the  form 

t^^-^o  (f )  +  U-^^-'g.  {B+-+  9^>n  (I)  =  0, 

where  the  coefficients  g  are  rational  (and  can  be  taken  as  integral)  functions 
of  ^.  If  q  be  the  highest  degree  of  ^  in  any  of  them,  the  equation  can  be 
transformed  into 

where  the  coefficients  f  are  rational  integral  functions  of  U  of  degree  not 
higher  than  27w. 

Let  r)  denote  p  (v)  and  ^  denote  p  (u  +  v)  ;  then  the  corresponding  values 
of  the  function  are  determined  by  the  equations 

v'fo(V)+v'^-'f^{V)+ +f,(V)  =  0, 

and  ^Vo{W)+i:^-\fAW)+ +f^{W)=0. 

By  using  Ex.  4,  §  131,  it  is  easy  to  shew  that  the  relation  between  ^,  rj,  ^  is 

^(i  (^  +  V  +  OH^ -  vf  -  8 {^  +  V  +  0  {4^(e  +  r)- g.{^  +  V) -  ^9s} 

+  (4p  +  4|^77  +  4r7-  -  g.y  =  0. 


150.]  AN    ADDITION- THEOREM  347 

The  elimination  of  |,  77,  f  from  the  three  equations  leads  as  before  to 

an  alsrebraical  equation 

G{U,  V,  W)=0, 

of  finite  degree  and  Avith  coefficients  independent  of  u  and  v.     Therefore  in 
this  case  also  the  function  U  possesses  an  algebraical  addition-theorem. 

If,  however,  all  the  quantities  N  be  zero,  the  equation  defining  U  is  of 
the  form 

t/-/Ao  (?)  +  U'-'K  (I)  +  •  • .  +  /'m  (I)  =  0  ; 
and  a  similar  argument  then  leads   to   the  inference   that    U  possesses  an 
algebraical  addition-theorem. 

The  proposition  is  thus  completely  established. 

151.  The  generalised  converse  of  the  preceding  proposition  now  suggests 
itself:  what  are  the  classes  of  functions  of  one  variable  that  possess  an 
algebraical  addition-theorem  ?  The  solution  is  contained  in  Weierstrass's 
theorem  : — 

An  analytical  function  (f)  (»),  icJiich  possesses  an  algebraical  addition- 
theorem,  is  either 

(i)     an  algebraic  function   of  u  ;   or 

(ii)    an    algebraic   function    of    e  '^  ,    where    w    is    a    suitably    chosen 

constant;  or 
(iii)   an  algebraic  function  of  the  elliptic  function  (^  {u),  the  periods — or 
the  invariants  g^  and  g^ — being  suitably  chosen  constants. 

Let  U  denote  (/>  {u). 

For  a  given  general  value  of  u,  the  function  U  may  have  m  values  where, 
for  functions  in  general,  there  is  nob  a  necessary  limit  to  the  value  of  m  ;  it 
will  be  proved  that,  when  the  function  possesses  an  algebraical  addition- 
theorem,  the  integer  m  must  be  finite. 

For  a  given  general  value  of  U,  that  is,  a  value  of  U  when  its  argument 
is  not  in  the  immediate  vicinity  of  a  branch-point  if  there  be  branch-points, 
the  variable  u  may  have  p  values,  where  p  may  be  finite  or  may  be  infinite. 

Similarly  for  given  general  values  of  v  and  of  F,  which  will  be  used  to 
denote  0  {v). 

First,  let  p  be  finite.  Then  because  u  has  p  values  for  a  given  value 
of  U  and  v  has  p  values  for  a  given  value  of  V,  and  since  neither  set  is 
affected  by  the  value  of  the  other  function,  the  sum  u  +  v  has  p"  values 
because  any  member  of  the  set  m  can  be  combined  with  any  member  of  the 
set  v;  and  this  number  p"-  of  values  of  u-\-v  is  derived  for  a  given  value 
of  U  and  a  given  value  of  V. 

Now  in  forming  the  function  (^{u  +  v),  which  will  be  denoted  by  W,  we 
have  m  values  of  W  for  each  value  oi  u  +  v  and  therefore  we  have  mp""  values 


348  WEIERSTRASS   ON    FUNCTIONS  [151. 

of  W  for  the  whole  set,  that  is,  for  a  given  value  of  U  and  a  given  value  of  V. 
Hence  the  equation  between  U,  V,  W  is  of  degree*  mp'  in  W,  necessarily 
finite  when  the  equation  is  algebraical ;  and  therefore  m  is  finite. 

Because  in  is  finite,  U  has  a  finite  number  m  of  values  for  a  given  value 
of  u ;  and,  because  p  is  finite,  u  has  a  finite  number  p  of  values  for  a  given 
value  of  U.  Hence  U  is  determined  in  terms  of  u  by  an  algebraical  equation 
of  degree  m,  the  coefficients  of  which  are  rational  integral  functions  of 
degree  p ;  and  therefore   U  is  an  algebraic  function  of  u. 

152.  Next,  let  p  be  infinite  ;  then  (see  Note,  p.  350)  the  system  of  values 
may  be  composed  of  (i)  a  single  simply-infinite  series  of  values  or  (ii)  a  finite 
number  of  simply-infinite  series  of  values  or  (iii)  a  simply-infinite  number  of 
simply-infinite  series  of  values,  say,  a  single  doubly-infinite  series  of  values 
or  (iv)  a  finite  number  of  doubly-infinite  series  of  values  or  (v)  an  infinite 
number  of  doubly-infinite  series  of  values :  where,  in  (v),  the  infinite  number 
is  not  restricted  to  be  simply-infinite. 

Taking  these  alternatives  in  order,  we  first  consider  the  case  where  the 
p  values  of  u  for  a  given  general  value  of  U  constitute  a  single  simply -infinite 
series.  They  may  be  denoted  by  f{u,  n),  where  n  has  a  simply-infinite  series 
of  values  and  the  form  of  /  is  such  that  f{u,  0)  =■  u. 

Similarly,  the  p  values  of  v  for  a  given  general  value  of  V  may  be  denoted 
by  f(v,  n'),  where  n  has  a  simply-infinite  series  of  values.  Then  the  different 
values  of  the  argument  for  the  function  W  are  the  set  of  values  given  by 

f{u,  n)  +  f{v,  n), 
for  the  simply-infinite  series  of  values  for  n  and  the  similar  series  of  values 
for  n'. 

The  values  thus  obtained  as  arguments  of  W  must  all  be  contained  in 

the  series  f{u+v,  n"),  where  n"  has  a  simply-infinite  series  of  values  ;  and, 

in  the  present  case,  f{u  +  v,  n")  cannot   contain   other   values.      Hence   for 

some  values  of  n  and  some  values  of  n,  the  total  aggregate  being  not  finite, 

the  equation 

f{u,  n)  +  f{v,  n')=f{u  -F  v,  n") 

must  hold,  for  continuously  varying  values  of  u  and  v. 

In  the  first  place,  an  interchange  of  u  and  v  is  equivalent  to  an  inter- 
change of  n  and   n    on  the   left-hand  side;   hence  n"  is   symmetrical  in  n 

and  n'.     Again,  we  have 

9/  {u,  n)  _  df(:Li  +  v,  n") 
du       ~      d(u  +  v) 
_  df(v,  n) 
dv 
*  The  degree  for  special  functions  may  be  reduced,  as  in  Cor.  1,  Prop.  XIII.,  §  118 ;  but  in  no 
case  is  it  increased.     Similarly  modifications,  in  the  way  of  finite  reductions,  may  occur  in  the 
succeeding  cases ;  but  they  will  not  be  noticed,  as  they  do  not  give  rise  to  essential  modification 
in  the  reasoning. 


152.]  POSSESSING   AN    ADDITION-THEOREM  349 

SO  that  the  form  of  f(ii,  n)  is  such  that  its  first  derivative  with  regard  to  u  is 
independent  of  u.  Let  6  {n)  be  this  value,  where  6  (n),  independent  of  ^t,  may- 
be dependent  on  n ;  then,  since 

ou 
we  have  f{^>'>  '0  —  ^''^  ('0  +  "^  0^)> 

'\lr(n)  being  independent  of  u.  Substituting  this  expression  in  the  former 
equation,  we  have  the  equation 

ae  (n)  +  y^r  (n)  +  vO  {n)  +  y\r  (nf)  =  {u  +v)e  (?i")  +  yjr  (n"), 

which  must  be  true  for  all  values  of  u  and  v ;  hence 

d{n)  =  e{n")  =  e(n'), 

so  that  6  (n)  is  a  constant  and  equal  to  its  value  when  oi  =  0.  But  when  7i  is 
zero,  f{ti,  0)  is  u ;  so  that  ^ (0)  =  1  and  i/r  (0)  —  0,  and  therefore 

f(u,  n)  =  u  +  yfr^n), 
where  yfr  vanishes  with  n. 

The  equation  defining  yfr  is 

-v/r(??)  +  -v|r(;i')  =  -»|r(7i"); 

for  values  of  ?i  from  a  singly-infinite  series  and  for  values  of  n  from  the  same 
series,  that  series  is  reproduced  for  n".     Since  1/^(7?)  vanishes  with  n,  we  take 

and  therefore  n^  (n)  +  nx  (^^')  =  '^"%  (^^")- 

Again,  when  n  vanishes,  the  required  series  of  values  of  n"  is  given  by  taking 

n"  =  n ;  and,  when  n  does  not  vanish,  n"  is  symmetrical  in  n  and  n ,  so  that 

we  have 

n"  =  n  +  n  -\-  nn'X, 

where  A.  is  not  infinite  for  zero  or  finite  values  of  n  or  n'.     Thus 
n^  (?0  +  y^'x  (^0  =  0^  +  *^'  +  nn'X)  x  (n  +  n  +  nn'X). 
Since  the  left-hand  side  is  the  sum  of  two  functions  of  distinct  and  inde- 
pendent magTiitudes,  the  form  of  the  equation  shews  that  it  can  be  satisfied 

only  if 

\^0,  so  that  n"  =n  +  n' ; 

and  ■  X  ('0  =  %  (jO 

=  X  ('^')' 
so  that  each  is  a  constant,  say  co ;  then 

/  (m,  ?? )  =  u  +  ncD, 
which  is  the  form  that  the  series  must  adopt  when  the  series  f(u  +  v,  n")  is 
obtained  by  the  addition  of /(w,  n)  and  f(v,  n). 


350  WEIERSTRASS   ON    FUNCTIONS  [152. 

It  follows  at  once  that  the  single  series  of  arguments  for  W  is  obtained, 
as  one  simply-infinite  series,  of  the  form  u  +  v  +  n"w.  For  "each  of  these 
arguments  we  have  m  values  of  W,  and  the  set  of  m  values  of  W  is 
the  same  for  all  the  different  arguments;  that  is,  W  has  m  values  for  a 
given  value  of  U  and  a  given  value  of  V.  Moreover,  U  has  m  values  for  each 
argument  and  likewise  F;  hence,  as  the  equation  between  U,  V,  W  is  of 
a  degree  that  is  necessarily  finite  because  the  equation  is  algebraical,  the 
integer  m  is  finite. 

It  thus  appears  that  the  function  U  has  a  finite  number  m  of  values  for 
each  value  of  the  argument  u,  and  that  for  a  given  value  of  the  function  the 
values  of  the  argument  form  a  simply-periodic  series  represented  by  u  +  nw. 

But  the  function  tan  —  is  such  that,  for  a  given  value,  the  values  of  the 
ft) 

argument  are  represented    by  the   series  u  +  noo ;    hence  for  each  value  of 
tan  —  there  are  m  values  of  U,  and  for  each  value  of  U  there  is  one  value 


of  tan  — .  It  therefore  follows,  by  §§  113,  114,  that  between  U  and  tan  — 
there  is  an  algebraical  relation  which  is  of  the  first  degree  in  tan  —  and  the 
mth  degree  in  U,  that  is,  U  is  an  algebraic  function  of  tan  — .     Hence  U  is 


an  algebraic  function  also  of  e  '"  . 

Note.  This  result  is  based  upon  the  supposition  that  the  series  of  argu- 
ments, for  which  a  branch  of  the  function  has  the  same  value,  can  be  arranged 
in  the  form/(w,  n),  where  n  has  a  simply-infinite  series  of  integral  values.  If, 
however,  there  were  no  possible  law  of  this  kind — the  foregoing  proof  shews 
that,  if  there  be  one  such  law,  there  is  only  one  such  law,  with  a  properly 
determined  constant  to — then  the  values  would  be  represented  hj  u^,  U2, . . . ,  Up 
with  p  infinite  in  the  limit.  In  that  case,  there  would  be  an  infinite  number  of 
sets  of  values  for  u  +  v  of  the  type  u^  +  v^,  where  X,  and  fi  might  be  the  same 
or  might  be  different ;  each  set  would  give  a  branch  of  the  function  W,  and 
then  there  would  be  an  infinite  number  of  values  of  W  corresponding  to  one 
branch  of  U  and  one  branch  of  V.  The  equation  between  U,  V  and  W  would 
be  of  infinite  degree  in  W,  that  is,  it  would  be  transcendental  and  not  alge- 
braical. The  case  is  excluded  by  the  hypothesis  that  the  addition-theorem  is 
algebraical,  and  therefore  the  equation  between  U,  V  and  W  is  algebraical. 

153.  Next,  let  there  be  a  number  of  simply-infinite  series  of  values  of 
the  argument  of  the  function,  say  q,  where  q  is  greater  than  unity  and 
may  be  either  finite  or  infinite.  Let  ti^,  ih,  ..-,  Uq  denote  typical  members 
of  each  series. 

Then  all  the  members  of  the  series  containing  Uj  must  be  of  the  form 
y;  (^<,l,  n),  for  an  infinite  series  of  values  of  the  integer  n.    Otherwise,  as  in  the 


153.]  POSSESSING    AN    ADDITION-THEOREM  351 

preceding  Note,  the  sum  of  the  values  in  the  series  of  arguments  a  and  of 
those  in  the  same  series  of  arguments  v  would  lead  to  an  infinite  number  of 
distinct  series  of  values  of  the  argument  u  +  v,  with  a  corresponding  infinite 
number  of  values  W;  and  the  relation  between  U,  V,  W  would  cease  to  be 
algebraical. 

In  the  same  way,  the  members  of  the  corresponding  series  containing  v^ 
must  be  of  the  form  fi{v^,  n)  for  an  infinite  series  of  values  of  the  integer  n'. 
Among  the  combinations 

/i(«i,  n)+/i(yi>  n'y 

the  simply-infinite  series  /^  (Wj  +  v-^,  n")  must  occur  for  an  infinite  series 
of  values  of  n" ;   and  therefore,  as  in  the  preceding  case, 

y*!  (ttj ,  n)  =  t(i  -I-  ?i(«i , 

where  oo^  is  an  appropriate  constant.  Further,  there  is  only  one  series  of 
values  for  the  combination  of  these  two  series ;    it ,  is  represented  by 

^<l  +  i\  -\-  }i"oi-^. 
In  the  same  way,  the  members  of  the  series  containing  u^  can  be  repre- 
sented in  the  form  Wg  +  nw.2,  where  ajg  is  an  appropriate  constant,. which  may 
be  (but  is  not  necessarily)  the  same  as  coj ;  and  the  series  containing  u^,, 
when  combined  with  the  set  containing  V2,  leads  to  only  a  single  series 
represented  in  the  form  1/04  v^  +  n'w^.     And  so  on,  for  all  the  series  in  order. 

But  now,  since  a^  +  nhwc^  where  m^  is  an  integer,  is  a  value  of  u  for  a  given 
value  of  U,  it  follows  that  U (u.,  +  m2co.,)  =  U  (iio)  identically,  each  being  equal 
to  U.     Hence 

U  («i  +  m^fOj^  +  1112(02)  =  U  (u^  4-  ?"i&)i)  =  U  (ill)  =  U, 

and  therefore  Ui  +  nijCOi  +  m^coz  is  also  a  value  of  u  for  the  given  value  of  U, 
leading  to  a  series  of  arguments  which  must  be  included  among  the  original 
series  or  be  distributed  through  them.  Similarly  u^  +  2m,.Wy,  where  the 
coefficients  m  are  integers  and  the  constants  &>  are  properly  determined, 
represents  a  series  of  values  of  the  variable  u,  included  among  the  original 
series  or  distributed  through  them.  And  generally,  when  accouut  is  taken  of 
all  the  distinct  series  thus  obtained,  the  aggregate  of  values  of  the  variable  u 
can  be  represented  in  the  form  Wa  +  Swi^o)^,  for  X  =  1,  2,  ...,«,  where  k  is 
some  finite  or  infinite  integer. 

Three  cases  arise,  (a)  when  the  quantities  w  are  equal  to  one  another  or 
can  be  expressed  as  integral  multiples  of  only  one  quantity  co,  (b)  when  the 
quantities  o)  are  equivalent  to  two  quantities  fi^  and  Xlg  (the  ratio  of  which  is 
not  real),  so  that  each  quantity  co  can  be  expressed  in  the  form 

(Or  =  Pir^i  +  P2r^2, 

the  coefficients  p^,.,  p2r  being  finite  integers ;  (c)  when  the  quantities  co  are 
not  equivalent  to  only  two  quantities,  such  as  Xli  and  Og- 


352  FUNCTIONS   POSSESSING   AN  [153. 

For  case  (a),  each  of  the  k  infinite  series  of  values  u  can  be  expressed 
in  the  form  W;^  +  pco,  for  X  =  1,  2,  ...,  /c  and  integral  values  of  j9. 

First,  let  k  be  finite,  so  that  the  original  integer  q  is  finite.  Then  the 
values  of  the  argument  for  W  are  of  the  type 

Uf,  -\-  pw  -\rV^  +  p'o3, 

that  is,  U),  ^-v^-\r  p"o), 

for  all  combinations  of  A,  and  /x  and  for  integral  values  of  p".  There  are  thus 
k'^  series  of  values,  each  series  containing  a  simply-infinite  number  of  terms 
of  this  type. 

For  each  of  the  arguments  in  any  one  of  these  infinite  series,  W  has  m 
values ;  and  the  set  of  m  values  is  the  same  for  all  the  arguments  in  one  and 
the  same  infinite  series.  Hence  W  has  mK^  values  for  all  the  arguments  in 
all  the  series  taken  together,  that  is,  for  a  given  value  of  U  and  a  given 
value  of  V.  The  relation  between  U,  V,  W  is  therefore  of  degree  m/c^, 
necessarily  finite  when  the  equation  is  algebraical ;    hence  m  is  finite. 

It  thus  appears  that  the  function  U  has  a  finite  number  m  of  values  for 
each  value  of  the  argument  u,  and  that  for  a  given  value  of  the  function  there 
are  a  finite  number  k  of  distinct  series  of  values  of  the  argument  of  the  form 

2i  +  p(o,  (o  being  the  same  for  all  the  series.     But  the  function  tan —  has 

one  value  for  each  value  of  u,  and  the  series  u  +  pw  represents  the  series  of 

ITU' 

values  of  w  for  a  given  value  of  tan  —  .     It  therefore  follows  that  there  are 

°  CO 

TTti  TTU 

m  values  of  U  for  each  value  of  tan  —  and  that  there  are  k  values  of  tan  — 

ft)  &) 

for  each  value  of  U ;  and  therefore  there  is  an  algebraical  relation  between 

U  and  tan  —  ,  which  is  of  degree  k  in  the  latter  and  of  degree  m  in  the 
w 

inu 
TTIC  

former.    Hence  U  is  an  algebraic  function  of  tan  —  and  therefore  also  of  e  "'  . 

Next,  let  K  be  infinite,  so  that  the  original  integer  q  is  infinite.  Then, 
as  in  the  Note  in  §  152,  the  equation  between  U,  V,  W  will  cease  to  be 
algebraical  unless  each  aggregate  of  values  u^  +  pco,  for  each  particular 
value  of  p  and  for  the  infinite  sequence  X=  1,  2,  ...,  k,  can  be  arranged  in  a 
system  or  a  set  of  systems,  say  a  in  number,  each  of  the  form  fp{u  +  pco,  pp) 
for  an  infinite  series  of  values  of  p^.  Each  of  these  implies  a  series  of  values 
fp(v  +p'co,  pp)  of  the  argument  of  V  for  the  same  series  of  values  of  pp  as  of 
Pp,  and  also  a  series  of  values  fp(it  -h  v  +  p"co,  pp")  of  the  argument  of  W  for 
the  same  series  of  values  of  pj'.     By  proceeding  as  in  §  152,  it  follows  that 

fp  {u  +  pco,  Pp)  =  a  +po3+  PpCOp  , 

where  cOp  is  an  appropriate  constant,  the  ratio  of  which  to  co  can  be  proved 


153.]  ALGEBRAICAL   ADDITION-THEOREM  353 

(as  in  §  106)  to  be  not  purely  real,  and  p^  has  a  simply-infinite  succession  of 
values.     The  integer  a  may  be  finite  or  it  may  be  infinite. 

When  CO  and  all  the  constants  to'  which  thus  arise  are  linearly  equivalent 
to  two  quantities  flj  and  Clor  so  that  the  terms  additive  to  it  can  be  expressed 
in  the  form  s-^Hi  +  531^2,  then  the  aggregate  of  values  -u  can  be  expressed 
in  the  form 

Up  +  pT^ni+p^n^, 

for  a  simply -infinite  series  for  pi  and  for  po',  and  p  has  a  series  of  values 
1,  2,  ...,o-.     This  case  is,  in  effect,  the  same  as  case  (h). 

When  o)  and  all  the  constants  co'  are  not  linearly  equivalent  to  only 
two  quantities,  such  as  Oj  and  fig)  we  have  a  case  which,  in  effect,  is  the 
same  as  case  (c). 

These  two  cases  must  therefore  now  be  considered. 

For  case  (h),  either  as  originally  obtained  or  as  derived  through  part 
of  case  (a),  each  of  the  (doubly)  infinite  series  of  values  of  u  can  be  expressed 
in  the  form 

u^^+p^n^  +  p^no^, 

for  X  =  1,  2,  ...,  a,  and  for  integral  values  of  p^  and  p.^.  The  integer  a  may  be 
finite  or  infinite ;  the  original  integer  q  is  infinite. 

First,  let  o-  be  finite.  Then  the  values  of  the  argument  for  W  are  of  the 
type 

Ul^  +  p^n^  -f  P2^-2  +   I'm  +  Pl^l   +  P2'^2, 

that  is,  Ux  +  v^  +  pi'D,^  +  p^'^^, 

for  all  combinations  of  A,  and  /x,,  and  for  integral  values  of  ja/'  and  p^'.  There 
are  thus  o--  series  of  values,  each  series  containing  a  doubly-infinite  number  of 
terms  of  this  type. 

For  every  argument  there  are  m  values  of  W ;  and  the  set  of  m  values  is 
the  same  for  all  the  arguments  in  one  and  the  same  infinite  series.  Thus  W 
has  m<j^  values  for  all  the  arguments  in  all  the  series,  that  is,  for  a  given  value 
of  U  and  a  given  value  of  F;  and  it  follows,  as  before,  from  the  consideration 
of  the  algebraical  relation,  that  m  is  finite. 

The  function  U  thus  has  m  values  for  each  value  of  the  argument  u ;  and 
for  a  given  value  of  the  function  there  are  a  series  of  values  of  the  argument, 
each  series  being  of  the  form  u^  -\-  p■S^■^  +  p^^^-^- 

Take  a  doubly-periodic  function  @  having  fli  and  Og  for  its  periods,  such  * 
that  for  a  given  value  of  0  the  values  of  its  arguments  are  of  the  foregoing 
form.     Whatever  be  the  expression  of  the    function,  it   is  of  the   order  a. 

*  All  that  is  necessary  for  this  purpose  is  to  construct,  by  the  use  of  Prop.  XII.,  §  118,  a 
function  having,  as  its  irreducible  simple  infinities,  a  series  of  points  a-^,  a^,  ...,  a^ — special 
values  of  Wj,  u^,  ...,  Uu — m  the  parallelogram  of  periods,  chosen  so  that  no  two  of  the  a- 
points  a  coincide. 

¥.  F.  .   23 


354  FUNCTIONS   POSSESSING   AN  [153. 

Then  U  has  m  values  for  each  vahie  of  @,  and  ©  has  one  value  for  each 
value  of  U;  hence  there  is  an  algebraical  equation  between  U  and  @,  of 
the  first  degree  in  the  latter  and  of  the  inth.  degree  in  U:  that  is,  U  is  an 
algebraic  function  of  ©.  But,  by  Prop.  XV.  §  119,  @  can  be  expressed  in 
the  form 

M  +  N^o  (u) 
T         ' 

where  L,  M,  N  are  rational  integral  functions  of  p  (u),  if  Clj  and  fig  be  the 
periods  of  'p  {u) ;  and  ^o' (u)  is  a  two-valued  algebraic  function  of  gj (u), 
so  that  @  is  an  algebraic  function  of  ^j(u).  Hence  also  U  is  an  algebraic 
function  of  (p  (ii),  the  periods  of  (^  (u)  being  jwoperly  chosen. 

This  inference  requires  that  a,  the  order  of  @,  be  greater  thanl. 
Because  U  has  m  values  for  an  argument  u,  the  symmetric  function  "^U 
has  one  value  for  an  argument  u  and  it  is  therefore  a  uniform  function. 
But  each  term  of  the  sura  has  the  same  value  for  a -\- p-^Q.^ -^ pSlo  as  for 
u ;  and  therefore  this  uniform  function  is  doubly-periodic.  The  number  of 
independent  doubly-infinite  series  of  values  of  u  for  a  uniform  doubly- 
periodic  function  is  at  least  two :  and  therefore  there  must  be  at  least  two 
doubly-infinite  series  of  values  of  u,  so  that  cr  >1.  Hence  a  function,  that 
possesses  an  addition-theorem,  cannot  have  only  one  doubly-infinite  series  of 
values  for  its  argument. 

If  o-  be  infinite,  there  is  an  infinite  series  of  values  of  u  of  the  form 
"x  +  Pi^i  +  P2^^  \  an  argument,  similar  to  that  in  case  (a),  shews  that  this  is, 
in  effect,  the  same  as  case  (c). 

It  is  obvious  that  cases  (ii),  (iii)  and  (iv)  of  §  152  are  now  completely 
covered;    case  (v)  of  §  152  is  covered  by  case  (c)  now  to  be  discussed. 

154.  For  case  (c),  we  have  the  series  of  values  u  represented  by  a  number 
of  series  of  the  form 

Ma.  +    2    nir(Or, 
)•  =  ! 

where  the  quantities  &>  are  not  linearly  equivalent  to  two  quantities  Hj  and 
Dg.     The  original  integer  q  is  infinite. 

Then,  by  §§  108,  110,  it  follows  that  integers  ni  can  be  chosen  in  an 
unlimited  variety  of  ways  so  that  the  modulus  of 

S    nirWr 
r  =  l 

is  infinitesimal,  and  therefore  in  the  immediate  vicinity  of  any  point  u^^ 
there  is  an  infinitude  of  points  at  which  the  function  resumes  its  value. 
Such  a  function  would,  as  in  previous  instances,  degenerate  into  a  mere 
constant,  unless  each  point  were  an  essential  singularity  (as  is  not  the  case) ; 
hence  the  combination  of  values  which  gives  rise  to  this  case  does  not  occur. 


154.]  ALGEBRAICAL   ADDITION-THEOREM  355 

All  the  possible  cases  have  been  considered :  and  the  truth  of  Weierstrass's 
theorem*  that  a  function,  which  has  an  algebraical  addition-theorem,  is  either 

an  algebraical  function  of  u,  or  of  e  "  (where  co  is  suitably  chosen),  or  of  <^  (u), 
where  the  periods  of  ^J{u)  are  suitably  chosen,  is  established;  and  it  has 
incidentally  been  established — it  is,  indeed,  essential  to  the  derivation  of  the 
theorem — that  a  function,  ivhich  has  an  algebraical  addition -theorein,  has  only 
a  finite  number  of  values  for  a  given  argument. 

It  is  easy  to  see  that  the  first  derivative  has  only  a  finite  number  of  values 
for  a  given  argument;  for  the  elimination  of  U  between  the  algebraical 
equations 

G(U,u)  =  0,     ^Cr'  +  ?^  =  0, 
^         ^  db  tiu 

leads  to  an  equation  in  U'  of  the  same  finite  degree  as  G  in  U. 

Further,  it  is  now  easy  to  see  that  if  the  analytical  function  (p  (u),  luhich 
possesses  an   algebraical  addition-theorem,  be  uniform,  then  it  is  a  rational 

iwu  .. 

function  either  of  u,  or  ofe*^,  or  of  ^j  (u)  and  ^j'  (u) ;  and  that  any  uniform 
function,  which  is  transcendental  in  the  sense  of  §  46  and  which  possesses  an 
algebraical  addition-theorem,  is  either  a  simply-periodic  function  or  a  doubly- 
periodic  function. 

The  following  examples  will  illustrate  some  of  the  inferences  in  regard  to  the  number 
of  values  of  (f)(u  +  v)  arising  from  series  of  values  for  ti  and  r. 

Fx:  1.     Let  U=u^  +  {2u  +  l)^. 

Evidently  m,  the  number  of  values  of  U  for  a  value  of  u,  is  4  ;  and,  as  the  rationalised 
form  of  the  equation  is 

the  value  of  p,  being  the  number  of  values  of  u  for  a  given  value  of  U,  is  2.  Thus  the 
equation  in  W  should  be,  by  §  151,  of  degree  (4.  22  =  )  16. 

This  equation  is  n  {3  ( Tf  2  _  1/2  _  72)  + 1  _  2kr]=0, 

r=l 

where  k,.  is  any  one  of  the  eight  values  of 

when  rationalised,  the  equation  is  of  the  16th  degree  in  W. 

Ex.  2.     Let  U—cosi(. 

Evidently  m  =  l  ;  the  values  of  u  for  a  given  value  of  U  are  contained  in  the  double 
series  u+2Trn,  —u-\-2ivn,  for  all  values  of  n  from  —  c»  to  +00.     The  values  of  u  +  v  are 

u  +  2Trn-\-v+2iTm,  that  is,  ^l  +  v  +  2^^p  ;     —u  +  2TT}i+v  +  27r?n,  that  is,  —u  +  v  +  2Trp  ; 

u  +  2Tr7i-  v  +  2Trni,  that  is,  u  —  v  +  2Trp  ;      -u+2irn  —  v+2Trm,  that  is,  —u-v-\-2ttp, 

*  The  theorem  has  been  used  by  Schwarz,  Ges.  Werke,  t.  ii,  pp.  260 — 268,  in  determining  all 
.  the  families  of  plane  isothermie  curves  which  are  algebraic  curves,  an  '  isothermic '  curve  being 
of  the  form  u  =  c,  where  m  is  a  function  satisfying  the  potential-equation 


23—2 


356  FUNCTIONS   POSSESSING   AN  [154. 

so  that  the  number  of  series  of  values  of  li  +  v  is  four,  each  series  being  simply-infinite. 
It  might  thus  be  expected  that  the  equation  between  U,  V,  W  would  be  of  degree 
(1.4  =  )  4  in  W ;   but  it  happens  that 

cos  {u  +  v)  =  cos  (  — It  — v), 
and  so  the  degree  of  the  equation  in  W  is  reduced  to  half  its  degree.     The  equation  is 

If  2  _  2  WUV+  U^+  F2  -1=0. 
Ex.  3.     Let  U=  sn  u. 

Evidently  m  =  l;  and  there  are  two  doubly-infinite  series  of  values  of  w  determined 
by  a  given  value  of  U,  having  the  form  u  +  2m(o  +  2m' a>',  q>  —  u  +  2mco  +  2m'co'.  Hence  the 
values  of  ii+v  are 

=        u+v  (mod.  2a),  2(b')  ;        =a)  —  u+v  (mod.  2<o,  2a)')  ; 
=  a)  +  u—v  {mod.  2(o,  2<o');       =    —it  —  v  {mod.  2a>,  2co')  ; 
four  in  number.     The  equation  may  therefore  be  expected  to  be  of  the  fourth  degree 
in  W ;   it  is 

4  {l-V^}  (1  -  F2)  (i_  W^)  =  {2-U-^-  F2-  W^  +  F-U^V'n¥^)l 

155.  But  it  must  not  be  supposed  that  any  algebraical  equation  between 
U,  V;  W,  which  is  symmetrical  in  U  and  V,  is  one  necessarily  implying  the 
representation  of  an  algebraical  addition-theorem.  Without  entering  into  a. 
detailed  investigation  of  the  formal  characteristics  of  the  equations  that  are 
suitable,  a  latent  test  is  given  by  implication  in  the  following  theorem,  also 
due  to  Weierstrass  : — 

If  an  analytical  function  possess  an  algebraical  addition-theorem,  an 
algebraical  equation  involving  the  function  and  its  first  derivative  with  regard 
to  its  argument  exists ;  and  the  coefficients  in  this  equation  do  not  involve  the 
argument  of  the  function. 

The  proposition  might  easily  be  derived  by  assuming  the  preceding 
proposition,  and  applying  the  known  results  relating  to  the  algebraical 
dependence  between  those  functions,  the  types  of  which  are  suited  to  the 
representation  of  the  functions  in  question,  and  their  derivatives ;  we  shall, 
however,  proceed  more  directly  from  the  equation  expressing  the  algebraical 
addition-theorem  in  the  form 

G{U,  V,  W)  =  0, 
which  may  be  regarded  as  a  rationally  irreducible  equation. 

Differentiating  with  regard  to  w,  we  have 

dU         dW 
and  similarly,  with  regard  to  v,  we  have 


from  which  it  follows  that 


dU         dV 


155.]  ALGEBRAICAL   ADDITION-THEOREM  857 

This  equation*  will,  in  general,  involve  W ;  in  order  to  obtain  an  equation 
free  from  W,  we  eliminate   W  between 

G^  =  0  and  ^-^^ZT'  =  |^F', 
oL  0  V 

the  elimination  being  possible  because  both  equations  are  of  finite  degree ; 
and  thus  in  any  case  we  have  an  algebraical  equation  independent  of  W  and 
involving  U,  U',  V,  V . 

Not  more  than  one  equation  can  arise  by  assigning  various  values  to  v,  a 
quantity  that  is  independent  of  u;  for  we  should  have  either  inconsistent 
equations  or  simultaneous  equations  which,  being  consistent,  determine  a 
limited  number  of  values  of  JJ  and  U'  for  all  values  of  u,  that  is,  only  a 
number  of  constants.  Hence  there  can  be  only  one  equation,  obtained  by 
assigning  varying  values  to  v  ;  and  this  single  equation  is  the  algebraical 
equation  between  the  function  and  its  first  derivative,  the  coefficients  being 
independent  of  the  argument  of  the  function. 

N'ote.  A  test  of  suitability  of  an  algebraical  equation  G  =  0  between 
three  variables  U,  V,  W  to  represent  an  addition-theorem  is  given  by  the 
condition  that  the  elimination  of  W  between 

oL  oV 

leads  to  only  a  single  equation  between  U  and  U'  for  different  values  of 
V  and  v. 

Ecc.     Consider  the  equation 

(2-  v-  Y-  Tr)2-4  (1-  r)(i-  r)(i-  n')=o. 

The  deduced  equation  involving  V  and  Y'  is 

(2 Fir-  r-  ir-f-  u)  u'={2 uw-  u-  Tr+  f)  p, 

so  that  W-       ^^-U)iV'  +  U') 

••'^  ^^^^^  ^-{%Y-l)U'-{iU-\)Y'- 

The  ehmination  of  IF  is  simple.     We  have 

1-TF=    {V+U-l){V'-Y') 


(2F-l)Z7'-(2  6^-l)F" 

and                           2 -  U-  Y-  W- 2  (r+ ^- 1)  {(1  -  F)  ^'-(1- C/)  F} 
-""^  ^     ^       (^-H-2 (2F^l)?7'-(2C^-l)F 

Neglecting  4  (F-|- £/"—!)  =  0,  which  is  an  irrelevant  equation,  and  multiplying  by 
(2  F- 1)  U' - (2U-  1)  F,  we  have 

{Y+  U-l)  {(1-  F)  U'-(l-U)  F'}2  =  (1-  Z7)  (l-Y){U'-  F')  {(2F-  1)  U'-{2U-1)  Y'},  ' 

and  therefore  V{U-Y)(l~  F)  U"-+  U{  F-  U)  {I  -  U)  F'2  =  0. 

*  It  is  permissible  to  adopt  any  subsidiary  irrational  or  non-algebraical  form  as  the  equivalent 
of  G  =  0,  provided  no  special  limitation  to  the  subsidiary  form  be  implicitly  adopted.  Thus,  if  W 
can  be  expressed  explicitly  in  terms  of  XJ  and  V,  this  resoluble  (but  irrational)  equivalent  of  the 
equation  often  leads  rapidly  to  the  equation  between  U  and  its  derivative. 


358  FUNCTIONS   WITH   AN   ADDITION-THEOREM  [155. 

When  the  irrelevant  factor  U—  V  is  neglected,  this  equation  gives 

Uil-[7)~  V{1-  F)' 

the  equation  required  :  and  this,  indeed,  is  the  necessary  form  in  which  the  equation 
involving  U  and  V  arises  in  general,  the  variables  being  combined  in  associate  pairs. 
Each  side  is  evidently  a  constant,  say  4a^ ;   and  then  we  have 

U'^  =  4a^U{l-U). 

Then  the  value  of  U  is  sin^  (aw  + /3),  the  arbitrary  additive  constant  of  integration 
being  /3  ;    by  substitution  in  the  original  equation,  ^  is  easily  proved  to  be  zero. 

156.     Again,  if  the  elimination  between 

G  =  0  and  l^jU'^l^^V 
oil  0  V 

be  supposed  to  be  performed  by  the  ordinary  algebraical  process  for  finding 
the  greatest  common  measure  of  G  and  U''  ^tj—  ^'orr?  regarded  as  functions 

of  W,  the  final  remainder  is  the  eliminant  which,  equated  to  zero,  is  the 
differential  equation  involving  U,  U',  V,  V'\  and  the  greatest  common  measure, 
equated  to  zero,  gives  the  simplest  equation  in  virtue  of  which  the  equations 

0  =  Q  and  ^rjf  U'  =  ?r^  V  subsist.     It  will  be  of  the  form 

oU  o  V 

f(W,  U,  V,  U',  V')  =  0. 

If  the  function  have  only  one  value  for  each  value  of  the  argument,  so  that  it 
is  a  uniform  function,  this  last  equation  can  give  only  one  value  for  W;  for  all 
the  other  magnitudes  that  occur  in  the  equation  are  uniform  functions  of 
their  respective  arguments.  Since  it  is  linear  in  W,  the  equation  can  be 
expressed  in  the  form 

W=R(U,V,  U',  V), 

where  R  denotes  a  rational  function.     Hence*  : — 

A  uniform  analytical  function  cf)  (u),  tvhich  possesses  an  algebraical 
addition-theorem,  is  such  that  (f>  {u  +  v)  can  be  expressed  rationally  in  tet^ms 
of  0  (u),  </)'  (u),  (b  (v)  and  (f)'  (v). 

It  need  hardly  be  pointed  out  that  this  result  is  not  inconsistent  with  the 
fact  that  the  algebraical  equation  between  0  (u  +  v),  (p  (u)  and  (f>  (v)  does  not, 
in  general,  express  (f)  {u  +  ^;)  as  a  rational  function  of  6  (w)  and  ^  (v).  And  it 
should  be  noticed  that  the  rationality  of  the  expression  of  (j)  (u  +  v)  in  terms 
of  ^  {u),  (j)  {v),  (f)'  (ii),  (f)'  {v)  is  characteristic  of  functions  with  an  algebraical 
addition-theorem.  Instances  do  occur  of  functions  such  that  (f){u  +  v)  can  be 
expressed,  not  rationally,  in  terms  of  cf)  (u),  (f)  (v),  (f)'  (m),  </>'  (v) ;  they  do  not 
possess  an  algebraical  addition-theorem.  Such  an  instance  is  furnished  by 
^{u):  the  expression  of  ^(u  +  v),  given  in  Ex.  3  of  §  131,  can  be  modified  so 
as  to  have  the  form  indicated. 

*  The  theorem  is  due  to  Weierstrass ;  see  Schwarz,  §  2,  (I.e.  in  note  to  p.  344). 


CHAPTER  xry. 

Connection  of  Surfaces. 

157.  In  proceeding  to  the  discussion  of  multiform  functions,  it  was 
stated  (§  100)  that  there  are  two  methods  of  special  importance,  one  of 
which  is  the  development  of  Cauchy's  general  theory  of  functions  of  complex 
variables  and  the  other  of  which  is  due  to  E-iemann.  The  former  has  been 
explained  in  the  immediately  preceding  chapters ;  we  now  pass  to  the 
consideration  of  Riemann's  method.  But,  before  actually  entering  upon  it, 
there  are  some  preliminary  propositions  on  the  connection  of  surfaces  which 
must  be  established ;  as  they  do  not  find  a  place  in  treatises  on  geometry, 
an  outline  will  be  given  here,  though  only  to  that  elementary  extent  which 
is  necessary  for  our  present  purpose. 

In  the  integration  of  meromorphic  functions,  it  proved  to  be  convenient 
to  exclude  the  poles  from  the  range  of  variation  of  the  variable  by  means  of 
infinitesimal  closed  simple  curves,  each  of  which  was  thereby  constituted  a 
limit  of  the  region :  the  full  boundary  of  the  region  was  composed  of  the 
aggi'egate  of  these  non-intersecting  curves. 

Similarly,  in  dealing  with  some  special  cases  of  multiform  functions,  it 
proved  convenient  to  exclude  the  branch-points  by  means  of  infinitesimal 
curves  or  by  loops.  And-,  in  the  case  of  the  fundamental  lemma  of  §  16, 
the  region  over  which  integration  extended  was  considered  as  one  which 
possibly  had  several  distinct  curves  as  its  complete  boundary. 

These  are  special  examples  of  a  general  class  of  regions,  at  all  points 
within  the  area  of  which  the  functions  considered  are  monogenic,  finite,  and 
continuous  and,  as  the  case  may  be,  uniform  or  multiform  But,  important 
as  are  the  classes  of  functions  which  have  been.,  considered,  it  is  necessary  to 
consider  wider  classes  of  multiform  functions  and  to  obtain  the  regions  which 
are  appropriate  for  the  representation  of  the  variation  of  the  variable  in  each 
case.  The  most  conspicuous  examples  of  such  new  functions  are  the  algebraic 
functions,  adverted  to  in  §§  94 — 99  ;  and  it  is  chiefly  in  view  of  their  value 
and  of  the  value  of  functions  dependent  upon  them,  as  well  as  of  the  kind  of 
surface  on  which  their  variable  can  be  simply  represented,  that  we  now 
proceed  to  establish  some  of  the  topological  properties  of  surfaces  in  general. 

158.  A  surface  is  said  to  be  connected  when,  from  any  point  of  it  to  any 
other  point  of  it,  a  continuous  line  can  be  drawn  without  passing  out  of  the 


360 


CONNECTED 


[158. 


surface.  Thus  the  surface  of  a  circle,  that  of  a  plane  ring  such  as  arises  in 
Lambert's  Theorem,  that  of  a  sphere,  that  of  an  anchor-ring,  are  connected 
surfaces.  Two  non-intersecting  spheres,  not  joined  or  bound  together  in 
any  manner,  are  not  a  connected  surface  but  are  two  different  connected 
surfaces.  It  is  often  necessary  to  consider  surfaces,  which  are  constituted 
by  an  aggregate  of  several  sheets ;  in  order  that  the  surface  may  be  regarded 
as  connected,  there  must  be  junctions  between  the  sheets. 

One  of  the  simplest  connected  surfaces  is  such  a  plane  area  as  is  enclosed 
and  completel}^  bounded  by  the  circumference  of  a  circle.  All  lines  drawn 
in  it  from  one  internal  point  to  another  can  be  deformed  into  one  another ; 
any  simple  closed  line  lying  entirely  within  it  can  be  deformed  so  as  to  be 
evanescent,  without  in  either  case  passing  over  the  circumference ;  and  any 
simple  line  from  one  point  of  the  circumference  to  another,  when  regarded 
as  an  impassable  barrier,  divides  the  surface  into  two  portions.  Such  a 
surface  is  called*  simply  connected. 

The  kind  of  connected  surface  next  in  point  of  simplicity  is  such  a  plane 
area  as  is  enclosed  between  and  is  completely  bounded  by  the  circumferences 
of  two  concentric  circles.  All  lines  in  the  surface 
from  one  point  to  another  cannot  necessarily  be 
deformed  into  one  another,  e.g.,  the  lines  Zf^az  and 
zjbz ;  a  simple  closed  line  cannot  necessarily  be 
deformed  so  as  to  be  evanescent  without  crossing 
the  boundary,  e.g.,  the  line  azjjza ;  and  a  simple 
line  from  a  point  in  one  part  of  the  boundary 
to  a  point  in  another  and  different  part  of  the 
boundary,  such  as  a  line  AB,  does  not  divide  the 
surface  into  two  portions  but,  set  as  an  impassable  barrier,  it  makes  the 
surface  simply  connected. 

Again,  on  the  surface  of  an  anchor-ring,  a  closed  line  can  be  drawn  in 
two  essentially  distinct  ways,  abc,  ab'c,  such  that 
neither  can  be  deformed  so  as  to  be  evanescent 
or  so  as  to  pass  continuously  into  the  other. 
If  abc  be  made  the  only  impassable  barrier,  a 
line  such  as  a/37  cannot  be  deformed  so  as  to  be 
evanescent ;  if  ab'c  be  made  the  only  impassable 
barrier,  the  same  holds  of  a  line  such  as  a/3'7'. 
In  order  to  make  the  surface  simply  connected, 
two  impassable  barriers,  such  as  abc  and  ab'c, 
must  be  set. 

Surfaces,  like  the  flat  ring  or  the  anchor-ring, 


Fig.  36. 


*  Sometimes  the  term  monadelphic  is  used. 
hdngend. 


The  German  equivalent  is  einfach  zusainmen- 


158.] 


SURFACES 


361 


are   called*  multiply  connected;  the  establishment  of  barriers  has  made  it 
possible,  in  each  case,  to  modify  the  surface  into  one  which  is  simply  connected. 

159.  It  proves  to  be  convenient  to  arrange  surfaces  in  classes  according 
to  the  character  of  their  connection;  and  these  few  illustrations  suggest  that 
the  classification  may  be  made  to  depend,  either  upon  the  resolution  of  the 
surface,  by  the  establishment  of  barriers,  into  one  that  is  simply  connected, 
or  upon  the  number  of  what  may  be  called  independent  irreducible  circuits. 
The  former  mode — -that  of  dependence  upon  the  establishment  of  barriers — 
will  be  adopted,  thus  following  Riemannf ;  but  whichever  of  the  two  modes 
be  adopted  (and  they  are  not  necessarily  the  only  modes),  subsequent  demands 
require  that  the  two  be  brought  into  relation  with  one  another. 

The  most  effective  way  of  securing  the  impassability  of  a  barrier  is  to 
suppose  the  surface  actually  cut  along  the  line  of  the  barrier.  Such  a  section 
of  a  surface  is  either  a  cross-cut  or  a  loop-cut. 

If  the  section  be  made  through  the  interior  of  the  surface  from  one  point 


Fig.  37. 

of  the  boundary  to  another  point  of  the  boundary,  without  intersecting  itself 
or  meeting  the  boundary  save  at  its  extremities,  it  is  called  a  cross-cutX- 
Every  part  of  it,  as  it  is  made,  is  to  be  regarded  as  boundary  during  the 
formation  of  the  remainder ;  and  any  cross-cut,  once  made,  is  to  be  regarded 
as  boundary  during  the  formation  of  any  cross-cut  subsequently  made. 
Illustrations  are  given  in  fig.  37. 

The  definition  and  explanation  imply  that  the  surface  has  a  boundary. 
Some  surfaces,  such  as  a  complete  sphere  and  a  complete  anchor-ring,  do 
not  possess  a  boundary;  but,  as  will  be  seen  later  (§§  163,  168)  from  the 
discussion  of  the  evanescence  of  circuits,  it  is  desirable  to  assign  some 
boundary  in  order  to  avoid  merely  artificial  difiiculties  as  to  the  numerical 

*  Sometimes  the  term  pohjadelphic  is  used.  The  German  equivalent  is  viehrfach  zusammen- 
hdngend. 

t  "  Grundlagen  fiir  eine  allgemeine  Theorie  der  Functioneri  einer  veranderlichen  coraplexen 
Grosse,"  Kiemann's  Gesammelte  Werke,  pp.  9—12;  "Theorie  der  Abel'schen  Functionen,"  ib., 
pp.  84—89.  "When  reference  to  either  of  these  memoirs  is  made,  it  will  be  by  a  citation  of  the 
page  or  pages  in  the  volume  of  Eiemann's  Collected  Works. 

+  This  is  the  equivalent  used  for  the  German  word  Querschnitt;  French  writers  use  Section, 
and  Italian  writers  use  Trasversale  or  Taglio  trasversale. 


562 


CROSS-CUTS   AND   LOOP-CUTS 


[159. 


expression  of  the  connection.  This  assignment  usually  is  made  by  taking  for 
the  boundary  of  a  surface,  which  otherwise  has  no  boundary,  an  infinitesimal 
closed  curve,  practically  a  point ;  thus  in  the  figure  of  the  anchor-ring  (fig.  36) 
the  point  a  is  taken  as  a  boundary,  and  each  of  the  two  cross-cuts  begins  and 
ends  in  a. 

If  the  section  be  made  through  the  interior  of  the  surface  from  a  point 
not  on  the  boundary  and,  without  meeting  the  boundary  or  crossing  itself, 
return  to  the  initial  point,  (so  that  it  has  the  form  of  a  simple  curve  lying 


Fig.   38. 


entirely  in  the  surface),  it  is  called*  a  loop-cut  Thus  a  piece  can  be  cut 
out  of  a  bounded  spherical  surface  by  a  loop-cut  (fig.  38) ;  but  it  does 
not  necessarily  give  a  separate  piece  when  made  in  the  surface  of  an 
anchor-ring. 

It  is  evident  that  both  a  cross-cut  and  a  loop-cut  furnish  a  double 
boundary-edge  to  the  whole  aggregate  of  surface,  whether  consisting  of 
two  pieces  or  of  only  one  piece  after  the  section. 

Moreover,  these  sections  represent  the  impassable  barriers  of  the  pre- 
liminary explanations ;  and  no  specified  form  was  assigned  to  those  barriers. 
It  is  thus  possible,  within  certain  limits,  to  deform  a  cross-cut  or  a  loop-cut 
continuously  into  a  closely  contiguous  and  equivalent  position.  If,  for 
instance,  two  barriers  initially  coincide  over  any  finite  length,  one  or  other 
can  be  slightly  deformed  so  that  finally  they  intersect  only  in  a  point ;  the 
same  modification  can  therefore  be  made  in  the  sections. 

The  definitions  of  simple  connection  and  of  multiple  connection  will  nowf 
be  as  follows  : — 

A  surface  i,9  simply  connected,  if  it  he  resolved  into  tiuo  distinct  pieces  by 
every  cross-cut ;  but  if  there  be  any  cross-cut,  ivkich  does  not  resolve  it  into 
distinct  pieces,  the  surface  is  midtiply  connected. 

160.  Some  fundamental  propositions,  relating  to  the  connection  of 
surfaces,  may  now  be  derived. 

*  This  is  the  equivalent  used  for  the  German  word  Riickkehrscknitt  ;  French  writers  use  the 
word  Retrosection. 

t  Other  definitions  will  be  required,  if  the  classification  of  surfaces  be  made  to  depend  on 
methods  other  than  resolution  by  sections. 


160.]  PROPERTIES   OF    CONNECTED    SURFACES  363 

I.  Each  of  the  two  distinct  pieces,  into  which  a  simply  connected  surface  S 
is  resolved  by  a  cross-cut,  is  itself  simply  connected. 

If  either  of  the  pieces,  made  by  a  cross-cut  ah,  be  not  simply  connected, 
then  some  cross-cut  cd  must  be  possible  which  will  not  resolve  that  piece 
into  distinct  portions. 

If  neither  c  nor  d  lie  on  ah,  then  the  obliteration  of  the  cut  ah  will  restore 
the  original  surface  S,  which  now  is  not  resolved  by  the  cut  cd  into  distinct 
pieces. 

If  one  of  the  extremities  of  cd,  say  c,  lie  on  ah,  then  the  obliteration  of 
the  portion  ch  will  change  the  two  pieces  into  a  single  piece  which  is  the 
original  surface  8;  and  S  now  has  a  cross-cut  acd,  which  does  not  resolve 
it  into  distinct  pieces. 

If  both  the  extremities  lie  on  ah,  then  the  obliteration  of  that  part  of  ah 
which  lies  between  c  and  d  will  change  the  two  pieces  into  one ;  this  is  the 
original  surface  8,  now  with  a  cross-cut  acdh,  which  does  not  resolve  it  into 
distinct  pieces. 

These  are  all  the  possible  cases  should  either  of  the  distinct  pieces  of  8 
not  be  simply  connected ;  each  of  them  leads  to  a  contradiction  of  the  simple 
connection  of  8 ;  therefore  the  hypothesis  on  which  each  is  based  is  untenable, 
that  is,  the  distinct  pieces  of  8  in  all  the  cases  are  simply  connected. 

Corollary  1.  A  simply  connected  surface  is  resolved  hy  n  cross-cuts  into 
n+1  distinct  pieces,  each  simply  connected ;  and  an  aggregate  of  m,  simply 
connected  surfaces  is  resolved  hy  n  cross-cuts  into  n  +  m  distinct  pieces  each 
simply  connected. 

Corollary  2.  A  surface  that  is  resolved  into  tivo  distinct  simply  con- 
nected pieces  hy  a  cross-cut  is  simply  connected  hefore  the  resolution. 

Corollary  3.  If  a,  midtiply  connected  surface  he  resolved  into  two 
different  pieces  hy  a  cross-cut,  hoth  of  these  pieces  cannot  he  simply  connected. 

We  now  come  to  a  theorem*  of  great  importance: — 

II.  If  a  resolution  of  a  surface  hy  m  cross-cuts  into  n  distinct  simply 
connected  pieces  he  possible,  arid  also  a  different  resolution  of  the  same  surface 
hy  fi  cross-cuts  into  v  distinct  simply  connected  pieces,  then  m  —  n  =  /j,  —  v. 

Let  the  aggregate  of  the  w  pieces  be  denoted  by  8  and  the  aggregate  of 
the  V  pieces  by  2 :  and  consider  the  effect  on  the  original  surface  of  a  united, 
system  of  m  -\-  /j,  simultaneous  cross-cuts  made  up  of  the  two  systems  of  the 
7n  and  of  the  /x  cross-cuts  respectively.  The  operation  of  this  system  can  be 
carried  out  in  two  ways :  (i)  by  effecting  the  system  of  /j.  cross-cuts  on  >S^  and 

*  The  following  proof  of  this  proposition  is  substantially  due  to  Neumann,  p.  157.  Another 
proof  is  given  by  Eiemaun,  pp.  10,  11,  and  is  amplified  by  Durege,  Elemente  der  Theorie  der 
Functionen,  pp.  183—190 ;    and  another  by  Lippich,   see  Durege,  pp.  190—197. 


364  EFFECT    OF    CROSS-CUTS   ON  [160. 

(ii)  by  effecting  the  system  of  m  cross-cuts  on  ^ :  with  the  same  result  on  the 
original  surface. 

After  the  explanation  of  §  159,  we  may  justifiably  assume  that  the  lines 
of  the  two  systems  of  cross-cuts  meet  only  in  points,  if  at  all :  let  S  be  the 
number  of  points  of  intersection  of  these  lines.  Whenever  the  direction  of  a 
cross-cut  meets  a  boundary  line,  the  cross-cut  terminates  ;  and  if  the  direction 
continue  beyond  that  boundary  line,  that  produced  part  must  be  regarded  as 
a  new  cross-cut. 

Hence  the  new  system  of  /x  cross-cuts  applied  to  S  is  effectively 
equivalent  to  /x  -(-  S  new  cross-cuts.  Before  these  cuts  were  made,  S  was 
composed  of  n  simply  connected  pieces ;  hence,  after  they  are  applied,  the 
new  arrangement  of  the  original  surface  is  made  up  of  /i  -f-  (/x  -I-  S)  simply 
connected  pieces. 

Simihirly,  the  new  system  of  in  cross-cuts  applied  to  2  will  give  an 
arrangement  of  the  original  surface  made  up  of  v  +  {m-\-  8)  simply  connected 
pieces.     These  two  arrangements  are  the  same :  and  therefore 

n  •{■  fx  +  h  =  V  +  m  +  h, 
so  that  m  —  n  =  jju  —  v. 

It  thus  appears  that,  if  by  any  system  of  q  cross-cuts  a  multiply 
connected  surface  be  resolved  into  a  number  p  of  pieces  distinct  from  one 
another  and  all  simply  connected,  the  integer  q  —  p  is  independent  of  the 
particular  system  of  the  cross-cuts  and  of  their  configuration.  The  integer 
q—p  is  therefore  essentially  associated  with  the  character  of  the  multiple 
connection  of  the  surface :  and  its  in  variance  for  a  given  surface  enables  us 
to  arrange  surfaces  according  to  the  value  of  the  integer. 

No  classification  among  the  multiply  connected  surfaces  has  yet  been 
made :  they  have  merely  been  defined  as  surfaces  in  which  cross-cuts  can 
be  made  that  do  not  resolve  the  surface  into  distinct  pieces. 

It  is  natural  to  arrange  them  in  classes  according  to  the  number  of  cross- 
cuts which  are  necessary  to  resolve  the  surface  into  one  of  simple  connection 
or  a  number  of  pieces  each  of  simple  connection. 

For  a  simply  connected  surface,  no  such  cross-cut  is  necessary :  then 
q  =  0,  p  =  l,  and  in  general  q  —  p  =  —  l.  We  shall  say  that  the  connectivity* 
is  unity.  Examples  are  furnished  by  the  area  of  a  plane  circle,  and  by  a 
spherical  surface  with  one  hole"f*. 

A  surface  is  called  doubly-connected  when,  by  one  appropriate  cross-cut, 
the  surface  is  changed  into  a  single  surface  of  simple  connection :  then  g  =  1, 
p  =  l  for  this  particular  resolution,  and  therefore  in  general,  q  —  p  =  0.     We 

*  Sometimes  ordei-  of  connection,  sometimes  adelphic  order;  the  German  word,  that  is  used, 
is  Grundzahl. 

t  The  hole  is  made  to  give  the  surface  a  boundary  (§  163). 


160.] 


THE   CONNECTIVITY 


365 


shall  say  that  the  connectivity  is  2.     Examples  are  furnished  by  a  plane  ring 
and  by  a  spherical  surface  with  two  holes. 

A  surface  is  called  triply-connected  when,  by  two  appropriate  cross-cuts, 
the  surface  is  changed  into  a  single  surface  of  simple  connection :  then  g  =  2, 
p  =  I  for  this  particular  resolution  and  therefore,  in  general,  q  —  p  =  1.  We 
shall  say  that  the  connectivity  is  3.  Examples  are  furnished  by  the  surface 
of  an  anchor-ring  with  one  hole  in  it*,  and  by  the  surfaces f  in  figure  39,  the 
surface  in  (2)  not  being  in  one  plane  but  one  part  beneath  another. 


And,  in  general,  a  surface  will  be  said  to  be  iV^-ply  connected  or  its 
connectivity  will  be  denoted  by  N,  if,  by  N  —  1  appropriate  cross-cuts,  it  can 
be  changed  into  a  single  surface  that  is  simply  connected  :|:.  For  this 
particular  resolution  q  =  N  —1,  p  =  1 :    and  therefore  in  general 

q-p  =  N  -2, 
or  N  =  q  —  p  +  2. 

Let  a  cross-cut  I  be  drawn  in  a  surface  of  connectivity  N.  There  are 
two  cases  to  be  considered,  according  as  it  does  not  or  does  divide  the  surface 
into  distinct  pieces. 

First,  let  the  surface  be  only  one  piece  after  /  is  drawn :  and  let  its 
connectivity  then  be  N'.  If  in  the  original  surface  q  cross-cuts  (one  of 
which  can,  after  the  preceding  proposition,  be  taken  to  be  I)  be  drawn 
dividing  the  surface  into  p  simply  connected  pieces,  then 

N'=q-p  +  2. 
To  obtain  these  p  simply  connected  pieces  from  the  surface  after  the  cross-cut 
I,  it  is  evidently  sufficient  to  make  the  q  —  1  original  cross-cuts  other  than  I ; 
that  is,  the  modified  surface  is  such  that  hy  q  —1  cross-cuts  it  is  resolved  into 
p  simply  connected  pieces,  and  therefore 

N'  =  {q-l)-p  +  2. 
Hence  iV'  =  iV— 1,  or  the  connectivity  of  the  surface  is  diminished  by  unity, 

*  The  hole  is  made  to  give  the  surface  a  boundary  (§  163). 

t  Riemann,  p.  89. 

X  A  few  writers  estimate  the  connectivity  of  such  a  surface  as  N-1,  the  same  as  the  number 
of  cross-cuts  which  can  change  it  into  a  single  surface  of  the  simplest  rank  of  connectivity :  the 
estimate  in  the  text  seems  preferable. 


366 


CONNECTIVITY    AS    AFFECTED   BY 


[160. 


Secondly,  let  the  surface  be  two  pieces  after  I  is  drawn,  of  connectivities 
i\^i  and  N^  respectively.  Let  the  appropriate  N-^  —  1  cross-cuts  in  the  former, 
and  the  appropriate  N^  -  1  in  the  latter,  be  drawn  so  as  to  make  each 
a  simply  connected  piece.  Then,  together,  there  are  two  simply  connected 
pieces. 

To  obtain  these  two  pieces  from  the  original  surface,  it  will  suffice  to 
make  in  it  the  cross-cut  I,  the  iVj  —  1  cross-cuts,  and  the  iVg  —  1  cross-cuts, 
that  is,  1  4-  (iV"i  —  1)  +  {No  —  1)  or  iVj  -f  iVg  —  1  cross-cuts  in  all.  Since  these, 
when  made  in  the  surface  of  connectivity  N,  give  two  pieces,  we  have 


and  therefore 


N,  +  N.^-^N+1. 


If  one  of  the  pieces  be  simply  connected,  the  connectivity  of  the  other  is  N ; 
so  that,  if  a  simply  connected  piece  of  surface  be  cut  off  a  multiply  connected 
surface,  the  connectivity  of  the  remainder  is  unchanged.     Hence  : — 

III.  If  a  cross-cut  he  made  in-  a  surface  of  connectivity  N  and  if  it  do 
not  divide  it  into  separate  pieces,  the  connectivity  of  the  modified  surface  is 
N  —1  ;  hut  if  it  divide  the  surface  into  two  separate  pieces  of  connectivities  Ni 
and  iVa,  then  N^  +  N^  =  N+l. 


Illustrations  are  shewn,  in  fig.  40,  of  the  effect  of  cross-cuts  on  the  two 
surfaces  in  fig.  39. 

IV.  In  the  same  way  it  may  be  proved  that,  if  s  cross-cuts  he  made  in  a 
surface  of  connectivity  N  and  divide  it  into  r  +  1  separate  pieces  (where  r^s) 
of  connectivities  N^,  iV'a,  ... ,  Nr+i  respectively,  then 

N,+N^-\-  ...  +  N,+,  =  N+2r-s, 

a  more  general  result  including  both  of  the  foregoing  cases. 

Thus  far  we  have  been  considering  only  cross-cuts :  it  is  now  necessary 
to  consider  loop-cuts,  so  far  as  they  affect  the  connectivity  of  a  surface  in 
which  they  are  made. 


160.]  CROSS-CUTS   AND   LOOP-CUTS  367 

A  loop-cut  is  changed  into  a  cross-cut,  if  from  A  any  point  of  if  a  cross-cut 
be  made  to  any  point  C  in  a  boundary-curve  of 
the  original  surface,  for  CAhdA  (fig.  41)  is  then 
evidently  a  cross-cut  of  the  original  surface ;  and 
CA  is  a  cross-cut  of  the  surface,  which  is  the  modi- 
fication of  the  original  surface  after  the  loop-cut 
has  been  made.  Since,  by  definition,  a  loop-cut 
does  not  meet  the  boundary,  the  cross-cut  CA  does 
not  divide  the  modified  surface  into  distinct  pieces ; 
hence,  according  as  the  effect  of  the  loop-cut  is,  \  '^' 

or  is  not,  that  of  making  distinct  pieces,  so  will 
the  effect  of  the  whole  cross-cut  be,  or  not  be,  that  of  making  distinct  pieces. 

161.  Let  a  loop-cut  be  drawn  in  a  surface  of  connectivity  N;  as  before 
for  a  cross-cut,  there  are  two  cases  for  consideration,  according  as  the  loop-cut 
does  or  does  not  divide  the  surface  into  distinct  pieces. 

First,  let  it  divide  the  surface  into  two  distinct  pieces,  say  of  connectivities 
-A^i  and  Nn  respectively.  Change  the  loop-cut  into  a  cross-cut  of  the  original 
surface  by  drawing  a  cross-cut  in  either  of  the  pieces,  say  the  second,  from 
a  point  in  the  course  of  the  loop-cut  to  some  point  of  the  original  boundary. 
This  cross-cut,  as  a  section  of  that  piece,  does  not  divide  it  into  distinct 
pieces:  and  therefore  the  connectivity  is  now  N.j  {=  N^—  1).  The  effect  of 
the  whole  section,  which  is  a  single  cross-cut,  of  the  original  surface  is  to 
divide  it  into  two  pieces,  the  connectivities  of  which  are  N-^  and  iV/ :  hence, 
by  §  160,  III., 

and  therefore  N^  +  N.^^  N  +  2. 

If  the  piece  cut  out  be  simply  connected,  say  iV^i  =  1,  then  the  connectivity 
of  the  remainder  is  N  +1.  But  such  a  removal  of  a  simply  connected  piece 
by  a  loop-cut  is  the  same  as  making  a  hole  in  a  continuous  part  of  the 
surface :  and  therefore  the  effect  of  making  a  simple  hole  in  a  continuous  part 
of  a  surface  is  to  increase  by  unity  the  connectivity  of  the  surface. 

If  the  piece  cut  out  be  doubly-connected,  say  N-^  =  2,  then  the  connect- 
ivity of  the  remainder  is  iV,  the  same  as  the  connectivity  of  the  original 
surface.  Such  a  portion  would  be  obtained  by  cutting  out  a  piece  with 
a  hole  in  it  which,  so  far  as  concerns  the  original  surface,  would  be  the  same 
as  merely  enlarging  the  hole — an  operation  that  naturally  would  not  affect 
the  connectivity. 

Secondly,  let  the  loop-cut  not  divide  the  surface  into  two  distinct  pieces : 
and  let  JSf'  be  the  connectivity  of  the  modified  surface.  In  this  modified 
surface  make  a  cross-cut  k  from  any  point  of  the  loop-cut  to  a  point  of  the 
boundary :  this  does  not  divide  it  into  distinct  pieces  and  therefore  the 
■connectivity  after  this  last  modification  is  N'  —  1.     But   the   surface  thus 


368  EFFECT   OF   LOOP-CUTS  [161. 

finally  modified  is  derived  from  the  original  surface  by  the  single  cross-cut, 
constituted  by  the  combination  of  k  with  the  loop-cut ;  this  single  cross-cut 
does  not  divide  the  surface  into  distinct  pieces,  and  therefore  the  connectivity 
after  the  modification  is  K  —  1.     Hence 

i\^'-l  =  A"-l, 

that  is,  N'  =  N,  or  the  connectivity  of  a  surface  is  not  affected  by  a  loop-cut 
which  does  not  divide  the  surface  into  distinct  pieces. 

Both  of  these  results  are  included  in  the  following  theorem  : — 

V.  If  after  any  number  of  loop-cuts  made  in  a  surface  of  connectivity  N, 
there  be  r-^1  distinct  pieces  of  surface,  of  connectivities  JSf^,  N^,  ...,  ^r+i,  then 

N,-\-N,-\-...+  Nr+^  =  J^+2r. 

Let  the  number  of  loop-cuts  be  s.  Each  of  them  can  be  changed  into  a 
cross-cut  of  the  original  surface,  by  drawing  in  some  one  of  the  pieces,  as 
may  be  convenient,  a  cross-cut  from  a  point  of  the  loop-cut  to  a  point  of 
a  boundary :  this  new  cross-cut  does  not  divide  the  piece  in  which  it  is  drawn 
into  distinct  pieces.  If  k  such  cross-cuts  (where  k  may  be  zero)  be  drawn  in 
the  piece  of  connectivity  iV',^,  the  connectivity  becomes  N^',  where 

-^^  in  ^^  -^  m       "^  J 
r+1  r+1  r+1 

hence  ^  iV^^  =   t  N.^-^k^   t  N^-s. 

m  =  \  m  =  l  m  =  1 

We  now  have  s  cross-cuts  dividing  the  surface  of  connectivity  N  into  r  +  1 
distinct  pieces,  of  connectivities  i\^i',  N2,  ...,  N,.',  J^^r+i  ',  and  therefore,  by 

^  160,  IV., 

iV/+  ...  -f  N;  +  Nr+^=N  +  2r  -  s, 

so  that  N,  +  N.-{- ...+  ^^r+^  =  ^''  +  2r. 

This  result  could  have  been  obtained  also  by  combination  and  repetition 
of  the  two  results  obtained  for  a  single  loop-cut. 

Thus  a  spherical  surface  with  one  hole  in  it  is  simply  connected :  when 
7h  —  1  other  different  holes  *  are  made  in  it,  the  edges  of  the  holes  being 
outside  one  another,  the  connectivity  of  the  surface  is  increased  by  n  —  1, 
that  is,  it  becomes  n.  Hence  a  spherical  surface  ivith  n  holes  in  it  is  n-ply 
connected. 

Ex.  Two  equal  anchor-rings,  which  are  horizontal  and  have  their  centres  in  the  same 
vertical  line,  are  connected  together  by  three  vertical  right  circular  cylinders.  Determine 
the  connectivity  of  the  solid  so  formed.  (Math.  Trip.,  Part  II.,  1893.) 

162.  Occasionally,  it  is  necessary  to  consider  the  effect  of  a  slit  made  in 
the  surface. 

*  These  are  holes  in  the  surface,  not  holes  bored  through  the  volume  of  the  sphere ;  one  of 
the  latter  would  give  two  holes  in  the  surface. 


162.]  BOUNDARIES  369 

If  the  slit  have  neither  of  its  extremities  on  a  boundary  (and  therefore  no 
point  on  a  boundary),  it  can  be  regarded  as  the  limiting  form  of  a  loop-cut 
which  makes  a  hole  in  the  surface.  Such  a  slit  therefore  (§  161)  increases  the 
connectivity  by  unity. 

If  the  slit  have  one  extremity  (but  no  other  point)  on  a  boundary,  it  can 
be  regarded  as  the  limiting  form  of  a  cross-cut,  which  returns 
on  itself  as  in  the  figure,  and  cuts  off  a  single  simply  con-       / 

nected  piece.     Such  a  slit  therefore  (§  160,  III.)  leaves  the      [_ ^_^ 

connectivity  unaltered.  I  "^""^ 

If  the  slit  have  both  extremities  on  boundaries,  it  ceases 
to  be  merely  a  slit :  it  is  a  cross-cut  the  effect  of  which  on 
the  connectivity  has  been  obtained.     We  do  not  regard  such  sections  as  slits. 

163.  In  the  preceding  investigations  relative  to  cross-cuts  and  loop-cuts, 
reference  has  continually  been  made  to  the  boundary  of  the  surface  con- 
sidered. 

The  houndary  of  a  surface  consists  of  a  line  returning  to  itself,  or  of  a 
system  of  lines  each  returning  to  itself.  Each  part  of  such  a  boundary-line 
as  it  is  drawn  is  considered  a  part  of  the  boundary,  and  thus  a  boundary-line 
cannot  cut  itself  and  pass  beyond  its  earlier  position,  for  a  boundary  cannot 
be  crossed :  each  boundary-line  must  therefore  be  a  simple  curve*. 

Most  surfaces  have  boundaries :  an  exception  arises  in  the  case  of  closed 
surfaces  whatever  be  their  connectivity.  It  was  stated  (|  159)  that  a 
boundary  is  assigned  to  such  a  surface  by  drawing  an  infinitesimal  simple 
curve  in  it  or,  what  is  the  same  thing,  by  making  a  small  hole.  The 
advantage  of  this  can  be  seen  from  the  simple  example  of  a  spherical 
surface. 

When  a  small  hole  is  made  in  any  surface  the  connectivity  is  increased 
by  unity :  the  connectivity  of  the  spherical  surface  after  the  hole  is  made 
is  unity,  and  therefore  the  connectivity  of  the  complete  spherical  surface 
must  be  taken  to  be  zero. 

The  mere  fact  that  the  connectivity  is  less  than  unity,  being  that  of  the 
simplest  connected  surfaces  with  which  we  have  to  deal, 
is  not  in  itself  of  importance.     But  let  us  return  for  a  /^CT'^^'^^^^^^X 

moment  to  the  suggested  method  of  determining  the       /  c  \ 

connectivity  by  means  of  the   evanescence   of  circuits      / - \ 

without  crossing  the  boundary.     When  the  surface  is      T -- """  / 

the  complete  spherical  surface  (fig.  43),  there  are  two       \  / 

essentially  distinct  ways  of  making  a  circuit  C  evan-         ^^         ,         y^ 
escent,  first,  by  making  it  collapse   into   the  point  a, 
secondly,  by  making    it  expand   over  the  equator  and 

*  Also  a  line  not  returning  to  itself  may  be  a  boundary ;  it  can  be  regarded  as  the  limit  of  a 
simple  curve  when  the  area  becomes  infinitesimal. 

F.    F.  24 


370  BOUNDARIES   AND  [163. 

then  collapse  into  the  point  b.  One  of  the  two  is  superfluous  :  it  introduces 
an  element  of  doubt  as  to  the  mode  of  evanescence  unless  that  mode  be 
specified — a  specification  which  in  itself  is  tantamount  to  an  assignment  of 
boundary.  And  in  the  case  of  multiply  connected  surfaces  the  absence  of 
boundary,  as  above,  leads  to  an  artificial  reduction  of  the  connectivity  by 
unity,  arising  not  from  the  greater  simplicity  of  the  surface  but  from  the 
possibility  of  carrying  out  in  two  ways  the  operation  of  reducing  any  circuit 
to  given  circuits,  which  is  most  effective  when  only  one  way  is  permissible. 
We  shall  therefore  assume  a  boundary  assigned  to  such  closed  surfaces  as  in 
the  first  instance  are  destitute  of  boundary. 

164.  The  relations  between  the  number  of  boundaries  and  the  connect- 
ivity of  a  surface  are  given  by  the  following  propositions. 

I.  The  boundary  of  a  simply  connected  surface  consists  of  a  single  line. 
When  a  boundary  consists  of  separate  lines,  then  a  cross-cut  can  be  made 

from  a  point  of  one  to  a  point  of  another.  By  proceeding  from 
P,  a  point  on  one  side  of  the  cross-cut,  along  the  boundary 
ac.c'a  we  can  by  a  line  lying  wholly  in  the  surface  reach  a 
point  Q  on  the  other  side  of  the  cross-cut :  hence  the  parts  of 
the  surface  on  opposite  sides  of  the  cross-cut  are  connected. 
The  surface  is  therefore  not  resolved  into  distinct  pieces  by  the 
cross-cut. 

A  simply  connected  surface  is  resolved  into  distinct  pieces  pi„_  ^^^ 

by  each  cross-cut  made  in  it :  such  a  cross-cut  as  the  foregoing 
is  therefore  not  possible,  that  is,  there  are  not  separate  lines  which  make  up 
its  boundary.    It  has  a  boundary  :  the  boundary  therefore  consists  of  a  single 
line. 

II.  A  cross-cut  either  increases  by  unity  or  diminishes  by  unity  the  number 
of  distinct  boundary -lines  of  a  multiply  connected  surface. 

A  cross-cut  is  made  in  one  of  three  ways :  either  from  a  point  a  of  one 
boundary-line  A  to  &  point  b  of  another  boundary-line  B ;  or  from  a  point  a 
of  a  boundary-line  to  another  point  a  of  the  same  boundary-line ;  or  from  a 
point  of  a  boundary-line  to  a  point  in  the  cut  itself. 

If  made  in  the  first  way,  a  combination  of  one  edge  of  the  cut,  the 
remainder  of  the  original  boundary  A,  the  other  edge  of  the  cut  and  the 
remainder  of  the  original  boundary  B  taken  in  succession,  form  a  single 
piece  of  boundary ;  this  replaces  the  two  boundary-lines  A  and  B  which 
existed  distinct  from  one  another  before  the  cross-cut  was  made.  Hence  the 
number  of  lines  is  diminished  by  unity.  An  example  is  furnished  by  a  plane 
ring  (ii.,  fig.  37,  p.  361). 

If  made  in  the  second  way,  the  combination  of  one  edge  of  the  cut  with 
the  piece  of  the  boundary  on  one  side  of  it  makes  one  boundarj^-line,  and  the 


164.]  CONNECTIVITY  371 

combination  of  the  other  edge  of  the  cut  with  the  other  piece  of  the  boundary 
makes  another  boundary-line.  Two  boundary-lines,  after  the  cut  is  made, 
replace  a  single  boundary-line,  which  existed  before  it  was  made :  hence  the 
number  of  lines  is  increased  by  unity.  Examples  are  furnished  by  the  cut 
surfaces  in  fig.  40,  p.  366. 

If  made  in  the  third  way,  the  cross-cut  may  be  considered  as  constituted 
by  a  loop-cut  and  a  cut  joining  the  loop-cut  to  the  boundary.  The  boundary- 
lines  may  now  be  considered  as  constituted  (fig.  41,  p.  367)  by  the  closed 
curve  ABD  and  the  closed  boundary  abda'c'e  ...eca;  that  is,  there  are  now 
two  boundary-lines  instead  of  the  single  boundary-line  c€...e'cc  in  the  uncut 
surface.     Hence  the  number  of  distinct  boundary-lines  is  increased  by  unity. 

Corollary.  A  loop-cut  increases  the  number  of  distinct  boundary -lines 
by  two. 

This  result  follows  at  once  from  the  last  discussion. 

III.  The  number  of  distinct  boundary-lines  of  a  surface  of  connectivity  N 
is  N  —  2k,  where  k  is  a  positive  integer  that  may  be  zero. 

Let  m  be  the  number  of  distinct  boundary-lines ;  and  let  iV^  —  1  appro- 
priate cross-cuts  be  drawn,  changing  the  surface  into  a  simply  connected 
surface.  Each  of  these  cross-cuts  increases  by  unity  or  diminishes  by  unity 
the  number  of  boundary-lines ;  let  these  units  of  increase  or  of  decrease  be 
denoted  by  Cj,  e^,  ... ,  6iv-i-  Each  of  the  quantities  e  is  +  1 ;  let  /fc  of  them  be 
positive,  and  N  —  1  —  k  negative.  The  total  number  of  boundary-lines  is 
therefore 

m  +  k-  (i\^ -  1  -  k). 

The  surface  now  is  a  single  simply  connected  surface,  and  there  is  therefore 
only  one  boundary-line  ;  hence 

m-{-k-{N-l-k)=l, 

so  that  m  =  N  —2k; 

and  evidently  k  is  an  integer  that  may  be  zero. 

Corollary   1.     A  closed  surface  with  a  single  boundary -line*  is  of  odd 

connectivity. 

For  example,  the  surface  of  an  anchor-ring,  when  bounded,  is  of  con- 
nectivity 3 ;  the  surface,  obtained  by  boring  two  holes  through  the  volume 
of  a  solid  sphere,  is,  when  bounded,  of  connectivity  5. 

If  the  connectivity  of  a  closed  surface  with  a  single  boundary  be  2p  -|-  1, 
the  surface  is  often  saidf  to  be  of  genus  p  (§  178,  p.  395). 

*  See  §  159. 

t  Sometimes  class.     The  German  word  is  Geschlecht ;  French  writers  use  the  word  genre,  and 
Italians  genere. 

24—2 


372  lhuilier's  [164. 

Corollary  2.  If  the  number  of  distinct  boundary -lines  of  a  surface  of 
connectivity  N  be  N,  any  loop-cut  divides  the  surface  into  ttvo  distinct  pieces. 

After  the  loop-cut  is  made,  the  number  of  distinct  boundary-lines  is 
JV-l-2;  the  connectivity  of  the  whole  of  the  cut  surface  is  therefore  not  less 
than  iV+  2.  It  has  been  proved  that  a  loop-cut,  which  does  not  divide  the 
surface  into  distinct  pieces,  does  not  affect  the  connectivity;  hence  as  the 
connectivity  has  been  increased,  the  loop-cut  must  divide  the  surface  into 
two  distinct  pieces.  It  is  easy,  by  the  result  of  §  161,  to  see  that,  after  the 
loop-cut  is  made,  the  sum  of  connectivities  of  the  two  pieces  is  1^+2,  so 
that  the  connectivity  of  the  whole  of  the  cut  surface  is  equal  to  N  +  2. 

Note.  Throughout  these  propositions,  a  tacit  assumption  has  been  made, 
which  is  important  for  this  particular  proposition  when  the  surface  is  the 
means  of  representing  the  variable.  The  assumption  is  that  the  surface  is 
bifacial  and  not  unifacial;  it  has  existed  implicitly  throughout  all  the 
geometrical  representations  of  variability :  it  found  explicit  expression  in 
I  4  when  the  plane  was  brought  into  relation  with  the  sphere :  and  a  cut 
in  a  surface  has  been  counted  a  single  cut,  occurring  on  one  side,  though  it 
would  have  to  be  counted  as  two  cuts,  one  on  each  side,  were  the  surface 
unifacial. 

The  propositions  are  not  necessarily  valid,  when  applied  to  unifacial 
surfaces.  Consider  a  surface  made  out  of  a  long  rectangular  slip  of  paper, 
which  is  twisted  once  (or  any  odd  number  of  times)  and  then  has  its  ends 
fastened  together.  This  surface  is  of  double  connectivity,  because  one 
section  can  be  made  across  it  which  does  not  divide  it  into  separate  pieces ; 
it  has  only  a  single  boundary-line,  so  that  Prop.  III.  just  proved  does  not 
apply.  The  surface  is  unifacial ;  and  it  is  possible,  without  meeting  the 
boundary,  to  pass  continuously  in  the  surface  from  a  point  P  to  another 
point  Q  which  could  be  reached  merely  by  passing  through  the  material 
at  P. 

We  therefore  do  not  retain  unifacial  surfaces  for  consideration. 

165.  The  following  proposition,  substantially  due  to  Lhuilier*,  may  be 
taken  in  illustration  of  the  general  theory. 

If  a  closed  surface  of  connectivity  2N  +  1  (or  of  genus  N)  be  divided  by 
circuits  into  any  number  of  simply  connected  portions,  each  in  the  form  of  a 
curvilinear  polygon,  and  if  F  be  the  number  of  polygons,  E  be  the  number  of 
edges  and  S  the  number  of  angular  points,  then 

2N=2  + E-F-S. 

Let  the  edges  E  be  arranged  in  systems,  a  system  being  such  that  any 
line  in  it  can  be  reached  by  passage  along  some  other  line  or  lines  of  the 

*  Gergonne,  Ann.  de  Math.,  t.  iii,  (1813),  pp.  181—186;  see  also  Mobius,  Ges.  Werke,  t.  ii, 
p.  468.     A  circuit  is  defined  in  §  166. 


165.]  THEOREM  373 

system  ;  let  k  be  the  number  of  such  systems*.  To  resolve  the  surface  into  a 
number  of  simply  connected  pieces  composed  of  the  F  polygons,  the  cross-cuts 
will  be  made  along  the  edges ;  and  therefore,  unless  a  boundary  be  assigned 
to  the  surface  in  each  system  of  lines,  the  first  cut  for  any  system  will  be 
a  loop-cut.  We  therefore  take  k  points,  one  in  each  system  as  a  boundary ; 
the  first  will  be  taken  as  the  natural  boundary  of  the  surface,  and  the 
remaining  k-\,  being  the  limiting  forms  of  ^'  -  1  infinitesimal  loop-cuts, 
increase  the  connectivity  of  the  surface  by  A^—  1,  that  is,  the  connectivity  now 
iB^N  +  k 

The  result  of  the  cross-cuts  is  to  leave  ^simply  connected  pieces  :  hence  Q, 
the  number  of  cross-cuts,  is  given  by 

Q  =  2N+k  +  F~2. 
At  every  angular  point  on  the  uncut  surface,  three  or  more  polygons  are 
contiguous.     Let  S,n  be  the  number  of  angular  points,  where  vi  polygons  are 
contiguous ;  then 

S=  S^-\-  S^-\-  S^+  ... 

Again,  the  number  of  edges  meeting  at  each  of  the  S.  points  is  three,  at 
each  of  the  Si  points  is  four,  at  each  of  the  ^g  points  is  five,  and  so  on ;  hence, 
in  taking  the  sum  35^3 -1- 45^4  4-  bS^-{- ...,  each  edge  has  been  counted  twice, 
once  for  each  extremity.     Therefore 

Consider  the  composition  of  the  extremities  of  the  cross-cuts ;  the  number 
of  the  extremities  is  2Q,  twice  the  number  of  cross-cuts. 

Each  of  the  k  points  furnishes  two  extremities;  for  each  such  point  is 
a  boundary  on  which  the  initial  cross-cut  for  each  of  the  systems  must  begin 
and  must  end.     These  points  therefore  furnish  2k  extremities. 

The  remaining  extremities  occur  in  connection  with  the  angular  points. 
In  making  a  cut,  the  direction  passes  from  a  boundary  along  an  edge,  past 
the  point  along  another  edge  and  so  on,  until  a  boundary  is  reached ;  so  that 
on  the  first  occasion  when  a  cross-cut  passes  through  a  point,  it  is  made  along 
two  of  the  edges  meeting  at  the  point.  Every  other  cross-cut  passing  through 
that  point  must  begin  or  end  there,  so  that  each  of  the  8^  points  will  furnish 
one  extremity  (corresponding  to  the  remaining  one  cross-cut  through  the 
point),  each  of  the  S^  points  will  furnish  two  extremities  (corresponding  to 
the  remaining  two  cross-cuts  through  the  point),  and  so  on.  The  total 
number  of  extremities  thus  provided  is 

Hence  ^Q  =  2k^  8,  +  'lS,  +  ^S,+ ... 

=  2k  +  2E-28, 

*  The  value  of  /c  is  1  for  the  proposition  and  is  greater  than  1  for  the  Corollary. 


374  CIRCUITS  ON                                             [165. 

or  Q  =  k  +  E  -  S, 

which,  combined  with  Q  =  2N  +  k  +  F-2, 

leads  to  the  relation  2N=2  +  E-F-S. 

The  simplest  case  is  that  of  a  sphere,  when  Euler's  relation  F+8  =  E  +  2 
is  obtained.  The  case  next  in  simplicity  is  that  of  an  anchor-ring,  for  which 
the  relation  is  F+8-=E. 

Corollary.  If  the  result  of  making  the  cross-cuts  along  the  various  edges 
he  to  give  the  F polygons,  not  simply  connected  areas  hut  areas  of  connectivities 
iVi  +  1,  i\^2+l)  •••)  -Z^j+l  respectively,  then  the  connectivity  of  the  original 
surface  is  given  hy 

2N=2  +  E-F-S+  2  Nr- 

r=l 

166.  The  method  of  determining  the  connectivity  of  a  surface  by  means 
of  a  system  of  cross-cuts,  which  resolve  it  into  one  or  more  simply  connected 
pieces,  will  now  be  brought  into  relation  with  the  other  method,  suggested 
in  §  159,  of  determining  the  connectivity  by  means  of  irreducible  circuits. 

A  closed  line  drawn  on  the  surface  is  called  a  circuit. 

A  circuit,  which  can  be  reduced  to  a  point  by  continuous  deformation 
without  crossing  the  boundary,  is  called  reducihle ;  a  circuit,  which  cannot  be 
so  reduced,  is  called  irreducihle. 

An  irreducible  circuit  is  either  (i)  simple,  when  it  cannot  without  crossing 
the  boundary  be  deformed  continuously  into  repetitions  of  one  or  more 
circuits ;  or  (ii)  multiple,  when  it  can  without  crossing  the  boundary  be 
deformed  continuously  into  repetitions  of  a  single  circuit ;  or  (iii)  compound, 
when  it  can  without  crossing  the  boundary  be  deformed  continuously  into 
combinations  of  different  circuits,  that  may  be  simple  or  multiple.  The 
distinction  between  simple  circuits  and  compound  circuits,  that  involve  no 
multiple  circuits  in  their  combination,  depends  upon  conventions  adopted  for 
each  particular  case. 

A  circuit  is  said  to  be  reconcileahle  with  the  system  of  circuits  into  a 
combination  of. which  it  can  be  continuously  deformed. 

If  a  system  of  circuits  be  reconcileahle  with  a  reducible  circuit,  the 
system  is  said  to  be  reducible. 

Let  a  simple  circuit  be  denoted  by  a  single  letter,  say,  A,  B,  C, ....  A 
multiple  circuit,  composed  of  n  repetitions  of  a  simple  circuit  A,  can  then  be 
denoted  by  A'K  A  compound  circuit,  composed  of  a  simple  circuit  A  followed 
by  another  simple  circuit  B,  can  be  denoted  hy  AB:  the  order  of  the  symbols 
being  of  importance.  As  circuits  thus  have  their  symbols  associated  in  the 
manner  of  (non-commutative  algebraical)  factors,  the  symbol  1  will  represent 


166.]  CONNECTED  SURFACES  375 

a  reducible  circuit ;  for  a  circuit  causing  no  change  must  be  represented  by 
a  factor  causing  no  change. 

There  are  two  directions,  one  positive  and  the  other  negative,  in  which 
a  circuit  can  be  described.  Let  it  be  described  first  in  the  positive  direction 
and  afterwards  in  the  negative  direction :  the  circuit,  compounded  of  the 
two  descriptions,  is  easily  seen  to  be  continuously  deformable  to  a  point, 
and  it  therefore  is  reducible.  Similarly,  if  the  circuit  is  described  first  in 
the  negative  direction  and  afterwards  in  the  positive  direction,  the  compound 
circuit  thus  obtained  is  reducible.  Accordingly,  if  a  simple  circuit  described 
positively  be  represented  by  A,  the  same  circuit  described  negatively  can  be 
represented  by  A~'^,  the  symbols  of  the  circuits  obeying  the  associative  law. 

A  compound  circuit,  reconcileable  with  a  system  of  simple  irreducible 
circuits  A,  B,  C,  ...,  would  be  represented  by  A'^B^A"-'B^' ...  GyA"-" ...,  where 
o,  /3,  a',  /3',  . . . ,  7,  a"  are  integers  positive  or  negative. 

In  order  to  estimate  circuits  on  a  multiply  connected  surface,  it  is 
sufficient  to  know  a  system  of  irreducible  simple  circuits.  Such  a  system  is 
naturally  to  be  considered  complete  when  every  other  circuit  on  the  surface 
is  reconcileable  with  the  s^^stem.  It  also  may  be  supposed  to  contain  the 
smallest  possible  number  of  simjjle  circuits ;  for  any  one,  which  is  reconcile- 
able with  the  rest,  can  be  omitted  without  affecting  the  completeness  of  the 
system. 

167.     Such  a  system  is  indicated  by  the  following  theorems : — 

I.  No  irreducible  simjtle  circuit  can  he  drawn  on  a  simply  comiected 
surface  *. 

If  possible,  let  an  irreducible  circuit  C  be  drawn  in  a  simply  connected 
surface  with  a  boundary  B.  Make  a  loop-cut  along  C,  and  change  it  into 
a  cross-cut  by  making  a  cross-cut  A  from  some  point  of  C  to  a  point  of  B ; 
this  cross-cut  divides  the  surface  into  two  simply  connected  pieces,  one  of 
which  is  bounded  by  B,  the  two  edges  of  A,  and  one  edge  of  the  cut  along  G, 
and  the  other  of  which  is  bounded  entirely  by  the  cut  along  G. 

The  latter  surface  is  smaller  than  the  original  surface;  it  is  simply 
connected  and  has  a  single  boundary.  If  an  irreducible  simple  circuit  can 
be  drawn  on  it,  we  proceed  as  before,  and  again  obtain  a  still  smaller  simply 
connected  surface.  In  this  way,  we  ultimately  obtain  an  infinitesimal 
element ;  for  every  cut  divides  the  surface,  in  which  it  is  made,  into 
distinct  pieces.  Irreducible  circuits  cannot  be  drawn  in  this  element;  and 
therefore  its  boundary  is  reducible.  This  boundary  is  a  circuit  in  a  larger 
portion  of  the  surface :  the  circuit  is  reducible  so  that,  in  that  larger  portion, 
no  irreducible  circuit  is  possible  and  therefore  its  boundary  is  reducible. 
This   boundary  is  a  circuit    in  a  still    larger   portion,    and    the    circuit    is 

*  All  surfaces  considered  are  supposed  to  be  bounded. 


376  RELATIONS   BETWEEN    CONNECTIVITY  [167. 

reducible :  so  that  in  this  still  larger  portion  no  irreducible  circuit  is  possible 
and  once  more  the  boundary  is  reducible. 

Proceeding  in  this  way,  we  find  that  no  irreducible  simple  circuit  is 
possible  in  the  original  surface. 

Corollary.  No  irreducible  circuit  can  he  drawn  on  a  simply  connected 
surface. 

II.  A  complete  system  of  irreducible  simple  circuits  for  a  surface  of 
connectivity  N  contains  N  —  1  simple  circuits,  so  that  every  other  circuit  on 
the  surface  is  reconcileable  with  that  system. 

Let  the  surface  be  resolved  by  cross-cuts  into  a  single  simply  connected 
surface :    N  —1  cross-cuts  will  be  necessary.     Let  CD  be 
any  one  of  them  :  and  let  a  and  h  be  two  points  on  the  ,g 

opposite  edges  of  the  cross-cut.     Then  since  the  surface  is  / 

simply  connected,  a  line  can  be  drawn  in  the  surface  from  '^ 

a  to   h  without  passing    out    of  the    surface    or   without  \ 

meeting  a  part  of  the  boundary,  that  is,  without  meeting  V 

any  other   cross-cut.     The    cross-cut   CD  ends   either   in  Fig.  4-5. 

another  cross-cut  or  in  a  boundary ;  the  line  ae ...  fb 
surrounds  that  other  cross-cut  or  that  boundary  as  the  case  may  be :  hence, 
if  the  cut  CD  be  obliterated,  the  line  ae . .  .fba  is  irreducible  on  the  surface  in 
which  the  other  i\^  —  2  cross-cuts  are  made.  But  it  meets  none  of  those  cross- 
cuts; hence,  when  they  are  all  obliterated  so  as  to  restore  the  unresolved 
surface  of  connectivity  N,  it  is  an  irreducible  circuit.  It  is  evidently  not 
a  repeated  circuit ;  hence  it  is  an  irreducible  simple  circuit.  Hence  the 
line  of  an  irreducible  simple  circuit  on  an  unresolved  surface  is  given  by 
a  line  passing  from  a  point  on  one  edge  of  a  cross-cut  in  the  resolved  surface 
to  a  point  on  the  opposite  edge. 

Since  there  are  N ~\  cross-cuts,  it  follows  that  iV  —  1  irreducible  simple 
circuits  can  thus  be  obtained :  one  being  derived  in  the  foregoing  manner 
from  each  of  the  cross-cuts,  which  are  necessary  to  render  the  surface  simply 
connected.  It  is  easy  to  see  that  each  of  the  irreducible  circuits  on  an 
unresolved  surface  is,  by  the  cross-cuts,  rendered  impossible  as  a  circuit  on 
the  resolved  surface. 

But  every  other  irreducible  circuit  C  is  reconcileable  with  the  N  —1 
circuits,  thus  obtained.  If  there  be  one  not  reconcileable  with  these  N  —\ 
circuits,  then,  when  all  the  cross-cuts  are  made,  the  circuit  G  is  not  rendered 
impossible,  if  it  be  not  reconcileable  with  those  which  are  rendered  impossible 
by  the  cross-cuts :  that  is,  there  is  on  the  resolved  surface  an  irreducible 
circuit.  But  the  resolved  surface  is  simply  connected,  and  therefore  no 
irreducible  circuit  can  be  drawn  on  it :  hence  the  hypothesis  as  to  C,  which 
leads  to  this  result,  is  not  tenable. 


167.] 


AND   IRREDUCIBLE    CIRCUITS 


377 


Thus  every  other  circuit  is  reconcileable  with  the  system  of  i\^  —  1  circuits  : 
and  therefore  the  system  is  complete*. 

This  mfethod  of  derivation  of  the  circuits  at  once  indicates  how  far  a 
system  is  arbitrary.  Each  system  of  cross-cuts  leads  to  a  complete  system  of 
irreducible  simple  circuits,  and  vice  versa ;  as  the  one  system  is  not  unique, 
so  the  other  system  is  not  unique. 

168.  It  follows  that  the  niunher  of  simple  irreducible  circuits  in  any 
complete  system  must  he  the  same  for  the  same  surface:  this  number  is  iV^—  1, 
where  iF  is  the  connectivity  of  the  surface.  Let  A^,  A^,  ...,  ^j^r-i;  B^,  B^,  ..., 
B 2,7-1 ;  be  two  distinct  complete  systems ;  then  we  have 

B,=  U,(A,A,...A^_,), 
where  Tig  means  the  symbolic  product  representing  that  circuit  compounded 
of  the  system  A-^,  ...,  -4^y_i  with  which  Bg  is  reconcileable;  and 

Ar  =  U;{B,B,...B,^_,) 
with  a  similar  significance  for  11/. 

Further  any  circuit,  that  is  reconcileable  ivith  one  cotnplete  system,  is 
reconcileable  with  any  other  complete  system.  For  if  X  denote  a  circuit 
reconcileable  with  A^,  A^,  ...,  A^^i,  we  have 

X  =  U{A,A,...A^_,): 
whence,  taking  account  of  the  reconcileability  of  each  circuit  A  with  the 
complete  system  B^,  B2,  ...,  Bj^t-^,  we  have 

x  =  n(n/n/...nVi) 

=  U''(B,B,...B^_,), 

thus  proving  the  statement. 

For  the  general  question,  Jordan's  memoir,  "  Des  contours  traces  sur  les  surfaces," 
Liouville,  2™«  Ser.,  t.  xi,  (1866),  pp.  110 — 130,  may  be  consulted. 

Ex.  1.  On  a  doubly  connected  surface,  one  irreducible  simple  circuit  can  be  drawn. 
It  is  easily  obtained  by  first  resolving  the  surface  into  one  that  is  simply  connected — 


Fig.  46,  (i). 

*  If  the  number  of  independent  irreducible  simple  circuits  be  adopted  as  a  basis  for  the 
definition  of  the  connectivity  of  a  surface,  the  result  of  the  proposition  would  be  taken  as 
the  definition :  and  the  resolution  of  the  surface  into  one,  which  is  simply  connected,  would 
then  be  obtained  by  developing  the  preceding  theory  in  the  reverse  order. 


378 


EXAMPLES 


[168. 


a  single  cross-cut  CD  is  effective  for  this  purpose — and  then  by  drawing  a  curve  aeh  in  the 
surface  from  one  edge  of  the  cross-cut  to  the  other.  All  other  irreducible  circuits  on  the 
unresolved  surface  are  reconcileable  with  the  circuit  aeba. 

Ex.  2.     On  a  triply  connected  surface,  two  independent  irreducible  circuits  can  be 
drawn.     Thus  in  the  figure  Ci  and  C2  will  form  a  complete  system.     The  circuits   C3. 


Fig.  46,  (ii). 

and  C4  are  also  irreducible  :  they  can  evidently  be  deformed  into  Ci  and  €'2  and  reducible 
circuits  by  continuous  deformation  :  in  the  algebraical  notation  adopted,  we  have 

But  C3  and  C4  are  not  simple  circuits  :  hence  they  are  not  suited  for  the  construction 
of  a  complete  system. 

E.r.   3.     Another  example  of  a  triply  connected  surface  is  given   in   fig.   47.     Two 
irreducible   simple   circuits   are    Cj   and    Co.      Another    irreducible    circuit   is    C3  ;    this 


Fig.  47. 


can  be  reconciled  with  Cj  and   C2  by  drawing  the  point  a  into  coincidence  with  the 
intersection  of  Cj  and  C2,  and  the  point  c  into  coincidence  with  the  same  point. 

Ex.  4.  As  a  last  example,  consider  the  surface  of  a  solid  sphere  with  n  holes  bored 
through  it.  The  connectivity  is  2n  +  l  :  hence  2??  independent  irreducible  simple  circuits 
can  be  drawn  on  the  surface.  The  simplest  complete  system  is  obtained  by  taking  2n 
curves  :  made  up  of  a  set  of  n,  each  round  one  hole,  and  another  set  of  n,  each  through 
one  hole. 


168.]  DEFORMATION  OF  CONNECTED  SURFACES  379 

A  resolution  of  this  surface  is  given  by  taking  cross-cuts,  one  round  each  hole  (making 
the  circuits  through  the  holes  no  longer  possible)  and  one  through  each  hole  (making  the 
circuits  round  the  holes  no  longer  possible). 


Fig.  48. 
The  simplest  case  is  that  for  which  n  =  l:  the  surface  is  equivalent  to  the  anchor-ring. 

169.  Surfaces  are  at  present  being  considered  in  view  of  their  use  as  a 
means  of  representing  the  value  of  a  complex  variable.  The  foregoing  inves- 
tigations imply  that  surfaces  can  be  classed  according  to  their  connectivity ; 
and  thus,  having  regard  to  their  designed  use,  the  question  arises  as  to 
whether'all  surfaces  of  the  same  connectivity  are  equivalent  to  one  another, 
so  as  to  be  transformable  into  one  another. 

Moreover,  a  surface  can  be  physically  deformed  and  still  remain  suitable  for 
representation  of  the  variable,  provided  certain  conditions  are  satisfied.  We 
thus  consider  geometrical  transformation  as  well  as  physical  deformation ;  but 
we  are  dealing  only  with  the  general  results  and  not  with  the  mathematical 
relations  of  deformed  inextensible  surfaces,  which  are  discussed  in  treatises 
on  Differential  Geometry*. 

It  is  evident  that  continuity  is  necessary  for  both  :  discontinuity  would 
imply  discontinuity  in  the  representation  of  the  variable.  Points  that  are 
contiguous  (that  is,  separated  only  by  small  distances  measured  in  the  surface) 
must  remain  contiguous  f:  and  one  point  in  the  unchanged  surface  must 
correspond  to  only  one  point  in  the  changed  surface.  Hence  in  the  continuous 
deformation  of  a  surface  there  may  be  stretching  and  there  may  he  bending ; 
but  there  must  be  no  tearing  and  there  must  be  no  joining. 

For  instance,  a  single  untwisted  ribbon,  if  cut,  comes  to  be  simply  connected.  If  a 
twist  through  180°  be  then  given  to  one  end  and  that  end  be  then  joined  to  the  other, 
we  shall  have  a  once-twisted  ribbon,  which  is  a  surface  with  only  one  face  and  only  one 
edge ;   it  cannot  be  looked  upon  as  an  equivalent  of  the  former  surface. 

*  See  Darboux's  Theorie  generale  des  surfaces,  Books  vii  and  viii,  for  the  fullest  discussion. 
Some  account  is  given  in  Chapter  x  of  my  Lectures  on  the  differential  geometry  of  curves  and 
surfaces. 

t  Distances  between  points  must  be  measured  along  the  surface,  not  through  space;  the 
distance  between  two  points  is  a  length  which  one  point  would  traverse  before  reaching  the 
position  of  the  other,  the  motion  of  the  point  being  restricted  to  take  place  in  the  surface. 
Examples  will  arise  later,  in  Eiemann's  surfaces,  in  which  points  that  are  contiguous  in  space 
are  separated  by  finite  distances  on  the  surface. 


380  DEFORMATION   OF   SURFACES  [169. 

A  spherical  surface  with  a  single  hole  can  have  the  hole  stretched  and  the  surface 
flattened,  so  as  to  be  the  same  as  a  bounded  portion  of  a  plane  :  the  two  surfaces  are 
equivalent  to  one  another.  Again,  in  the  spherical  surface,  let  a  large  indentation  be 
made  :  let  both  the  outer  and  the  inner  surfaces  be  made  spherical ;  and  let  the  mouth  of 
the  indentation  be  contracted  into  the  form  of  a  long,  narrow  hole  along  a  part  of  a  great 
circle.  When  each  point  of  the  inner  surface  is  geometrically  moved  so  that  it  occupies 
the  position  of  its  reflexion  in  the  diametral  plane  of  the  hole,  the  final  form*  of  the 
whole  surface  is  that  of  a  two-sheeted  surface  with  a  junction  along  a  line  ;  it  is  a 
spherical  winding- surface,  and  is  equivalent  to  the  simply  connected  spherical  surface. 

170.  It  is  sufficient,  for  the  pnrj)ose  of  representation,  that  the  two 
surfaces  should  have  a  point-to-point  transformation :  it  is  not  necessary 
that  physical  deformation,  without  tears  or  joins,  should  be  actually  possible. 
Thus  a  ribbon  with  an  even  number  of  twists  would  be  as  effective  as  a 
limited  portion  of  a  cylinder,  or  (what  is  the  same  thing)  an  untwisted 
ribbon  :  but  it  is  not  possible  to  deform  the  one  into  the  other  physically  f. 

It  is  easy  to  see  that  either  deformation  or  transformation  of  the  kind 
considered  luill  change  a  bifacial  surface  into  a  lifacial  surface ;  that  it  tuill 
not  alter  the  connectivity,  for  it  will  not  change  irreducible  circuits  into 
reducible  circuits,  and  the  number  of  independent  irreducible  circuits  deter- 
mines the  connectivity  :  and  that  it  will  not  alter  the  number  of  boundary 
curves,  for  a  boundary  will  be  changed  into  a  boundary.  These  are  necessary 
relations  between  the  two  forms  of  the  surface :  it  is  not  difficult  to  see  that 
they  are  sufficient  for  correspondence.  For  if,  on  each  of  two  bifacial  surfaces 
with  the  same  number  of  boundaries  and  of  the  same  connectivity,  a  complete 
system  of  simple  irreducible  circuits  be  drawn,  then,  when  the  members  of  the 
systems  are  made  to  correspond  in  pairs,  the  full  transformation  can  be  effected 
by  continuous  deformation  of  those  corresponding  irreducible  circuits.  It 
therefore  follows  that : — 

The  necessary  and  sufficient  conditions,  that  two  bifacial  surfaces  may  be 
equivalent  to  one  another  for  the  representation  of  a  variable,  are  that  the  two 
surfaces  should  be  of  the  same  connectivity  and  shoidd  have  the  same  number 
of  boundaries. 

As  already  indicated,  this  equivalence  is  a  geometrical  equivalence : 
deformation  may  be  (but  is  not  of  necessity)  physically  possible. 

Similarly,  the  presence  of  one  or  of  several  knots  in  a  surface  makes  no 
essential  difference  in  the  use  of  the  surface  for  representing  a  variable. 
Thus  a  long  cylindrical  surface  is  changed  into  an  anchor-ring  when  its  ends 
are  joined  together;  but  the  changed  surface  would  be  equally  effective 
for  purposes  of  representation  if  a  knot  were  tied  in  the  cylindrical  surface 
before  the  ends  are  joined. 

*  Clifford,  Coll.  Math.  Papers,  p.  250. 

t  The  difference  between  the  two  cases  is  that,  in  physical  deformation,  the  surfaces  are  the 
surfaces  of  continuous  matter  and  are  impenetrable  ;  while,  in  geometrical  transformation,  the 
surfaces  may  be  regarded  as  penetrable  without  interference  with  the  continuity. 


170.]  REFERENCES  881 

But  it  need  hardly  be  pointed  out  that  though  surfaces,  thus  twisted  or 
knotted,  are  equivalent  for  the  purpose  indicated,  they  are  not  equivalent  for 
all  topological  enumerations. 

Seeing  that  bifacial  surfaces,  with  the  same  connectivity  and  the  same 
number  of  boundaries,  are  equivalent  to  one  another,  it  is  natural  to  adopt, 
as  the  surface  of  reference,  some  simple  surface  with  those  characteristics  ; 
thus  for  a  surface  of  connectivity  2p  +  1  with  a  single  boundary,  the  surface 
of  a  solid  sphere,  bounded  by  a  point  and  pierced  through  with  p  holes,  could 
be  adopted. 

Klein  calls*  such  a  surface  of  reference  a  Normal  Surface. 

It  has  been  seen  that  a  bounded  spherical  surface  and  a  bounded  simply  connected 
part  of  a  plane  are  equivalent — they  are,  moreover,  physically  deformable  into  one 
another. 

An  untwisted  closed  ribbon  is  equivalent  to  a  bounded  piece  of  a  plane  with  one  hole 
in  it — they  are  deformable  into  one  another :  but  if  the  ribbon,  previous  to  being  closed, 
have  undergone  an  even  number  of  twists  each  through  180°,  they  are  still  equivalent 
but  are  not  physically  deformable  into  one  another.  Each  of  the  bifacial  surfaces  is 
doubly  connected  (for  a  single  cross-cut  renders  each  simply  connected)  and  each  of  them 
has  two  boundaries.  If  however  the  ribbon,  previous  to  being  closed,  have  undergone 
an  odd  number  of  twists  each  through  180°,  the  surface  thus  obtained  is  not  equivalent 
to  the  single-holed  portion  of  the  plane  ;   it  is  unifacial  and  has  only  one  boundary. 

A  spherical  surface  pierced  in  n  +  \  holes  is  equivalent  to  a  bounded  j)ortion  of  the 
plane  with  n  holes  ;  each  is  of  connectivity  n  +  1  and  has  n-\-\  boundaries.  The  spherical 
surface  can  be  deformed  into  the  plane  sui-face  by  stretching.one  of  its  holes  into  the  form 
of  the  outside  boundary  of  the  plane  surface. 

Ex.  Prove  that  the  surface  of  a  bounded  anchor-ring  can  be  physically  deformed  into 
the  surface  in  fig.  47,  p.  378. 


For  continuation  and  fuller  development  of  the  subjects  of  the  present  chapter,  the 
following  references,  in  addition  to  those  which  have  been  given,  will  be  found  useful : — 

Klein,  Math.  Ann.,  t.  vii,  (1874),  pp.  548—557  ;   ib.,  t.  ix,  (1876),  pp.  476—482. 

Lippich,  Math.  A7in.,  t.  vii,  (1874),  pp.  212 — 229  ;  Wiener  Sitzungsb.,  t.  Ixix,  (ii), 
(1874),  pp.  91—99. 

Durfege,  Wiener  Sitzungsb.,  t.  Ixix,  (ii),  (1874),  pp.  115 — 120  ;  and  section  9  of  his 
treatise,  quoted  on  p.  363,  note. 

Neumann,  chapter  vii  of  his  treatise,  quoted  on  p.  5,  note. 

Dyck,  Math.  Ann.,  t.  xxxii,  (1888),  pp.  457—512,  ib.,  t.  xxxvii,  (1890),  pp.  273—316; 
at  the  beginning  of  the  first  part  of  this  investigation,  a  valuable  series  of 
references  is  given. 

Dingeldey,   Topologische  Studien,  (Leipzig,  Teubner,  1890). 

Mair,  Quart.  Joitrn.  of  Math.,  vol.  xxvii,  (1895),  jjp.  1 — 35. 

*  Ueber  Riemami's  Theorie  der  algebraischen  Functionen  und  ihrer  Integrale,  (Leipzig, 
Teubner,  1882),  p.  26.  This  tract  has  been  translated  into  English  by  Miss  Hardcastle, 
(Cambridge,  Macmillan  and  Bowes,  1893). 


j 


CHAPTER    XV. 

RiEM Ann's  Surfaces. 

171.  The  method  of  representing  a  variable  by  assigning  to  it  a  position 
in  a  plane  or  on  a  sphere  is  effective  when  properties  of  uniform  functions 
of  that  variable  are  discussed.  But  when  multiform  functions,  or  integrals 
of  uniform  functions  occur,  the  method  is  effective  only  when  certain  parts  of 
the  plane  are  excluded,  due  account  being  subsequently  taken  of  the  effect 
of  such  exclusions  ;  and  this  process,  the  extension  of  Cauchy's  method,  was 
adopted  in  Chapter  IX. 

There  is  another  method,  referred  to  in  §  100  as  due  to  Riemann,  of  an 
entirely  different  character.  In  Riemann's  representation,  the  region,  in 
which  the  variable  z  exists,  no  longer  consists  of  a  single  plane  but  of 
a  number  of  planes;  they  are  distinct  from  one  another  in  geometrical 
conception,  yet,  in  order  to  preserve  a  representation  in  which  the  value  of 
the  variable  is  obvious  on  inspection,  the  planes  are  infinitesimally  close  to 
one  another.  The  number  of  planes,  often  called  sheets,  is  the  same  as  the 
number  of  distinct  values  (or  branches)  of  the  function  w  for  a  general 
argument  z  and,  unless  otherwise  stated,  will  be  assumed  finite ;  each  sheet 
is  associated  with  one  branch  of  the  function,  and  changes  from  one  branch 
of  the  function  to  another  are  effected  by  making  the  ^-variable  change 
from  one  sheet  to  another,  so  that,  to  secure  the  possibility  of  change  of 
sheet,  it  is  necessary  to  have  means  of  passage  from  one  sheet  to  another. 
The  aggregate  of  all  the  sheets  is  a  surface,  often  called  a  Riemann's 
Surface. 

For  example,  consider  the  function 

the  cube  roots  being  independent  of  one  another.     It  is  evidently  a  nine-valued  function  ; 
the  number  of  sheets  in  the  appropriate  Riemann's  surface  is  therefore  nine. 

The  branch-points  are  2=0,  s=l,  2  =  oc.     Let  w  and  a  denote  a  cube-root  of  unity, 
independently  of  one  another  ;   then  the  values  of  z^  can  Be  represented  in   the  form 


171.] 


EXAMPLES    OF    RIEMANN  S    SURFACES 


383 


z'^,  cos^,  cu^s3  .    and  the  Vcxlues  of  (s  — 1)     ^  can   be  represented  in  the  form  (2  — 1)     ^, 
a^{z-l)~  '^,  a{z—l)~^.     The  nine  values  of  iv  can  be  symbolically  expressed  as  follows : — 


W'l 

1 

1 

W2 

a 

1 

W3 

0)2 

1 

Wi 

1 

a^ 

W5 

(0 

a2 

VJq 

a>2 

a-' 

2^7 

1 

a 

IVs 

0) 

a 

Wq 

0)2 

a 

Fig.  49. 


Fig.  50. 


where  the  symbols  opposite  to  v  give  the  coefficients  of  2^  and  of  (s—  1)     ^  respectively. 

Now  when  z  describes  a  small  simple  circuit  positively  round  the  origin,  the  groups 
in  cyclical  order  are  Wi,  W2,  'iVs  ;  ?i'4,  W5,  Wg  ;  %'7,  Wg,  Wg.  And  therefore,  in  the  immediate 
vicinity  of  the  origin,  there  must  be  means  of  passage  to  enable 
the  z-point  to  make  the  corresponding  changes  from  sheet  to  - — 
sheet.  Taking  a  section  of  the  whole  surface  near  the  origin  so  ZZ 
as  to  indicate  the  passages  and  regarding  the  right-hand  sides  ^^^ 
as  the  part  from  which  the  2-variable  moves  when  it  describes  a  i^ 
,  circuit  positively,  the  passages  must  be  in  character  as  indicated 
in  fig.  49.  And  it  is  evident  that  the  further  description  of 
small  simple  circuits  round  the  origin  will,  with  these  passages,  lead  to  the  proper  values : 
thus  W5,  which  after  the  single  description  is  the  value  of  w^,  becomes  Wq  after  another 
description,  and  it  is  evident  that  a  point  in  the  W5  sheet  passes  into  the  Wq  sheet. 

When  z  describes  a  small  simple  circuit  positively  round  the  point  1,  the  groups  in 
cyclical  order  are   Wi,  w^,  w- ;   w.>,  w^,  «'g  ;    103,  Wg,  ivq:    and 

therefore,  in  the  immediate  vicinity  of  the  point  1,  there  must  ~V/  | 

be   means  of  passage   to   render   possible   the   corresponding  ■— ^ —  5 

changes  of  z  from  sheet  to  sheet.     Taking  a  section  as  before  —y\  s 

near  the  point  1  and  with  similar  convention  as  to  tlie  positive 
direction  of  the  2-path,  the  jjassages  must  be  in  character  as 
indicated  in  fig.  50. 

Similarly  for  infinitely  large  values  of  z. 

If  then  the  sheets  ctin  be  so  joined  as  to  give  these  possibilities  of  passage  and  also 
give  combinations  of  them  corresponding  to  combinations  of  the  simple  jjaths  indicated, 
then  there  will  be  a  surface  to  any  point  of  which  will  correspond  one  and  only  one  value 
of  w :  and  when  the  value  of  w  is  given  for  a,  point  z  in  an  ordinary  plane  of  variation, 
then  that  value  of  w  will  determine  the  sheet  of  the  surface  in  which  the  point  z  is  to 
be  taken.  A  surface  will  then  have  been  constructed  such  that  the  function  w,  which  is 
multiform  for  the  single-plane  representation  of  the  variable,  is  uniform  for  variations 
in  the  many-sheeted  surface. 

Again,  for  the  simple  example  arising  from  the  two-valued  function,  defined  by  the 
equation 

^o  =  {{z-a){z-b){z-c)}--^ 

the  branch-points  are  a,  6,  c,  qo  ;  and  a  small  simple  circuit  round  any  one  of  these 
four  points  interchanges  the  two  values.  The  Riemann's  surface  is  two-sheeted  and 
there  must  be  means  of  passage  between  the  two  sheets  in  the  vicinity  of  a,  that  of  b, 
that  of  c,  and  at  the  infinite  part  of  the  plane. 

These  examples  are  sufficient  to  indicate  the  main  problem.     It  is  the 
construction  of  a  surface  in  which   the  independent  variable   can  move  so 


384  SHEETS  OF  RiEM Ann's  surface  [171. 

that,  for  variations  of  z  in  that  surface,  the  multiformity  of  the  function  is 
changed  to  uniformity.  From  the  nature  of  the  case,  the  character  of  the 
surface  will  depend  on  the  character  of  the  function :  and  thus,  though  all 
the  functions  are  uniform  with  their  appropriate  surfaces,  these  surfaces  are 
widely  various.  Evidently  for  uniform  functions  of  z  the  appropriate  surface 
on  the  above  method  is  the  single  plane  already  adopted. 

172.  The  simplest  classes  of  functions   for  which   a  Riemann's  surface 

is  useful  are  (i)  those   called  (§  94)   algebraic  functions,  that  is,  multiform 

functions  of  the  independent  variable  defined  by  an  algebraical  equation  of 

the  form 

f{w,z)  =  Q, 

which  is  of  finite  degree,  say  n,  in  w ;  and  (ii)  those  usually  called  Ahelian 
functions,  which  arise  through  integrals  connected  with  algebraic  functions. 

Of  such  an  algebraic  function  there  are,  in  general,  n  distinct  values ; 
but  for  the  special  values  of  z,  that  are  the  branch-points,  two  or  more  of  the 
values  coincide.  The  appropriate  Riemann's  surface  is  composed  of  n  sheets ; 
one  branch,  and  only  one  branch,  of  w  is  associated  with  a  sheet.  The 
variable  z,  in  its  relation  to  the  function,  is  determined  not  merely  by  its 
modulus  and  argument  but  also  by  its  sheet ;  that  is,  in  the  language  of  the 
earlier  method,  we  take  account  of  the  path  by  which  z  acquires  a  value. 
The  particular  sheet  in  which  z  lies  determines  the  particular  branch  of  the 
function.  Variations  of  z,  which  occur  within  a  sheet  and  do  not  coincide 
with  points  lying  in  regions  of  passage  between  the  sheets,  lead  to  variations 
in  the  value  of  the  branch  of  w  associated  with  the  sheet ;  a  return  to  an 
initial  value  of  z,  by  a  path  that  nowhere  lies  within  a  region  of  passage, 
leaves  the  ^-point  in  the  same  sheet  as  at  first  and  so  leads  to  the  initial 
branch  (and  to  the  initial  value  of  the  branch)  of  w.  But  a  return  to  an 
initial  value  of  ^^  by  a  path,  which,  in  the  former  method  of  representation, 
would  enclose  a  branch-point,  implies  a  change  of  the  branch  of  the  function 
according  to  the  definite  order  prescribed  by  the  branch-point.  Hence  the 
final  value  of  the  variable  z  on  the  Riemann's  surface  must  lie  in  a  sheet  that 
is  different  from  that  of  the  initial  (and  arithmetically  equal)  value  ;  and 
therefore  the  sheets  must  be  so  connected  that,  in  the  immediate  vicinity  of 
branch-points,  there  are  means  of  passage  from  one  sheet  to  another,  securing 
the  proper  interchanges  of  the  branches  of  the  function  as  defined  by  the 
equation. 

173.  The  first  necessity  is  therefore  the  consideration  of  the  mode  in 
which  the  sheets  of  a  Riemann's  surface  are  joined :  the  mode  is  indicated 
by  the  theorem  that  sheets  of  a  Riemann's  surface  are  joined  along  lines. 

The  junction  might  be  made  either  at  a  point,  as  with  two  spheres  in 
contact,  or  by  a  common  portion  of  a  surface,  as  with  one  prism  lying  on 


Fig.  51. 


173.]  JOINED   ALONG   BRANCH-LINES  385 

another,  or  along  lines;  but  whatever  the  character  of  the  junction  be,  it 
must  be  such  that  a  single  passage  across  it  (thereby  implying  entrance  to 
the  junction  and  exit  from  it)  must  change  the  sheet  of  the  variable. 

If  the  junction  were  at  a  point,  then  the  ^r- variable  could  change  from  one 
sheet  into  another  sheet,  only  if  its  path  passed  through  that  point :  any 
other  closed  path  would  leave  the  ^^-variable  in  its  original  sheet.  A  small 
closed  curve,  infinitesiraally  near  the  point  and  enclosing  it  and  no  other 
branch-point,  is  one  which  ought  to  transfer  the  variable  to  another  sheet, 
because  it  encloses  a  branch-point:  and  this  is  impossible  with  a  point-junction 
when  the  path  does  not  pass  through  the  point.  Hence  a  junction  at  a  point 
only  is  insufficient  to  provide  the  proper  means  of  passage  from  sheet  to 
sheet. 

If  the  junction  were  effected  by  a  common  portion 
of  surface,    then    a    passage    through    it    (implying    an        '"'''     ,.  .a 
entrance  into  that  portion  and  an  exit  from  it)  ought  to 
change  the  sheet.     But,  in  such  a  case,  closed  contours 
can  be  constructed  which  make  such  a  passage  without 
enclosing  the  branch-point  a:    thus  the  junction  would  cause  a  change  of 
sheet    for   certain    circuits    the    description    of   which    ought    to    leave    the 
^-variable  in  the  original  sheet.     Hence  a  junction  by  a  continuous  area  of 
surface  does  not  provide  the  proper  means  of  passage  from  sheet  to  sheet. 

The   only  possible  junction  which   remains  is  a  line. 

The  objection  in  the  last  case  does  not  apply  to  a  closed      — *—f — i 

contour  which  does  not  contain  the  branch-point ;  for  the  /---"' 

line    cuts   the    curve   twice   and   there   are   therefore   two  ^^^'  ^^• 

crossings ;  the  second  of  them  makes  the  variable  return  to  the  sheet  which 
the  first  crossing  compelled  it  to  leave. 

Hence  the  junction  between  any  two  sheets  takes  place  along  a  line. 

Such  a  line  is  called*  a  hranch-line.  The  branch-points  of  a  multiform 
function  lie  on  the  branch-lines,  after  the  foregoing  explanations ;  and  a 
branch-line  can  be  crossed  by  the  variable  only  if  the  variable  change  its 
sheet  at  crossing,  in  the  sequence  prescribed  by  the  branch-point  of  the 
function  which  lies  on  the  line.  Also,  the  sequence  is  reversed  when  the 
branch-line  is  crossed  in  the  reversed  direction. 

Thus,  if  two  sheets  of  a  surface  be  connected  along  a  branch-line,  a  point  which 
crosses  the  line  from  the  first  sheet  must  pass  into  the  second  and  a  point  which  crosses 
the  line  from  the  second  sheet  must  pass  into  the  first. 

Again,  if,  along  a  common  direction  of  branch-line,  the  first  sheet  of  a  surface  be 
connected  with  the  second,  the  second  with  the  third,  and  the   third  with  the  first, 

*  Sometimes  cross-line,  sometimes  branch-section.  The  German  title  is  Verzioeigungschnitt ; 
the  French  is  ligne  de  passage  ;    see  also  the  note  on  the  equivalents  of  branch-point,  p.  17.. 

P.  F.  25 


386  PROPERTIES   OF   BRANCH-LINES  [173. 

a  point  which  crosses  the  line  from  the  first  sheet  in  one  direction  must  pass  into  the 
second  sheet,  but  if  it  cross  the  hue  in  the  other  direction  it  must  pass  into  the  third 
sheet. 

A  branch-point  does  not  necessarily  affect  all  the  branches  of  a  function : 
when  it  affects  only  some  of  them,  the  corresponding  property  of  the  Niemann's 
surface  is  in  evidence  as  follows.  Let  z  =  a  determine  a  branch-point  affecting, 
say,  only  r  branches.  Take  n  points  a,  one  in  each  of  the  sheets ;  and  through 
them  draw  n  lines  cah,  having  the  same  geometrical  position  in  the  respective 
sheets.  Then  in  the  vicinity  of  the  point  a  in  each  of  the  r  sheets,  associated 
with  the  r  affected  branches,  there  must  be  means  of  passage  from  each  one 
to  all  the  rest  of  them ;  and  the  lines  cab  can  conceivabty  be  the  branch-lines 
with  a  properly  established  sequence.  The  point  a  does  not  affect  the  other 
n  —  ?'  branches :  there  is  therefore  no  necessity  for  means  of  passage  in  the 
vicinity  of  a  among  the  remaining  n  —  r  sheets.  In  each  of  these  remaining 
sheets,  the  point  a  and  the  line  cab  belong  to  their  respective  sheets  alone : 
for  them,  the  point  a  is  not  a  branch-point  and  the  line  cah  is  not  a  branch - 
line. 

174.  Several  essential  properties  of  the  branch-lines  are  immediate 
inferences  from  these  conditions. 

I.  A  free  end  of  a  branch-line  in  a  surface  is  a  branch-point. 

Let  a  simple  circuit  be  drawn  round  the  free  end  so  small  as  to  enclose  no 
branch-point  (except  the  free  end,  if  it  be  a  branch-point).  The  circuit  meets 
the  branch-line  once,  and  the  sheet  is  changed  because  the  branch-line  is 
crossed ;  hence  the  circuit  includes  a  branch-point  which  therefore  can  be 
only  the  free  end  of  the  line. 

Note.  A  branch-line  may  terminate  in  the  boundary  of  the  surface, 
and   then  the  extremity  need  not  be  a  branch-point. 

II.  When  a  branch-line  extends  beyond  a  branch-point  lying  in  its  course, 
the  sequence  of  interchange  is  not  the  same  on  the  tivo  sides  of  the  point. 

If  the  sequence  of  interchange  be  the  same  on  the  two  sides  of  the  branch- 
point, a  small  circuit  round  the  point  would  first  cross  one  part  of  the  branch- 
line  and  therefore  involve  a  change  of  sheet  and  then,  in  its  course,  would 
cross  the  other  part  of  the  branch-line  in  the  other  direction  which,  on  the 
supposition  of  unaltered  sequence,  would  cause  a  return  to  the  initial  sheet. 
In  that  case,  a  cii-cuit  round  the  branch-point  would  fail  to  secure  the  proper 
change  of  sheet.  Hence  the  sequence  of  interchange  caused  by  the  branch- 
line  cannot  be  the  same  on  the  two  sides  of  the  point. 

III.  If  two  branch-lines  with  different  sequences  of  interchange  have  a 
common  extremity,  that  point  is  either  a  branch-point  or  an  extremity  of  at 
least  one  other  branch-line. 


174.]  SYSTEM   OF   BRANCH-LINES  387 

If  the  point  be  not  a  branch-point,  then  a  simple  curve  enclosing  it,  taken 
so  small  as  to  include  no  branch-point,  must 
leave  the  variable  in  its  initial  sheet.    Let  A 
be  such  a  point,  AB  and  AC  he  two  branch- 
lines  having  A  for  a  common  extremity ;  let  TV^^ ^ ^ 

the  sequence  be  as  in  the  figure,  taken  for  a 
simple  case;  and  suppose  that  the  variable 
initially  is  in  the  rth  sheet.  A  passage  across  AR  makes  the  variable 
pass  into  the  sth  sheet.  If  there  be  no  branch-line  between  AB  and  AG 
having  an  extremity  at  A,  and  if  neither  7i  nor  m  be  s,  then  the  passage 
across  AG  makes  no  change  in  the  sheet  of  the  variable  and,  therefore,  in 
order  to  restore  r  before  AB,  at  least  one  branch-line  must  lie  in  the  angle 
between  AC  and  AB,  estimated  in  the  positive  trigonometrical  sense. 

If  either  n  or  wi,  say  n,  be  s,  then  after  passage  across  AG,  the  point  is  in 
the  ??ith  sheet ;  then,  since  the  sequences  are  not  the  same,  m  is  not  r  and 
there  must  be  some  branch-line  between  AC  and  AB  to  make  the  point 
return  to  the  7'th  sheet  on  the  completion  of  the  circuit. 

If  then  the  point  A  be  not  a  branch-point,  there  must  be  at  least  one 
other  branch-line  having  its  extremity  at  A.     This  proves  the  proposition. 

Corollary  1.  If  both  of  two  branch-lines  extend  beyond  a  point  of  inter- 
section, which  is  not  a  branch-point,  and  if  no  other  branch-line  pass  through 
the  point,  then  either  no  sheet  of  the  surface  has  both  of  them  for  branch-lines, 
or  they  are  branch-lines  for  tivo  sheets  that  are  the  same. 

Corollary  2.  If  a  change  of  sequence  occur  at  any  point  of  a  branch- 
line,  then  either  that  point  is  a  branch-point  or  it  lies  also  on  some  other 
branch-line. 

Corollary  3.  No  part  of  a  branch-line  with  only  one  branch-point  on  it 
can  be  a  closed  curve. 

It  is  evidently  superfluous  to  have  a  branch-line  without  any  branch-point 
on  it. 

175.  On  the  basis  of  these  properties,  we  can  obtain  a  system  of  branch- 
lines  satisfying  the  requisite  conditions  which  are  : — 

(i)  the  proper  sequences  of  change  from  sheet  to  sheet  must  be 
secured  by  a  description  of  a  simple  circuit  round  a  branch- 
point :  if  this  be  satisfied  for  each  of  the  branch-points,  it 
will  evidently  be  satisfied  for  any  combination  of  simple  circuits, 
that  is,  for  any  path  whatever  enclosing  one  or  more  branch- 
points. 

(ii)  the  sheet,  in  which  the  variable  re-assumes  its  initial  value  after 
describing  a  circuit,  that  encloses  no  branch-point,  must  be  the 
initial  sheet. 

25—2 


388  ^  SYSTEM   OF    BRANCH-LINES  [175. 

In  the  ^■-plane  of  Cauchy's  method,  let  lines  be  drawn  from  any  point  /,  not 
a  branch-point  in  the  first  instance,  to  each  of  the  branch- points,  as  in  fig.  19, 
p.  185,  so  that  the  joining  lines  do  not  meet  except  at  /:  and  suppose  the 
?z-sheeted  Riemann's  surface  to  have  branch-lines  coinciding  geometrically 
with  these  lines,  as  in  §  173,  and  having  the  sequence  of  interchange  for 
passage  across  each  the  same  as  the  order  in  the  cycle  of  functional  values 
for  a  small  circuit  round  the  branch-point  at  its  free  end.  No  line  (or  part 
of  a  line)  can  be  a  closed  curve ;  the  lines  need  not  be  straight,  but  they 
will  be  supposed  drawn  as  direct  as  possible  to  the  points  in  angular 
succession. 

The  first  of  the  above  requisite  conditions  is  satisfied  by  the  establish- 
ment of  the  sequence  of  interchange. 

To  consider  the  second  of  the  conditions,  it  is  convenient  to  divide 
circuits  into  two  kinds,  (a)  those  which  exclude  /,  (yS)  those  which  include  /, 
no  one  of  either  kind  (for  our  present  purpose)  including  a  branch-point. 

A  closed  circuit,  excluding  /  and  all  the  branch-points,  must  intersect  a 
branch-line  an  even  number  of  times,  if 
it  intersect  the  line  in  real  points.  Let 
the  figure  (fig.  54)  represent  such  a  case: 
then  the  crossings  at  A  and  B  counter- 
act one  another  and  so  the  part  be- 
tween A  and  B  may  without  effect  be 
transferred  across  IB3  so  as  not  to  cut 
the  branch-line  at  all.  Similarly  for 
the  points  C  and  I) :  and  a  similar 
transference  of  the  part  now  between 
C  and  D  may  be  made  across  the 
branch-line  without  effect :  that  is,  the 
circuit  can,  without  effect,  be  changed 
so  as  not  to  cut  the  branch-line  IB3  at  all.  A  similar  change  can  be  made 
for  each  of  the  branch-lines :  and  so  the  circuit  can,  without  effect,  be  changed 
into  one  which  meets  no  branch-line  and  therefore,  on  its  completion,  leaves 
the  sheet  unchanged. 

A  closed  circuit,  including  /  but  no  branch-point,  must  meet  each  branch- 
line  an  odd  number  of  times.  A  change  similar  in  character  to  that  in 
the  previous  case  may  be  made  for  each  branch-line :  and  without  affecting 
the  result,  the  circuit  can  be  changed  so  that  it  meets  each  branch-line  only 
once.  Now  the  effect  produced  by  a  branch-line  on  the  function  is  the  same 
as  the  description  of  a  simple  loop  round  the  branch-point  which  with  / 
determines  the  branch-line :  and  therefore  the  effect  of  the  circuit  at  present 
contemplated  is,  after  the  transformation  which  does  not  affect  the  result,  the 
same  as  that  of  a  circuit,  in  the  previously  adopted  mode  of  representation. 


175.]  FOE   A   SURFACE  389 

enclosing  all  the  branch-points.  But,  by  Cor.  III.  of  §  90,  the  effect  of  a 
circuit  which  encloses  all  the  branch-points  (including  ^^  =  oo  ,  if  it  be  a 
branch-point)  is  to  restore  the  value  of  the  function  which  it  had  at  the 
beginning  of  the  circuit :  and  therefore  in  the  present  case  the  effect  is  to 
make  the  point  return  to  the  sheet  in  which  it  lay  initially. 

It  follows  there^re  that,  for  both  kinds  of  a  closed  circuit  containing  no 
branch-point,  the  effect  is  to  make  the  ^•-variable  return  to  its  initial  sheet 
on  resuming  its  initial  value  at  the  close  of  the  circuit. 

Next,  let  the  point  /  be  a  branch-point;  and  let  it  be  joined  by  lines, 
as  direct*  as  possible,  to  each  of  the  other  branch-points  in  angular  suc- 
cession. These  lines  will  be  regarded  as  the  branch-lines ;  and  the  sequence 
of  interchange  for  passage  across  any  one  is  made  that  of  the  interchange 
prescribed  by  the  branch-point  at  its  free  extremity. 

The  proper  sequence  of  change  is  secured  for  a  description  of  a  simple 
closed  circuit  round  each  of  the  branch-points  other  than  /.  Let  a  small 
circuit  be  described  round  I;  it  meets  each  of  the  branch-lines  once  and 
therefore  its  effect  is  the  same  as,  in  the  language  of  the  earlier  method  of 
representing  variation  of  z,  that  of  a  circuit  enclosing  all  the  branch-points 
except  /.  Such  a  circuit,  when  taken  on  the  Neumann's  sphere,  as  in  Cor.  III., 
§  90  and  Ex.  2,  §  104,  may  be  regarded  in  two  ways,  according  as  one  or 
other  of  the  portions,  into  which  it  divides  the  area  of  the  sphere,  is  regarded 
as  the  included  area ;  in  one  way,  it  is  a  circuit  enclosing  all  the  branch- 
points except  /,  in  the  other  it  is  a  circuit  enclosing  /  alone  and  no  other 
branch-point.  Without  making  any  modification  in  the  final  value  of  w,  it 
can  (by  §  90)  be  deformed,  either  into  a  succession  of  loops  round  all  the 
branch-points  save  one,  or  into  a  loop  round  that  one ;  the  effect  of  these  two 
deformations  is  therefore  the  same.  Hence  the  effect  of  the  small  closed 
circuit  round  /  meeting  all  the  branch-lines  is  the  same  as,  in  the  other 
mode  of  representation,  that  of  a  small  curve  round  /  enclosing  no  other 
branch-point;  and  therefore  the  adopted  set  of  branch-lines  secures  the 
proper  sequence  of  change  of  value  for  description  of  a  circuit  round  1. 

The  first  of  the  two  necessary  conditions  is  therefore  satisfied  by  the 
present  arrangement  of  branch-lines. 

The  proof,  that  the  second  of  the  two  necessary  conditions  is  also  satisfied 
by  the  present  arrangement  of  branch-lines,  is  similar  to  that  in  the  preceding 
case,  save  that  only  the  first  kind  of  circuit  of  the  earlier  proof  is  possible. 

It  thus  appears  that  a  system  of  branch-lines  can  be  obtained  which 
secures  the  proper  changes  of  sheet  for  a  multiform  function :  and  therefore 
Riemann's  surfaces  can  be  constructed  for  such  a  function,  the  essential 
property  being  that  over  its  appropriate  surface  an  otherwise  multiform 
function  of  the  variable  is  a  uniform  function. 

*  The  reason  for  this  will  appear  in  §§  183,  184. 


890  EXAMPLES  [175. 

The  multipartite  character  of  the  function  has  its  influence  preserved  by 
the  character  of  the  surface  to  which  the  function  is  referred :  the  surface, 
consisting  of  a  number  of  sheets  joined  to  one  another,  may  be  a  multiply 
connected  surface. 

In  thus  proving  the  general  existence  of  appropriate  surfaces,  there  has 
remained  a  large  arbitrary  element  in  their  actual  contraction :  moreover, 
in  particular  cases,  there  are  methods  of  obtaining  varied  configurations  of 
branch-lines.  Thus  the  assignment  of  the  n  branches  to  the  n  sheets  has 
been  left  unspecified,  and  is  therefore  so  far  arbitrary :  the  point  I,  if  not  a 
branch-point,  is  arbitrarily  chosen  and  so  there  is  a  certain  arbitrariness  of 
position  in  the  branch-lines.  Naturally,  what  is  desired  is  the  simplest 
appropriate  surface  :  the  particularisation  of  the  preceding  arbitrary  qualities 
is  used  to  derive  a  canonical  form  of  the  surface. 

176.  The  discussion  of  one  or  two  simple  cases  will  help  to  illustrate  the 
mode  of  junction  between  the  sheets,  made  by  branch-lines. 

The  simplest  case  of  all  is  that  in  which  the  surface  has  only  a  single 
sheet:  it  does  not  require  discussion. 

The  case  next  in  simplicity  is  that  in  which  the  surface  is  two-sheeted : 
the  function  is  therefore  two-valued  and  is  consequently  defined  by  a 
quadratic  equation  of  the  form 

Lu-  +  2Mu  +  N  =  0, 
where  L,  M,  and  N  are  uniform  functions  of  z.     When  a  new  variable  w  is 
introduced,  defined  by  Lu  +  M=w,  so  that  values  of  w  and  of  u  correspond 

uniquely,  the  equation  is 

w^  =  M-'-LN=P(z). 

It  is  evident  that  every  branch-point  of  u  is  a  branch-point  of  iv,  and 
vice  versa ;  hence  the  Riemann's  surface  is  the  same  for  the  two  equations. 
Now  any  root  of  P  (z)  of  odd  degree  is  a  branch-point  of  w.     If  then 

P(z)  =  Q^{z)R{z), 
where  R  (z)  is  a  product  of  only  simple  factors,  every  factor  of  R  (z)  leads  to 
a  branch-point.  If  the  degree  of  R  (z)  be  even,  the  number  of  branch-points 
for  finite  values  of  the  variable  is  even,  and  ^  =  co  is  not  a  branch-point;  if  the 
degree  of  R  (z)  be  odd,  the  number  of  branch-points  for  finite  values  of  the 
variable  is  odd,  and  z  =  fx>  is  a  branch-point :  in  either  case,  the  number  of 
branch-points  is  even. 

There  are  only  two  values  of  w,  and  the  Riemann's  surface  is  two-sheeted: 
crossing  a  branch -line  therefore  merely  causes  a  change  of  sheet.  The  fi-ee 
ends  of  branch-lines  are  branch-points ;  a  small  circuit  round  any  branch- 
point causes  an  interchange  of  the  branches  w,  and  a  circuit  round  any  two 
branch-points  restores  the  initial  value  of  w  at  the  end  and  therefore  leaves 
the  variable  in  the  same  sheet  as  at  the  beginning.  These  are  the  essential 
requirements  in  the  present  case;  all  of  them  are  satisfied  by  taking  each 


176.]  OF  riemann's  surfaces  391 

branch-line  as  a  line  connecting  two  (and  only  tiuo)  of  the  branch-points.  The 
ends  of  all  the  branch-lines  are  free :  and  their  number,  in  this  method,  is 
one-half  that  of  the  (even)  number  of  branch-points.  A  small  circuit  round 
a  branch-point  meets  a  branch-line  once  and  causes  a  change  of  sheet;  a 
circuit  round  two  (and  not  more  than  two)  branch-points  causes  either  no 
crossing  of  branch-line  or  an  even  number  of  crossings  and  therefore  restores 
the  variable  to  the  initial  sheet. 

A  branch-line  is,  in  this  case,  usually  drawn  in  the  form  of  a  straight  line 
when  the  surface  is  plane :  but  this  form  is  not  essential  and  all  that  is 
desirable  is  to  prevent  intersections  of  the  branch-lines. 

IS^ote.  Junction  between  the  sheets  along  a  branch-line  is  easily  secured. 
The  two  sheets  to  be  joined  are  cut  along  the  branch-line.  One  edge  of  the 
cut  in  the  upper  sheet,  say  its  right  edge  looking  along  the  section,  is  joined 
to  the  left  edge  of  the  cut  in  the  lower  sheet ;  and  the  left  edge  in  the  upper 
sheet  is  joined  to  the  right  edge  in  the  lower. 

A  few  simple  examples  will  illustrate  these  remarks  as  to  the  sheets :  illustrations  of 
closed  circuits  will  arise  later,  in  the  consideration  of  integrals  of  multiform  functions. 

Kv.  I.     Let  w^=A(z-a){z-b), 

so  that  a  and  b  are  the  only  branch-points.  The  surface  is  two-sheeted  :  the  line  ab  may- 
be made  the  branch-line.  In  fig.  55  only  part  of  the  upper  sheet  is  shewn*,  as  likewise 
only  part  of  the  lower  sheet.  Continuous  lines  imply  what  is  visible ;  and  dotted  lines 
what  is  invisible,  on  the  supposition  that  the  sheets  are  opaque. 

The  circuit,  closed  in  the  surface  and  passing  round  a,  is  made  up  of  the  continuous 
line  in  the  upper  sheet  from  H  to  K :  the  point  crosses  the  branch-line  at  K  and  then 
passes  into  the  lower  sheet,  where  it  describes  the  dotted  line  from  K  to  H:  it  then  meets 
and  crosses  the  branch-line  at  H,  passes  into  the  upper  sheet  and  in  that  sheet  returns  to 
its  initial  jjoint.  Similarly  of  the  line  A  BC\  the  part  AB  lies  in  the  lower  sheet,  the  part 
BG  in  the  upper :  of  the  line  DG  the  part  DE  lies  in  the  upper  sheet,  the  part  EFG  in  the 
lower,  the  piece  FG  of  this  part  being  there  visible  beyond  the  boundary  of  the  retained 
portion  of  the  upper  surface. 

Ex.±     Let  \w'^=z^-a^. 

The  branch-points  (fig.  56)  are  A{  =  a),  B{  =  aa),  G{  =  aa^),  where  a  is  a  primitive  cube 
root  of  unity,  and  ?=  qo  .  The  branch -lines  can  be  made  by  BC,  Aco  ;  and  the  two-sheeted 
surface  is  a  surface  over  which  w  is  uniform.  Only  a  part  of  each  sheet  is  shewn  in  the 
figure  ;  a  section  also  is  made  at  i/"  across  the  surface,  cutting  the  branch-line  Aco  . 

Ex.  3.     Let  iv'^^z'', 

where  n  and  m  are  prime  to  each  other.  The  branch-points  are  z=Q  and  z  =  cc  ;  and  the 
branch-line  extends  from  0  to  oo  .  There  are  m  sheets  ;  if  we  associate  them  in  order  with 
the  branches  u\,  where 

for  s  =  l,  2,  ...,  HI,  then  the  first  sheet  is  connected  with  the  second  forwards,  the  second 
with  the  third  forwards,  and  so  on ;  the  mih  being  connected  with  the  first  forwards. 

*  The  form  of  the  three  figures  in  the  plate  opposite  p.  392  is  suggested  by  Holzmiiller, 
Einfuhrung  in  die  Theorie  der  isogonalen  Venoandschaften  und  der  conformen  Abhildungen, 
(Leipzig,  Teubner,  1882),  in  which  several  illustrations  are  given. 


592 


EXAMPLES   OF   RIEMANN  S   SURFACES 


[176. 


The  surface  is  sometimes  also  called  a  winding -surface  ;  and  a  branch-point  such  as 
z=0  on  the  surface,  where  a  number  m  of  sheets  j)ass  into  one  another  in  succession,  is 
also  called  a  ivinding -point  of  order  yn  -  1  (see  p.  17,  note).  An  illustration  of  the  surface 
for  m  =  3  is  given  in  fig.  57,  the  branch-line  being  cut  so  as  to  shew  the  branching :  what 
is  visible  is  indicated  by  continuous  lines  ;  what  is  in  the  second  sheet,  but  is  invisible, 
is  indicated  by  the  thickly  dotted  line  ;  what  is  in  the  third  sheet,  but  is  invisible,  is 
indicated  by  the  thinly  dotted  line. 

Ex.  4.  Consider  a  three-sheeted  surface  having  four  branch-points  at  a,  b,  c,  d ;  and 
let  each  point  interchange  two  branches,  say,  ^2,  lOs  Bit  a;   w,,  W3  at  6;    if 2,  W3  at  c; 


■?i'2,  w^  at  d;  the  points  being  as  in  fig.  58.     It  is  easy  to  verify  that  these  branch -points 
satisfy  the  condition  that  a  circuit,  enclosing  them  all,  restores  the  initial  value  of  w. 

The  branching  of  the  sheets  may  be  made  as  in  the  figure,  the  integers  on  the  two  sides 
of  the  line  indicating  the  sheets  that  are  to  be  joined  along  the  line. 

A  canonical  form  for  such  a  surface  can  be  derived  from  the  more  general  case  given 
later  (in  §§  186—189). 

Ex.  5.     Shew  that,  if  the  equation 

fi^v,z)=^0 

be  of  degree  M  in  ^v  and  be  irreducible,  all  the  71  sheets  of  the  surface  are  connected,  that 
is,  it  is  possible  by  an  appropriate  path  to  pass  from  any  sheet  to  any  other  sheet. 

For  if  not,  let  a  denote  any  arbitrary  value  of  z,  and  let  Ui,  U2,  ...,  m„  denote  the 
n  values  of  w  when  z  =  a.  Let  z  vary,  beginning  with  a  value  a  ;  let  the  variation  be 
restricted  solely  by  the  condition  that  z  does  not  acquire  a  value  giving  rise  to  a  branch- 
point, and  otherwise  be  perfectly  general ;  and  let  z  return  to  the  value  a.  If  it  is  not 
possible  to  pass  from  any  sheet  of  the  Riemann's  surface  to  any  other,  suppose  that  the 
first,  second,  ...,  mth  sheets  are  connected  with  one  another,  and  that  no  one  of  them 
is  connected  with  any  one  of  the  rest.  Then  whatever  be  the  variation  of  z,  and  whichever 
of  the  values  Ui,  U2,  ...,  u^  he  chosen  as  an  initial  value  of  w,  the  final  value  of  w  (when  z 
resumes  its  value  a)  will  be  one  of  the  set  iii,  U2,  ...,  «,„•  Hence  any  rational  symmetric 
function  of  Ui,  U2,  ...,  «,„  remains  unchanged  when  z,  after  varying  quite  arbitrarily, 
resumes  an  initial  value;  in  other  words,  that  symmetric  function  of  Ui,  U2,  ...,  w,„  is 
a  uniform  function  of  z,  which  (as  in  §  193)  is  a  rational  function  of  z.  Consequently, 
the  values  Ui,  «2)  •••)  «,„  of  w  are  the  roots  of  an  algebraical  eqiiation  f\  {w,  z)  =  0,  which 
is  polynomial  in  iv  and  s,  and  is  of  degree  m  in  tv.  But  these  values  of  w  are  roots 
of /(w,  z)=0;  hence  f{w,  z)  is  divisible  by /j  {;iv,  z),  contrary  to  the  given  condition  that 
f{iv,z)  =  0  is  irreducible. 

Corollary  I.  When  /=0  is  irreducible,  it  is  possible  to  make  z  vary  from  an  initial 
value,  and  return  to  a,  in  such  a  way  that  any  assigned  initial  value  of  lu  shall  lead  to 
any  assigned  final  value  of  xv,  among  the  n  values  which  it  has  for  z  =  a. 

Corollary  II.  If  3  =  a,  u'=A,  and  z  =  ^,  w  =  B  are  any  two  positions  on  the  Riemann's 
surface  corresponding  to  an  equation  f{w,z)  =  0,  and  if  a  path   exists  in  the   surface 


c ' 


-^  b  ^, 


Fig.  o5. 


Fiff.  .56. 


)\ 


Ym.  57. 


17 6. J  SPHEKICAL   RIEM ANN'S   SURFACE  393 

joining  the  one  position  to  the  other,  then  /  is  either  an  irreducible  polynomial  or  is 
some  power  of  an  irreducible  polynomial. 

For  if  /  can  be  resolved  into  the  product  of  two  different  polynomials,  each  of  them 
equated  to  zero  would  give  rise  to  a  Riemann's  surface  ;  and  the  two  surfaces  would  not 
be  connected,  so  that  it  would  be  impossible  to  pass  from  any  position  on  one  of  them 
to  any  position  on  the  other.  If  therefore /is  resoluble,  its  component  polynomials  must 
be  one  and  the  same :  that  is,  on  the  given  hypothesis,  when  /  is  reducible,  it  is  a  power 
of  an  irreducible  polynomial. 

177.  It  is  not  necessary  to  limit  the  surface  representing  the  variable  to 
a  set  of  planes ;  and,  indeed,  as  with  uniform  functions,  there  is  a  convenience 
in  using  the  sphere  for  the  purpose. 

We  take  n  spheres,  each  of  diameter  unity,  touching  the  Riemann's  plane 
surface  at  a  point  A;  each  sphere  is  regarded  as  the  stereographic  projection 
of  a  plane  sheet,  with  regard  to  the  other  extremity  A'  of  the  spherical 
diameter  through  A.  Then,  the  sequence  of  these  spherical  sheets  being 
the  same  as  the  sequence  of  the  plane  sheets,  branch-points  in  the  plane 
surface  project  into  branch-points  on  the  spherical  surface :  branch-lines 
between  the  plane  sheets  project  into  branch-lines  between  the  spherical 
sheets  and  are  terminated  by  corresponding  points ;  and  if  a  branch-line 
extend  in  the  plane  surface  to  z=  cc  ,  the  corresponding  branch-line  in  the 
spherical  surface  is  terminated  at  A'. 

A  surface  will  thus  be  obtained  consisting  of  n  spherical  sheets;  like 
the  plane  Riemann's  surface,  it  is  one  over  which  the  ?i-valued  function  is  a 
uniform  function  of  the  position  of  the  variable  point. 

But  also  the  connectivity  of  the  n-sheeted  spherical  surface  is  the  same  as 
that  of  the  n-sheeted  plane  surface  tuith  which  it  is  associated. 

In  fact,  the  plane  surface  can  be  mechanically  changed  into  the  spherical 
surface  without  tearing,  or  repairing,  or  any  change  except  bending  and 
compression :  all  that  needs  to  be  done  is  that  the  n  plane  sheets  shall  be 
bent,  without  making  any  change  in  their  sequence,  each  into  a  spherical 
form,  and  that  the  boundaries  at  infinity  (if  any)  in  the  plane  sheet  shall 
be  compressed  into  an  infinitesimal  point,  being  the  South  pole  of  the 
corresponding  spherical  sheet  or  sheets.  Any  junctions  between  the  plane 
sheets  extending  to  infinity  are  junctions  terminated  at  the  South  pole.  As 
the  plane  surface  has  a  boundary,  which,  if  at  infinity  on  one  of  the  sheets,  is 
therefore  not  a  branch-line  for  that  sheet,  so  the  spherical  surface  has  a 
boundary  which,  if  at  the  South  pole,  cannot  be  the  extremity  of  a  branch- 
line. 

178.  We  proceed  to  obtain  the  connectivity  of  a  Riemann's  surface : 
it  is  determined  by  the  following  theorem : — 

Let  the  total  number  of  branch-points  in  a  Riemann's  n-sheeted  surface 
be  r ;  and  let  the  number  of  branches  of  the  function  interchanging  at  the  first 


394  CONNECTIVITY    OF   A  [178. 

point  he  m-^,  the  iiumher  interchanging  at  the  second  he  m^,  and  so  on.     Then 
the  connectivity  of  the  surface  is 

n  -  2w  +  3, 

where  fl  denotes  Wj  +  ?7io  +  . . .  +  nir  —  r. 

Take  *  the  surface  in  the  bounded  spherical  form,  the  connectivity  N  of 
which  is  the  same  as  that  of  the  plane  surface :  and  let  the  boundary  be  a 
small  hole  A  in  the  outer  sheet.  By  means  of  cross-cuts  and  loop-cuts,  the 
surface  can  be  resolved  into  a  number  of  distinct  simply  connected  pieces. 

First,  make  a  slice  bodily  through  the  sphere,  the  edge  in  the 
outside  sheet  meeting  A  and  the  direction  of  the 
slice  through  A  being  chosen  so  that  none  of  the 
branch-points  lie  in  any  of  the  pieces  cut  off.  Then  n 
parts,  one  from  each  sheet  and  each  simply  connected, 
are  taken  away.  The  remainder  of  the  surface  has  a 
cup-like  form ;  let  the  connectivity  of  this  remainder 
be  M. 

This  slice  has  implied  a  number  of  cuts. 

The   cut  made  in  the  outside  sheet  is  a  cross-cut, 
because   it   begins   and   ends   in   the  boundary  A.     It  'ig- 

divides  the  surface  into  two  distinct  pieces,  one  being 
the  portion  of  the  outside  sheet  cut  off,  and  this  piece  is  simply  connected ; 
hence,  by  Prop.  III.  of  §  160,  the  remainder  has  its  connectivity  still  repre- 
sented by  N. 

The  cuts  in  all  the  other  sheets,  caused  by  the  slice,  are  all  loop-cuts, 
because  they  do  not  anywhere  meet  the  boundary.  There  are  n  —  1  loop- 
cuts,  and  each  cuts  off  a  simply  connected  piece  ;  let  the  remaining  surface 
be  of  connectivity  M.     Hence,  by  Prop.  V.  of  §  161, 

M+n-l  =  N+2{n-l), 

and  therefore  M  =  N  +  n  —  1. 

In  this  remainder,  of  connectivity  M,  make  r  —  1  cuts,  each  of  which 
begins  in  the  rim  and  returns  to  the  rim,  and  is  to  be  made  through  the  n 
sheets  together ;  and  choose  the  directions  of  these  cuts  so  that  each  of  the 
r  resulting  portions  of  the  surface  contains  one  (and  only  one)  of  the  branch- 
points. 

Consider  the  portion  of  the  surface  which  contains  the  branch-point 
where  mi  sheets  of  the  surface  are  connected.  The  m^  connected  sheets 
constitute  a  piece  of  a  winding- surface  round  the  winding-point  of  order 
mi  —  1  ;    the    remaining   sheets   are    unaffected    by    the   winding-point,  and 

*  The  proof  is  founded  on  Neumann's,  Vorlesungen  ilber  Riemann's  Theorie  der  AbeVschen 
Integrale,  pp.  168 — 172. 


178. J  riemann's  surface  395 

therefore  the  parts  of  them  are  n  —  m-^  distinct  simply  connected  pieces. 
The  piece  of  winding-surface  is  simply  connected ;  because  a  circuit,  that 
does  not  contain  the  winding-point,  is  reducible  without  passing  over  the 
winding-point,  and  a  circuit,  that  does  contain  the  winding-point,  is  reducible 
to  the  winding-point,  so  that  no  irreducible  circuit  can  be  drawn.  Hence 
the  portion  of  the  surface  under  consideration  consists  of  n  —  7ni+l  distinct 
simply  connected  pieces. 

Similarly  for  the  other  portions.     Hence  the  total  number  of   distinct 
simply  connected  pieces  is 


S   (n 

q  =  l 

—  771 

,  +  i) 

=  nr  - 

r 
.    % 
q  =  l 

niq  +  r 

=  )ir  - 

n. 

But  in  the  portion  of  connectivity  if  each  of  the  r  — 1  cuts  causes,  in 
each  of  the  sheets,  a  cut  passing  from  the  boundary  and  returning  to  the 
boundary,  that  is,  a  cross-cut.  Hence  there  are  n  cross-cuts  from  each  of  the 
5 —  1  cuts,  and  therefore  n  (r  —  1)  cross-cuts  altogether,  made  in  the  portion  of 
surface  of  connectivity  M. 

The  effect  of  these  %(r—  1)  cross-cuts  is  to  resolve  the  portion  of  con- 
nectivity M  into  7ir  —  fi  distinct  simply  connected  pieces  ;  hence,  by  |  160, 

M=7i(r-l)-{7ir-n)+2, 

and  therefore  N=M -  {71- 1)  =  n  -  271  +  3, 

the  connectivity  of  the  Riemann's  surface. 

r 

The  quantity  O,  having  the  value  2  (niq  —  1),  may  be  called  the  rcuni- 

q  =  l 

ficatio7i  of  the   surface,  as   indicating  the  aggregate  sum  of  the  orders  of 
the  different  branch-points. 

Note.  The  surface  just  considered  is  a  closed  surface  to  which  a  point 
has  been  assigned  for  boundary;  hence,  by  Cor.  1,  Prop.  III.,  §  164,  its 
connectivity  is  an  odd  integer.     Let  it  be  denoted  by  2p  +  1 ;   then 

2p  =  n-27i  +  2, 

and  2p  is  the  number  of  cross-cuts  which  change  the  Riemann's  surface  into 
one  that  is  simply  connected. 

The  integer  p  is  often  called  (Cor.  1,  Prop.  III.,  §  164)  the  geTius  of  the 
Riemann's  surface ;  and  the  equatioTi 

f{w,z)  =  0 

is  said  to  he  of  genus  p,  when  p  is  the  genus  of  the  associated  Miemanns 
surface. 


396  EXAMPLES    OF    CONNECTIVITY   OF  [178. 

The  genus  of  an  equation  is  discussed,  partly  in  association  with  Abel's 
Theorem  on  transcendental  integrals,  in  an  interesting  paper*  by  Baker,  who 
gives  a  simple  graphical  rule  to  determine  the  integer  when  the  coefficients 
are  general.     This  rule  is  given  in  the  example  at  the  end  of  §  182. 

Ex.  1.     When  the  equation  is 

iv"  =  \  (z  —  a)  (z~  b), 

we  have  a  two-sheeted  surface,  n  =  2.  There  are  two  branch-points,  s  =  a  and  z  =  b;  but 
2  =  CO  is  not  a  branch-point;  sotliatr  =  2.  At  each  of  the  branch-points  the  two  values  are 
interchanged,  so  that  mi  — 2,  m-2  =  'i;  thus  i2  =  2.  Hence  the  connectivity  =2  — 4-1-3  =  1, 
that  is,  the  surface  is  simply  connected. 

The  surface  can  be  deformed,  as  in  tlae  example  in  §  169,  into  a  sphere. 
Ex.  2.     When  the  equation  is 

we  have  n  =  %  There  are  four  branch-points,  viz.,  gj,  e^,  %,  Qo ,  so  that  r=4 ;  and  at  each 
of  them  the  two  values  of  w  are  interchanged,  so  that  mg  =  2  (for  s  =  1,  2,  3,  4),  and  therefore 
12  =  8  —  4  =  4.     Hence  the  connectivity  is  4  —  4-1-3,  that  is,  3  ;  and  the  value  of  jo  is  unity. 

Similarly,  the  surface  associated  with  the  equation^ 

iv'^=U{z), 

where  U  (z)  is  a  rational  integral  function  of  degree  2m  - 1  or  of  degree  2m,  is  of  con- 
nectivity 2j>i  + 1 ;  so  that  JO  =  m.     The  equation 

w'^^{l-z^)il-kh^) 

is  of  genus  p  =  l.  The  case  next  in  importance  is  that  of  the  algebraical  equation  leading 
to  the  hyperelliptic  functions,  when  t^is  either  a  quintic  or  a  sextic ;  and  then  p  =  2. 

Ex.  3.     Obtain  the  connectivity  of  the  Riemann's  surface  associated  with  the  equation 

w^  +  2^  —  3awz  =  l, 
where  a  is  a  constant,  (i)  when  a  is  zero,  (ii)  when  a  is  different  from  zero. 

Ex.  4.     Shew  that,  if  the  surface  associated  with  the  equation 

f(u',z)  =  0, 

have  /x  boundary-lines  instead  of  one,  and  if  the  equation  have  the  same  branch-points 
as  in  the  foregoing  proposition,  the  connectivity  is  Q,-2n  +  fx  +  2. 

Ex.  5.     Shew  that  the  genus  of  the  equation 

w^-z^z'^  +  z+l)  =  0 

is  1 ,  and  that  the  genus  of  the  equation 

tv^  +  z^=^5wz^ 
is  2.  (Raffy.) 

Discuss  the  genus  of  the  equation 

z(fi-5w'^{z'-+z  +  l}  +  5iv{z^-  +  z+lf-2z{z'^+z  +  iy=0. 

(Raffy:  Baker.) 

*  "  Examples  of  the  application  of  Newton's  polygon  to  the  theory  of  singular  points  of 
algebraic  functions,"  Camb.  Phil.  Trans.,  vol.  xv,  (1894),  pp.  403 — 405. 


178.]  riemann's  surfaces  397 

Ex.  6.     In  the  equation 

the  sum  of  the  positive  integers  n-^,  ...,  n^  is  divisible  by  n.     Shew  that  the  genus  p  of  the 
associated  Riemann's  surface  is  given  by 

where  X,  is  the  greatest  common  measure  of  Uq  and  n. 

Shew  also  that,  for  surfaces  of  a  given   genus  p,  associated  with  equations  of  the 
assigned  form,  s  cannot  be  greater  than  ip-i.  (Trinity  Fellowship,  1897.) 

Ex.  7.     Shew  that  the  values  of  ^  for  the  equations 

(i)     w^-z^  +  ^2ivz(ivh^~l)  =  Q: 

(ii)    {w'^-z^f-Awh^{wz-lf=0: 
are  7  and  3  respectively.  (Cayley.) 

Ex.  8.     Shew  that  the  genus  of  the  equation 

where  n  is  a  positive  integer  or  zero,  is  unity. 
Ex.  9.     Shew  that  the  genus  p  of  the  equation 

where  11  is  a  positive  integer,  is  given  as  follows  : — 

when  n  =  Qk-a,  then  p  =  'ik  —  a,  for  a  =  l,  2,  3; 
n  =  Qk  +  a,     ...     p  =  Sk       ,    ...   a  =  l,  2; 
...      n  =  Gk       ,     ...     p  =  Sk-l. 
Ex.  10.     Find  the  genus  of  the  equation 

w-  =  il-z^)il-kh% 
where  n  is  a  positive  integer  >  2. 

179.     The  consideration  of  irreducible  circuits  on    the  surface  at    once 
reveals  the    multiple   connection  of  the  surface,  the  numerical  measure  of 
which  has  been  obtained.     In  a  Riemann's  surface,  a  simple 
closed  circuit  cannot  he  deformed  over  a  branch-point.     Let  CX-  /q, 

A  he  a  branch-point,  and  let  AE...   be   the   branch-line      -• ^a(^' 


having  a  free  end  at  A.     Take  a  curve  ...GED...  crossing  '\  V  q/ 

the  branch-line  at  E  and  passing  into   a  sheet  different  pj„  gQ 

from  that  which  contains  the  portion  CE ;  and,  if  possible, 
let  a  slight  deformation  of  the  curve  be  made  so  as  to  transfer  the  portion 
GE  across  the  branch-point  A.  In  the  deformed  position,  the  curve 
...C'E'D' ...  does  not  meet  the  branch-line;  there  is,  consequently,  no 
change  of  sheet  in  its  course  near  A  and  therefore  E'D'...,  which  is  the 
continuation  of  ...G'E',  cannot  be  regarded  as  the  deformed  position  of  ED. 
The  two  paths  are  essentially  distinct ;  and  thus  the  original  path  cannot  be 
deformed  over  the  branch-point. 

It    therefore    follows    that    continuous    deformation    of  a   circuit  over  a 
branch-point  on  a  Riemann's  surface  is  a  geometrical  impossibility. 

Ex.     Trace  the  variation  of  the  curve  CED,  as  the  j3oiut  E  moves  up  to  ^-1  and  then 
returns  along  the  other  side  of  the  branch-line. 


398 


RESOLUTION    OF   RIEMANN  S   SURFACES 


[179. 


Hence  a  circuit  containing  two  or  more  (but  not  all)  of  the  branch-points 
is  irreducible ;  a  circuit  containing  all  the  branch-points  is  equivalent  to  a 
circuit  that  contains  none  of  them,  and  it  is  therefore  reducible. 

If  a  circuit  contain  only  one  branch-point,  it  can  be  continuously  deformed 
so  as  to  coincide  with  the  point  on  each  sheet  and  therefore,  being  deformable 
into  a  point,  it  is  a  reducible  circuit.  An  illustration  has  already  occurred  in 
the  case  of  a  portion  of  winding-surface  containing  a  single  winding-point 
(Ex.  3,  p.  391);  all  circuits  drawn  on  it  are  reducible. 

It  follows  from  the  preceding  results  that  the  Riemann's  surface  associated 
with  a  multiform  function  is  generally  one  of  multiple  connection;  we  shall 
find  it  convenient  to  know  how  it  can  be  resolved,  by  means  of  cross-cuts,  into 
a  simply  connected  surface.  The  representative  surface  will  be  supposed  a 
closed  surface  with  a  single  boundary;  its  connectivity,  necessarily  odd, being 
229  +  1^  the  number  of  cross-cuts  necessary  to  resolve  the  surface  into  one 
that  is  simply  connected  is  2^;  when  these  cuts  have  been  made,  the  simply 
connected  surface  then  obtained  will  have  its  boundary  composed  of  a  single 
closed  curve. 

One  or  two  simple  examples  of  resolution  of  special  Riemann's  surfaces  will  be  useful 
in  leading  up  to  the  general  explanation ;  in  the  examples  it  will  be  shewn  how,  in 
conformity  with  §  168,  the  resolving  cross-cuts  render  irreducible  circuits  impossible. 

Ex.  1.     Let  the  equation  be 

w-'-  =  A{z-a){z-h){z-c){z-d), 
where  a,  6,  c,  d  are  four  distinct  points,  all  of  finite  modulus.  The  surface  is  two-sheeted  ; 
each  of  the  points  a,  6,  c,  rf  is  a  branch-point  where  the  two  values  of  w  interchange ;  and 
so  the  surface,  assumed  to  have  a  single  boundary,  is  triply  connected,  the  value  of  p 
being  unity.  The  branch-lines  are  two,  each  connecting  a  pair  of  branch-points ;  let  them 
be  ah  and  cd. 

Two  cross-cuts  are  necessary  and  sufficient  to  resolve  the  surface  into  one  that  is 
simply  connected.  We  first  make  a  cross-cut, 
beginning  at  the  boundary  B,  (say  it  is  in  the 
upper  sheet),  continuing  in  that  sheet  and  re- 
turning to  B,  so  that  its  course  encloses  the 
branch-line  ah  (but  not  cd)  and  meets  no  branch- 
line.  It  is  a  cross-cut,  and  not  a  loop-cut,  for  it 
begins  and  ends  in  the  boundary ;  it  is  evidently 
a  cut  in  the  upper  sheet  alone,  and  does  not 
divide  the  surface  into  distinct  portions ;  and, 
once  made,  it  is  to  be  regarded  as  boundary  for 
the  partially  cut  surface. 

The  surface  in  its  present  condition  is  con- 
nected :  and  therefore  it  is  possible  to  pass  fi'om  one  edge  to  the  other  of  the  cut  just 
made.  Let  P  be  a  point  on  it ;  a  curve  that  passes  from  one  edge  to  the  other  is  indicated 
by  the  line  PQR  in  the  upper  sheet,  RS  in  the  lower,  and  SP  in  the  upper.  Along  this 
line  make  a  cut,  beginning  at  P  and  returning  to  P;  it  is  a  cross-cut,  partly  in  the 
upper  sheet  and  partly  in  the  lower,  and  it  does  not  divide  the  surface  into  distinct 
portions. 


Fig.  61. 


179.] 


BY   CROSS-CUTS 


399 


Two  cross-cuts  in  the  triply  connected  surface  have  now  been  made ;  neither  of  them, 
as  made,  divides  the  surface  into  distinct  portions,  and  each  of  them  when  made  reduces 
the  connectivity  by  one  unit ;  hence  the  surface  is  now  simply  connected.  It  is  easy  to 
see  that  the  boundary  consists  of  a  single  line  not  intersecting  itself;  for  beginning 
at  P,  we  have  the  outer  edge  of  PBT,  then  the  inner  edge  of  FQRSP,  then  the  inner 
edge  of  PTB,  and  then  the  outer  edge  of  PSRQP,  returning  to  P. 

The  required  resolution  has  been  eflfected. 

Before  the  surface  was  resolved,  a  number  of  irreducible  circuits  could  be  drawn ;  a 
complete  system  of  irreducible  circuits  is  composed  of  two,  by  §  168.  Such  a  system  may 
be  taken  in  various  ways ;  let  it  be  composed  of  a  simple  curve  C  lying  in  the  ujnper  sheet 
and  containing  the  points  a  and  b,  and  a  simple  curve  D,  lying  partly  in  the  upper 
and  partly  in  the  lower  sheet  and  containing  the  points  a  and  c;  each  of  these  curves 
is  irreducible,  because  it  encloses  two  branch-points.  Every  other  irreducible  circuit 
is  reconcileable  with  these  two ;  the  actual  reconciliation  in  particular  cases  is  effected 
most  simply  when  the  surface  is  taken  in  a  sphei'ical  form. 

The  irreducible  circuit  C  on  the  unresolved  surface  is  impossible  on  the  resolved 
surface  owing  to  the  cross-cut  SPQRS ;  and  the  irreducible  circuit  Z>  on  the  unresolved 
surface  is  impossible  on  the  resolved  surface  owing  to  the  cross-cut  PTB.  It  is  easy 
to  verify  that  no  irreducible  circuit  can  be  drawn  on  the  resolved  surface. 

In  practice,  it  is  conveniently  effective  to  select  a  complete  system  of  irreducible 
simple  circuits  and  then  to  make  the  cross-cuts  so  that  each  of  them  renders  one  circuit 
of  the  system  impossible  on  the  resolved  surface. 

A\v.  2.     If  the  equation  be 

=  4(2-ei)(z-<?2)(2-e3), 
the  branch-points  are  gj,  62,  63  and  qc  .     When  the  two-sheeted  surface  is  spherical,  and  the 
branch-lines  are  taken  to  be  (i)  a  line  joining  ej,  62!  ^'^d  (ii)  a  line  joining  63  to  the  South 
pole,  the  discussion  of  the  surface  is  similar  in  detail  to  that  in  the  preceding  example. 

Ex.  3.     Let  the  equation  be 

iv''  =  Az(l-z)U-z)(X-z)(fjL-z), 
and  for  simplicity  suppose  that  k,  X,  fx  are  real  quantities  subject  to  the  inequalities 

1  <K    <X</X<QO. 

The  associated  surface  is  two-sheeted  and  has  a  boundary  assigned  to  it ;  assuming 
that  its  sheets  are  planes,  we  shall  take  some  point  in  the  finite  part  of  the  upper  sheet, 
not  being  a  branch-point,  as  the  boundary.  There  are  six  branch -points,  viz.,  0,  1,  k, 
X,  /i,  00  at  each  of  which  the  two  values  of  tv  interchange ;  and  so  the  connectivity  of 
the  surface  is  5,  and  its  genus  is  2.  The  branch-lines  can  be  taken  as  three,  this  being 
the  simplest  arrangement ;  let  them  be  the  lines  joining  0,  1 ;  k,  X ;  /x,  go  . 


Fig.  62. 

Four  cross-cuts  are  necessary  to  resolve  the  surface  into  one  that  is  simply  connected 
and  has  a  single  boundary.     They  may  be  obtained  as  follows. 


400  GENERAL   RESOLUTION  [179. 

Beginning  at  the  boundary  Z,.  let  a  cut  LHA  be  made  entirely  in  the  upper  sheet 
along  a  line  which,  when  complete,  encloses  the  points  0  and  1  but  no  other  branch-points ; 
let  the  cut  return  to  L.  This  is  a  cross-cut  and  it  does  not  divide  the  surface  into 
distinct  pieces ;  hence,  after  it  is  made,  the  connectivity  of  the  modified  surface  is  4,  and 
there  are  two  boundary-lines,  being  the  two  edges  of  the  cut  LHA, 

Beginning  at  a  point  A  in  LHA,  make  a  cut  along  ABO  in  the  upper  sheet  until 
it  meets  the  branch-line  /j,oo ,  then  in  the  lower  sheet  along  GSD  until  it  meets  the 
branch-line  01,  and  then  in  the  upper  sheet  from  D  returning  to  the  initial  point  A. 
This  is  a  cross-cut  and  it  does  not  divide  the  surface  into  distinct  pieces  ;  hence,  after  it 
is  made,  the  connectivity  of  the  modified  surface  is  3,  and  it  is  easy  to  see  that  there 
is  only  one  boundary-edge,  similar  to  the  single  boundary  in  Ex.  1  when  the  surface 
in  that  example  has  been  completely  resolved. 

Make  a  loop-cut  EFG  along  a  line,  enclosing  the  points  k  and  X  but  no  other  branch- 
points ;  and  change  it  into  a  cross-cut  by  making  a  cut  from  E  to  some  point  B  of  the 
boundary.  This  cross-cut  can  be  regarded  as  BEFGE,  ending  at  a  point  in  its  own 
earlier  course.  As  it  does  not  divide  the  surface  into  distinct  pieces,  the  connectivity 
is  reduced  to  2  ;   and  there  are  two  boundary-lines. 

Beginning  at  a  point  O  make  another  cross-cut  GQPEG,  as  in  the  figure,  enclosing 
the  two  branch-points  X  and  fi  and  lying  partly  in  the  upper  sheet  and  partly  in  the 
lower.  It  does  not  divide  the  surface  into  distinct  pieces  :  the  connectivity  is  reduced 
to  unity,  and  there  is  a  single  boundary-line. 

Four  cross-cuts  have  been  made  ;  and  the  surface  has  been  resolved  into  one  that 
is  simply  connected. 

It  is  easy  to  verify  : — 

(i)  that  neither  in  the  upper  sheet,  nor  in  the  lower  sheet,  nor  partly  in  the 
upper  sheet  and  partly  in  the  lower,  can  an  irreducible  circuit  be  drawn 
in  the  resolved  surface  ;  and 
(ii)  that,  owing  to  the  cross-cuts,  the  simplest  irreducible  circuits  in  the  unresolved 
surface — viz.  those  which  enclose  0,  1  ;  1,  <  ;  k,  X  ;  X,  /x  ;  respectively — are 
rendered  impossible  in  the  resolved  surface. 

The  equation  in  the  present  example,  and  the  Riemanu's  surface  associated  with  it, 
lead  to  the  theory  of  hyperelliptic  functions*. 

180.  The  last  example  suggests  a  method  of  resolving  any  two-sheeted 
surface  into  a  surface  that  is  simply  connected. 

The  number  of  its  branch-points  is  necessarily  even,  say  2p  +  2.  The 
branch-lines  can  be  made  to  join  these  points  in  pairs,  so  that  there  will  be 
p  +  1  oi  them.  To  determine  the  connectivity  (§  178),  we  have  n  =  2  and, 
since  two  values  are  interchanged  at  every  branch-point,  n  =  2p  +  2;  so 
that  the  connectivity  is  2p  + 1.  Then  2p  cross-cuts  are  necessary  for  the 
required  resolution  of  the  surface. 

We  make  cuts  round  p  of  the  branch-lines,  that  is,  round  all  of  them  but 
one ;  each  cut  is  made  to  enclose  two  branch-points,  and  each  lies  entirely  in 
the  upper  sheet.  These  are  cuts  corresponding  to  the  cuts  LHA  and  EFG 
in  fig.  62 ;  and,  as  there,  the  cut  round  the  first  branch-line  begins  and  ends 

*  One  of  the  most  direct  discussions  of  the  theory  from  this  point  of  view  is  given  by  Prj'm, 
Neue  Theorie  der  ultraelliptischen  Functionen,  (BerHn,  Mayer  and  Miiller,  2nd  ed.,  1885). 


180.]  OF  riemann's  surfaces  401 

in  the  boundary,  so  that  it  is  a  cross-cut.     All  the  remaining  cuts  are  loop- 
cuts  at  present.     This  system  of  p  cuts  we  denote  by  ai,  a.2, ...,  ap. 

We  make  other  p  cuts,  one  passing  from  the  inner  edge  of  each  of  the  p 
cuts  a  already  made  to  the  branch-line  which  it  surrounds,  then  in  the  lower 
sheet  to  the  {p  +  l)th  branch-line,  and  then  in  the  upper  sheet  returning  to 
the  point  of  the  outer  edge  of  the  cut  a  at  which  it  began.  This  system  of 
cuts  corresponds  to  the  cuts  ADSCBA  and  GQPRG  in  fig.  62.  Each  of  them 
can  be  taken  so  as  to  meet  no  one  of  the  cuts  a  except  the  one  in  which  it 
begins  and  ends ;  and  they  can  be  taken  so  as  not  to  meet  one  another. 
This  system  of  p  cuts  we  denote  by  h^,  b^,  ...,hp,  where  6,.  is  the  cut  which 
begins  and  ends  in  a^.  All  these  cuts  are  cross-cuts,  because  they  begin  and 
end  in  boundary-lines. 

Lastly,  we  make  other  p  -1  cuts  from  a,,  to  b,._^,  for  r  =  2,  S,  ...,p,  all  in 
the  upper  sheet ;  no  one  of  them,  except  at  its  initial  and  its  final  points, 
meets  any  of  the  cuts  already  made.     This  system  of  p  —  1  cuts  we  denote 

by  Ca,  Cs,  ...,Cp. 

Because  br-i  is  a  cross-cut,  the  cross-cut  c,-  changes  a^  (hitherto  a  loop-cut) 
into  a  cross-cut  when  Cr  and  a^  are  combined  into  a  single  cut. 

It  is  evident  that  no  one  of  these  cuts  divides  the  surface  into  distinct 
pieces ;  and  thus  we  have  a  system  of  2p  cross-cuts  resolving  the  two-sheeted 
surface  of  connectivity  2p  +  1  into  a  surface  that  is  simply  connected.  The 
cross-cuts  in  order*  are 

«!,  bi,  Ca  and  ag,  b^,  Cg  and  as,  b^,  ...,Cp  and  a^,  bp. 

181.  This  resolution  of  a  general  two-sheeted  surface  suggests^  Riemann's 
general  resolution  of  a  surface  with  any  (finite)  number  of  sheets. 

As  before,  we  assume  that  the  surface  is  closed  and  has  a  single  boundary 
and  that  its  genus  is  p,  so  that  2/)  cross-cuts  are  necessary  for  its  resolution 
into  one  that  is  simply  connected. 

Make  a  cut  in  the  surface  such  as  not  to  divide  it  into  distinct  pieces ; 
and  let  it  begin  and  end  in  the  boundary.  It  is  a  cross-cut,  say  a-^;  it 
changes  the  number  of  boundary-lines  to  2,  and  it  reduces  the  connectivity 
of  the  cut  surface  to  2,p. 

Since  the  surface  is  connected,  we  can  pass  in  the  surface  along  a 
continuous  line  from  one  edge  of  the  cut  a^  to  the  opposite  edge.  Along 
this  line  make  a  cut  b^ :  it  is  a  cross-cut,  because  it  begins  and  ends  in 
the  boundary.  It  passes  from  one  edge  of  a^  to  the  other,  that  is,  from  one 
boundary-line  to  another.     Hence,  as  in  Prop.  II.  of  §  164,  it  does  not  divide 

*  See  Neumann,  pp.  178 — 182;  Prym,  Zur  Theorie  der  Functionen  in  einer  ziveibldttrigen 
Flache,  (1866). 

t  Riemann,  Ges.  V/erke,  pp.  122,  123 ;    Neumann,  pp.  182—185. 

F.  F.  26 


402 


CANONICAL  EESOLUTION   OF   SURFACE 


[181. 


the  surface  into  distinct  pieces ;  it  changes  the  number  of  boundaries  to  1 , 
and  it  reduces  the  connectivity  to  2p  —  1. 

The  problem  is  now  the  same  as  at  first,  except  that  now  only 
2p  —  2  cross-cuts  are  necessary  for  the  required  resolution.  We  make  a 
loop-cut  ttg,  not  resolving  the  surface  into  distinct  pieces,  and  a  cross-cut 
Ci  from  a  point  of  a^  to  a  point  on  the  boundary  at  h^;  then  Cj  and  ag,  taken 
together,  constitute  a  cross-cut  that  does  not  resolve  the  surface  into  distinct 
pieces.  It  therefore  reduces  the  connectivity  to  2^9  —  2,  and  leaves  two  pieces 
of  boundary. 

The  surface  being  connected,  we  can  pass  in  the  surface  along  a  continuous 
line  from  one  edge  of  a^  to  the  opposite  edge.  Along  this  line  we  make  a 
cut  h^,  evidently  a  cross-cut,  passing,  like  h^  in  the  earlier  case,  from  one 
boundary-line  to  the  other.  Hence  it  does  not  divide  the  surface  into 
distinct  pieces ;  it  changes  the  number  of  boundaries  to  1,  and  it  reduces 
the  connectivity  to  2p  —  3. 

Proceeding  in  p  stages,  each  of  two  cross-cuts,  we  ultimately  obtain  a 
simply  connected  surface  with  a  single  boundary;  and  the  general  effect  on 
the  original  unresolved  surface  is  to  have  a  system  of  cross-cuts  somewhat  of 
the  form 


Fig.  63. 

The  foregoing  resolution  is  called  the  canonical  resolution  of  a  Riemann's 
surface. 

Ex.  1.     Construct  the  Kiemann's  surface  for  the  equation 

w^  +  z^  —  Zawz  =  \, 
both  for  a  =  0  and  for  a  different  from  zero;  and  resolve  it  by  cross-cuts  into  a  simply 
connected  surface  with  a  single   boundary,  shewing  a  complete  system  of  irreducible 
simple  circuits  on  the  unresolved  surface. 

Ex.  2.     Shew  that  the  Riemann's  surface  for  the  equation 

{z  -c){z-  d) 
is  of  genus  j9  =  2;   indicate  the  possible  systems  of  branch-lines,  and,  for  each  system, 
resolve  the  surface  by  cross-cuts  into  a  simply  connected  surface  with  a  single  boundary. 

(Burnside.) 
Ex.  3.     Find  the  connectivity  of  the  surface  associated  with  the  equation 

wh  =  {z-\f{z  +  lf; 
draw  a  possible  system  of  branch-lines,  and  dissect  the  surface  so  as  to  reduce  it  to 
a  simply  connected  one.  (Math.  Trip.,  Part  II.,  1897.) 


182.]  DEFICIENCY  AND  GENUS  EQUAL  403 

182.  Among  algebraical  equations  with  their  associated  Riemann's 
surfaces,  two  general  cases  of  great  importance  and  comparative  simplicity 
distinguish  themselves.  The  first  is  that  in  which  the  surface  is  two- 
sheeted;  round  each  branch-point  the  two  branches  interchange.  The 
second  is  that  in  which,  while  the  surface  has  a  finite  number  of  sheets 
greater  than  two,  all  the  branch-points  are  of  the  first  order,  that  is,  are 
such  that  round  each  of  them  only  two  branches  of  the  function  interchange. 
The  former  has  already  been  considered,  in  so  far  as  concerns  the  surface ; 
we  now  proceed  to  the  consideration  of  the  latter. 

The  equation  is  /('^'  ^)  ~  0, 

of  degree  n  in  lu ;  and,  for  our  present  purpose,  it  is  convenient  to  regard 
/"=0  as  an  equation  corresponding  to  a  generalised  plane  curve  of  degree  n, 
so  that  no  term  in  /  is  of  dimensions  higher  than  n. 

The  total  number  of  branch-points  has  been  proved,  in  §  98,  to  be 

n  (n-l)-28-  2k, 

where  8  is  the  number  of  points  which  are  the  generalisation  of  double 
points  on  the  curve  with  non-coincident  tangents,  and  k  is  the  number 
of  double  points  on  the  curve  with  coincident  tangents.  Round  each  of 
these  branch-points,  two  branches  of  w  interchange  and  only  two,  so  that 
each  of  the  numbers  in^  of  §  178   is   equal   to   2 ;    hence   the  ramification 

O  is 

2  {n.{n  -  1)  -  28  -  2/c|  -  [n  (n  -  1)  -  2g  -  2k], 

that  is,  n  =  7i{n-l)-2S-2K. 

The  connectivity  of  the  surface  is  therefore 

n{7i-l)-2S-2K-2n  +  S; 
and  therefore  the  genus  p  of  the  surface  is 

1  (n  -l)(n-2)-S-K. 

NoAv  this  integer  is  known  *  as  the  deficiency  of  the  curve  ;  and  therefore 
it  appears  that  the  deficiency  of  the  curve  is  the  same  as  the  genus  of  the 
Riemann's  surface  associated  with  its  equation,  and  also  is  the  same  as  the 
genus  of  its  equation. 

Moreover,  the  number  of  branch-points  of  the  original  equation  is  fl,  that 

is, 

=  2p  +  2n-2 

=  2{p  +  (n-l)]. 

Note.  The  equality  of  these  numbers,  representing  the  deficiency  and  the  genus,  is 
one  among  many  reasons  that  lead  to  the  close  association  of  algebraic  functions  (and 

*  Salmon's  Higher  Plane  Curves,  §§  44,  83 ;  Clebsch's  Vorlesunyen  iiber  Geometrie,  (edited 
by  Lindemann),  t.  i,  pp.  351—429,  the  German  word  used  instead  of  deficiency  being  Geschlecht. 
The  name  'deficiency'  was  introduced  by  Cayley  in  1865:  see  Ptoc.  Lond.  Math.  Soc,  vol.  i, 
"On  the  transformation  of  plane  curves." 

26—2 


404  DEFORMATION   OF  [182. 

of  functions  deiDendent  on  them)  with  the  theory  of  plane  algebraic  curves,  in  the 
investigations  of  Nother,  Brill,  Clebsch  and  others,  referred  to  in  §§  191,  242;  and  in  the 
paper  by  Baker,  quoted  in  §  178.  Baker's  rule  for  determining  the  number  is  embodied 
in  the  following  question  ;   and  a  number  of  simple  examples  are  given  in  his  paper. 

Ex.  A  plane  {A)  of  rectangular  Cartesian  coordinates  is  ruled  with  lines  parallel  to 
the  axes,  at  unit  distances  apart,  and  the  angular  points  of  the  squares  obtained  are  called 
unit  points.  Corresponding  to  every  term  Ar^^x'y^  in  the  equation  of  an  irreducible 
plane  curve  (which  is  referred  to  rectangular  Cartesian  axes,  the  origin  being  a  multiple 
point  of  the  curve),  the  point  (r,  s)  is  marked  on  the  plane  {A)  and  called  a  curve  point. 
The  outermost  of  the  curve  points  are  joined  by  finite  straight  lines  so  as  to  form  a 
convex  polygon,  enclosing  the  other  curve  points  and  having  a  curve  point  at  each  vertex. 
Considering  first  the  sides  of  this  polygon  which  are  nearest  to  the  origin  and  limited  by 
the  axes  of  coordinates,  and  assuming  that  all  the  unit  points  upon  these  sides  are  also 
curve  points  and  that  all  these  sides  are  inclined  to  one  of  the  axes  at  an  angle  greater 
than  jTT,  prove  that  the  sum  of  the  number  of  double  points  and  cusps  to  which  the 
singularity  is  equivalent  is  equal  to  the  number  of  unit  points  between  these  sides  and 
the  axes  of  coordinates  together  with  the  number  of  unit  points  upon  these  sides  less 
two.  Considering  next  the  complete  polygon  and  assuming  the  curve  to  ha^'e  only  three 
singularities,  namely  at  the  origin  and  at  infinity  on  the  two  axes  of  coordinates,^  and 
excluding  exceptional  relations  between  the  coefficients  of  the  terms  entering  in  the 
equation  of  the  curve,  prove  that  its  deficiency  is  equal  to  the  whole  number  of  unit 
points  actually  within  the  polygon.  (Math.  Trip.,  Part  II.,  1893.) 

183.  With  a  view  to  the  construction  of  a  canonical  form  of  Riemann's 
surface  of  genus  'p  for  the  equation  under  consideration,  it  is  necessary  to 
consider  in  some  detail  the  relations  between  the  branches  of  the  functions 
as  they  are  affected  by  the  branch-points. 

The  effect  produced  on  any  value  of  the  function  by  the  description  of  a 
small  circuit,  enclosing  one  branch-point  (and  only  one),  is  known.  But 
when  the  small  circuit  is  part  of  a  loop,  the  effect  on  the  value  of  the 
function  with  which  the  loop  begins  to  be  described  depends  upon  the  form 
of  the  loop;  and  various  results  (e.g.  Ex.  1,  §  104)  are  obtained  by  taking 
different  loops.  In  the  first  form  (§  175)  in  which  the  branch-lines  were 
established  as  junctions  between  sheets,  what  was  done  was  the  equivalent 
of  drawing  a  number  of  straight  loops,  which  had  one  extremity  common  to 
all  and  the  other  free,  and  of  assigning  the  law  of  junction,  according  to  the 
law  of  interchange  determined  by  the  description  of  the  loop.  As,  however, 
there  is  no  necessary  limitation  to  the  forms  of  branch-lines,  we  may  draw 
them  in  other  forms,  always,  of  course,  having  branch-points  at  their  free 
extremities ;  and  according  to  the  variation  in  the  form  of  the  branch-line, 
(that  is,  according  to  the  variation  in  the  form  of  the  corresponding  loop 
or,  in  other  words,  according  to  the  deformation  of  the  loop  over  other 
branch-points  from  some  form  of  reference),  there  will  be  variation  in  the  law^ 
of  junction  along  the  branch-lines. 

There  is  thus  a  large  amount  of  arbitrary  character  in  the  forms  of  the 
branch-lines,  and  consequently  in  the  laws  of  junction  along  the  branch-lines, 


183.]  LOOPS  405 

of  the  sheets  of  a  Riemann's  surface.  Moreover,  the  assignment  of  the  n 
branches  of  the  function  to  the  n  sheets  is  arbitrary.  Hence  a  consider- 
able amount  of  arbitrary  variation  in  the  configuration  of  a  Riemann's 
surface  is  possible  within  the  limits  imposed  by  the  invariance  of  its 
connectivity.  The  canonical  form  will  be  established  by  making  these 
arbitrary  elements  definite. 

184.  After  the  preceding  explanation  and  always  under  the  hypothesis 
that  the  branch-points  are  simple,  we  shall  revert  temporarily  to  the  use  of 
loops  and  shall  ultimately  combine  them  into  branch-lines. 

When,  with  an  ordinary  point  as  origin,  we  construct  a  loop  round  a 
branch-point,  two  and  only  two  of  the  values  of  the  function  ,are  affected 
by  that  particular  loop ;  they  are  interchanged  by  it ;  but  a  different  form  of 
loop,  from  the  same  origin  round  the  same  branch-point,  might  affect  some 
other  pair  of  values  of  the  function. 

To  indicate  the  law  of  interchange,  a  symbol  will  be  convenient.  If  the 
two  values  interchanged  by  a  given  loop  be  w.;,  and  w„,,,  the  loop  will  be 
denoted  by  im ;  and  i  and  m  will  be  called  the  numbers  of  the  symbol  of  that 
loop. 

For  the  initial  configuration  of  the  loops,  we  shall  (as  in  §  175)  take  an 
ordinary  point  0 :  we  shall  make  loops  beginning  at  0,  forming  them  in  the 
sequence  of  angular  succession  of  the  branch-points  round  0  and  drawing  the 
double  linear  part  of  the  loop  as  direct  as  possible  from  0  to  its  branch-point : 
and,  in  this  configuration,  we  shall  take  the  law  of  interchange  by  a  loop  to 
be  the  law  of  interchange  by  the  branch-point  in  the  loop. 

In  any  other  configuration,  the  symbol  of  a  loop  round  any  branch-point 
depends  upon  its  form,  that  is,  depends  upon  the  deformation  over  other 
branch-points  which  the  loop  has  suffered  in  passing  from  its  initial  form. 
The  effect  of  such  deformation  must  first  be  obtained :  it  is  determined  by 
the  following  lemma  : — 

When  one  loop  is  deformed  over  another,  the  symbol  of  the  deformed  loop 
is  unaltered,  if  neither  of  its  numbers  or  if  both  of  its  numbers  occur  in  the 
symbol  of  the  unmoved  loop ;  but  if,  before  deformation,  the  symbols  have  one 
number  common,  the  neiu  symbol  of  the  deformed  loop  is  obtained  from  the  old 
symbol  by  substituting,  for  the  common  number,  the  other  number  in  the  symbol 
of  the  unmoved  loop. 

The  sufficient  test,  to  which  all  such  changes  must  be  subject,  is  that  • 
the  effect  on  the  values  of  the  function  at  any  point  of  a  contour  enclosing 
both  branch-points  is  the  same  at  that  point  for  all  deformations  into  two 
loops.  Moreover,  a  complete  circuit  of  all  the  loops  is  the  same  as  a  contour 
enclosing  all  the  branch-points;  it  therefore  (Cor.  III.  §  90)  restores  the  initial 
value  with  which  the  circuit  began  to  be  described. 


406 


MODIFICATION 


[184. 


Obviously  there  are  three  cases. 

First,  when  the  symbols  have  no  number  common :  let  them  be  11171,  rs. 
The  branch-point  in  the  loop  rs  does  not  affect  Wm  or  Wn :  it  is  thus  effectively 
not  a  branch-point  for  either  of  the  values  w^  and  Wn',  and  therefore  (§  91) 
the  loop  mn  can  be  deformed  across  the  point,  that  is,  it  can  be  deformed 
across  the  loop  mn. 

Secondly,  when  the  symbols  are  the  same :  the  symbol  of  the  deformed 
loop  must  be  unaltered,  in  order  that  the  contour  embracing  only  the  two 
branch-points  may,  as  it  should,  restore  after  its  complete  description  each  of 
the  values  affected. 

Thirdly,  when  the  symbols  have  one  number  common :  let  0  be  any 
point  and  let  the  loops  be  OA,  OB  in  any  given  position  such  as  (i),  fig.  64, 
with  symbols  mr,  nr  respectively.  Then  OB  may  be  deformed  over  OA  as 
in  (ii),  or  OA  over  OB  as  in  (iii). 

A 


Fig.  64. 


•  The  effect  at  0  of  a  closed  circuit,  including  the  points  A  and  B  and 
described  positively  beginning  at  0,  is,  in  (i)  which  is  the  initial  configura- 
tion, to  change  Wm,  into  tUr,  w^  into  w^ 


w« 


into  Wm',  this  effect  on  the 
values  at  0,  unaltered,  must  govern  the  deformation  of  the  loops. 

The  two  alternative  deformations  (ii)  and  (iii)  will  be  considered  separately. 

When,  as  in  (ii),  OB  is  deformed  over  OA,  then  OA  is  unmoved  and 
therefore  unaltered :  it  is  still  mr.  Now,  beginning  at  0  with  w„„  the  loop 
OA  changes  w-,^  into  w,. :  the  whole  circuit  changes  tu,n  into  Wr,  so  that  OB 
must  now  leave  Wr  unaltered.  Again,  beginning  with  Wn,  it  is  unaltered  by 
OA,  and  the  whole  circuit  changes  Wn  into  Wm :  hence  OB  must  change  tUn 
into  Wm,  that  is,  the  symbol  of  OB  must  be  m??.  And,  this  being  so,  an 
initial  Wr  at  0  is  changed  by  the  whole  circuit  into  Wn,  as  it  should  be. 
Hence  the  new  symbol  mn  of  the  deformed  loop  OB  in  (ii)  is  obtained  from 
the  old  symbol  by  substituting,  for  the  common  number  r,  the  other  number 
m  in  the  symbol  of  the  unmoved  loop  OA. 

We  may  proceed  similarly  for  the  deformation  in  (iii) ;  or  the  new  symbol 
may  be  obtained  as  follows.  The  loop  OA  in  (iii)  may  be  deformed  to  the 
form  in  (iv)  without  crossing  any  branch-point  and  therefore  without 
changing   its   symbol.     When  this   form   of  the    loop  is   described  in  the 


I 


184.]  OF  LOOPS  407 

positive  direction,  Wn  initially  at  0  is  changed  into  tu,.  after  the  first  loop 
OB,  for  this  loop  has  the  position  of  OB  in  (i),  then  it  is  changed  into  Wm 
after  the  loop  OA,  for  this  loop  has  the  position  of  OA  in  (i),  and  then  iu,n  is 
unchanged  after  the  second  (and  inner)  loop  OB.  Thus  iVn  is  changed  into 
w^,  so  that  the  sjnnbol  is  mn,  a  symbol  which  is  easily  proved  to  give  the 
proper  results  with  an  initial  value  w,n  or  w,.  for  the  whole  contour.  This 
change  is  as  stated  in  the  theorem,  which  is  therefore  proved. 

Ex.  If  the  deformation  from  (i)  to  (ii)  be  called  superior,  and  that  from  (i)  to  (iii) 
inferior,  then  x  successive  superior  deformations  give  the  same  loop-configuration, 
in  symbols  and  relative  order  for  positive  description,  as  Q  —  x  successive  inferior 
deformations. 

Corollary.  A  loop  can  be  jmssed  unchanged  over  two  loops  that  have  the 
same  symbol. 

Let  the  common  symbol  of  the  unmoved  loops  be  mn.  If  neither  number 
of  the  deformed  loop  be  m  or  n,  passage  over  each  of  the  loops  mn  makes  no 
difference,  after  the  lemma ;  likewise,  if  its  symbol  be  mn.  If  only  one  of  its 
numbers,  say  ?i,  be  in  vin,  its  symbol  is  ?ir,  where  r  is  ditferent  from  m.  When 
the  loop  nr  is  deformed  over  the  first  loop  mn,  its  new  symbol  is  mr ;  when 
this  loop  ??!?'  is  deformed  over  the  second  loop  mn,  its  new  symbol  is  nr,  that 
is,  the  final  symbol  is  the  same  as  the  initial  symbol,  or  the  loop  is  unchanged. 

185.  The  initial  configuration  of  the  loops  is  used  by  Clebsch  and 
Gordan  to  establish  their  simple  cycles  and  thence  to  deduce  the  periodi- 
city of  the  Abelian  integrals  connected  with  the  equation  f(tu,  z)  =  0, 
without  reference  to  the  Riemann's  surface ;  and  this  method  of  treating 
the  functions  that  arise  through  the  equation,  always  supposed  to  have 
merely  simple  branch -points,  has  been  used  by  Casorati*  and  Lurothf. 

We  can  pass  from  any  value  of  iv  at  the  initial  point  0  to  any  other 
value  by  a  suitable  series  of  loops ;  because,  were  it  possible  to  inter- 
change the  values  of  only  some  of  the  branches,  an  equation  could  be 
constructed  which  had  those  branches  for  its  roots.  The  fundamental 
equation  could  then  be  resolved  into  this  equation  and  an  equation  having 
the  rest  of  the  branches  for  its  roots :  that  is,  the  fundamental  equation 
would  cease  to  be  irreducible. 

We  begin  then  with  any  loop,  say  one  connecting  w^  with  Wg.  There 
will  be  a  loop,  connecting  the  value  iv^  with  either  lu^  or  w^;  there  will  be 
a  loop,  connecting  the  value  lu^  with  either  w^,  lu^,  or  w^;  and  so  on,  until 
we  select  a  loop,  connecting  the  last  value  lUn  with  one  of  the  other  values. 
Such  a  set  of  loops,  n—\  in  number,  is  coWedi  fundamental. 

A  passage  round   the  set  will  not  at  the  end  restore  the  branch  with 
which  the  description  began.     When  we  begin  with   any  value,  any  other 
value  can  be  obtained  after  the  description  of  properly  chosen  loops  of  the  set. 
*  Annali  di  Matematica,  2"^  Ser.,  t.  iii,  (1870),  pp.  1—27. 
t  Ahh.  d.  K.  bay.  Akad.,  t.  xvi,  i  Abtb.,  (1887),  pp.  199—241. 


408  CYCLES  OF  LOOPS  [185. 

Any  other  loop,  when  combined  with  a  set  of  fundamental  loops,  gives 
a  system  the  description  of  suitably  chosen  loops  of  which  restores  some 
initial  value;  only  two  values  can  be  restored  by  the  description  of  loops 
of  the  combined  system.  Thus  if  the  loops  in  order  be  12,  13,  14,  ...,  In 
and  a  loop  qr  be  combined  with  them,  the  value  Wq  is  changed  into  Wj  by 
Iq,  into  Wr  by  Ir,  into  Wq  by  qi^;  and  similarly  for  w,..  Such  a  combination 
of  71  loops  is  called  a  simple  cycle. 

The  total  number  of  branch-points,  and  therefore  of  loops,  is  (§  182) 

2[p  +  (n-l)}; 
and  therefore  the  total  number  of  simple   cycles  is  2p  +  n  —  1.     But  these 
simple  cycles  are  not  independent  of  one  another. 

In  the  description  of  any  cycle,  the  loops  vary  in  their  operation 
according  to  the  initial  value  of  w :  and,  for  two  different  initial  values  of 
lu,  no  loop  is  operative  in  the  same  way.  For  otherwise  all  the  preceding 
and  all  the  succeeding  loops  would  operate  in  the  same  way  and  would 
lead,  on  reversal,  to  the  same  initial  value  of  w.  Hence  a  loop  of  a  given 
cycle  can  be  operative  in  only  two  descriptions,  once  when  it  changes,  say,  iVi 
into  Wj,  and  the  other  when  it  changes  Wj  into  ifj. 

Now  consider  the  circuit  made  up  of  all  the  loops.  When  w^  is  taken  as 
the  initial  value,  it  is  restored  at  the  end:  and  in  the  description  only  a 
certain  number  of  loops  have  been  operative  :  the  cycle  made  up  of  these  loops 
can  be  resolved  into  the  operative  parts  of  simple  cycles,  that  is,  into  simple 
cycles :  hence  one  relation  among  the  simple  cycles  is  given  by  the  considera- 
tion of  the  operative  loops  when  the  whole  system  of  the  loops  is  described 
with  an  initial  value. 

Similarly  when  any  other  initial  value  is  taken ;  so  that  apparently  there 
are  n  relations,  one  arising  from  each  initial  value.  These  n  relations  are  not 
independent :  for  a  simultaneous  combination  of  the  operations  of  all  the 
loops  in  all  the  circuits  leads  to  an  identically  null  effect  (but  no  smaller 
combination  would  be  effective),  for  each  loop  is  operative  twice  (and  only 
twice)  with  opposite  effects,  shewing  that  one  and  only  one  of  the  relations  is 
derivable  from  the  remainder.  Hence  there  are  n  —  1  independent  relations 
and  therefore*  the  number  of  independent  simple  cycles  is  2p. 

186.  We  now  proceed  to  obtain  a  typical  form  of  the  Riemann's  surface 
by  deforming  the  initial  configuration  of  the  loops  into  a  typical  configura- 
tion f.    The  final  arrangement  of  the  loops  is  indicated  by  the  two  theorems : — 

*  Clebsch  und  Goidan,  Theorie  der  AheVschen  Fanctionen,  p.  85. 

t  The  investigation  is  based  upon  the  following  memoirs  :— 

Liiroth,  "Note  iiber  Verzweigiingsschnitte  und  Querschnitte  in  einer  Eiemann'schen  Flache," 

Math.  Aim.,  t.  iv,  (1871),  pp.  181 — 184  ;  "  Ueber  die  kanonischen  Perioden  der  Abel'schen 

Integrale,"  Ahh.  d.  K.  hay.  Akad.,  t.  xv,  ii  Abth.,  (1885),  pp.  329—366. 

Clebscb,  "  Zur  Theorie  der  Eiemann'schen  FJachen,"'  3Iath.  Ann.,  t.  vi,  (1873),  pp.  216—230. 

Clifford,  "  On  the  canonical  form  and  dissection  of  a  Eiemaun's  Surface,"  Lond.  Math.  Soc. 

Proc,  vol.  viii,  (1877),  pp.  292—304. 


186.]  luroth's  theorem  409 

I.  The  loops  can  he  made  in  pairs  in  which  all  loop-symbols  are  of  the 
form  (m,  m  + 1),  for  m=  1,  2,...,  n  —  1.  (With  this  configuration,  Wi  can  be 
changed  by  a  loop  only  into  Wg,  W2  hy  a  loop  only  into  Wg,  and  so  on  in 
succession,  each  change  being  effected  by  an  even  number  of  loops.)  This 
theorem  is  due  to  Ltiroth. 

II.  The  loops  can  he  made  so  that  there  is  only  one  pair  12,  only  one 
pair  23,...,  only  one  pair  (n  —  2,  n—1),  and  the  remaining  p  +  1  pairs  are 
(n  —  1,  n).     This  theorem  is  due  to  Clebsch. 

187.  We  proceed  to  prove  Luroth's  theorem,  assuming  that  the  loops 
have  the  initial  configuration  of  §  184. 

Take  any  loop  12,  say  OA  :  beginning  it  with  w^,  describe  loops  positively 
and  in  succession;  then  as  the  value  w^  is  restored  sooner  or  later,  for  it 
must  be  restored  by  the  circuit  of  all  the  loops,  let  it  be  restored  first  by  a 
loop  OB,  the  symbol  of  OB  necessarily  containing  the  number  1.  Between  OA 
and  OB  there  may  be  loops  whose  symbols  contain  1  but  which  have  been 
inoperative.  Let  each  of  these  in  turn  be  deformed  so  as  to  pass  back  over 
all  the  loops  between  its  initial  position  and  OA  ;  and  then  finally  over  OA. 
Before  passing  over  OA  its  symbol  must  contain  1,  for  there  is  no  loop  over 
which  it  has  passed  that,  having  1  in  its  symbol,  could  make  it  drop  1  in  the 
passage;  but  it  cannot  contain  2,  for,  if  it  did,  the  effect  of  OA  and  the 
deformed  loop  would  be  to  restore  1,  an  effect  that  would  have  been 
caused  in  the  original  position,  contrar}^  to  the  hypothesis  that  OB  is  the 
first  loop  that  restores  1.  Hence  after  it  has  passed  over  OA  its  symbol 
no  longer  contains  1.- 

Next,  pass  OB  over  the  loops  between  its  initial  position  and  OA  but  not 
over  OA  :  its  symbol  must  be  12  in  the  deformed  position  since  Wi  is  restored 
by  the  loop  OB.  Then  OA  and  the  deformed  loop  OB  are  each  12;  hence  each 
of  the  loops,  between  the  new  position  and  the  old  position  of  OB,  can  be  passed 
over  OA  and  the  new  loop  OB  without  any  change  in  its  symbol.  There  are 
therefore,  behind  OA,  a  series  of  loops  that  do  not  affect  w-^.  Thus  the  loops 
are 

(a)   loops  behind  OA  not  affecting  lu^,         (h)   OA,  OB  each  12, 

(c)    other  loops  beyond  the  initial  position  of  OB. 
Begin  now  with  ^u.,  at  the  loop  OB  and  again  describe  loops  positively 
and  in  succession :  then  w.^  must  be  restored  sooner  or  later.     It  may  be 
only  after  OA  is  described,  so  that   there  has  been  a  complete  circuit   of 
all  the  loops ;    or  it  may  first  be  by  an  intermediate  loop,  say  OG. 

For  the  former  case,  when  OA  is  the  first  loop  by  which  lu^  is  restored, 
we  deform  as  follows.  Deform  all  loops  affecting  w^,  which  lie  between 
OB  and  OA,  in  the  positive  direction  from  OB  back  over  other  loops  and 
over   OB.     The  symbol  of  each  just  before  its  deformation  contains  1  but 


410  luroth's  theorem  [187. 

not  2,  and  therefore  after  its  deformation  it  does  not  contain  1.  Moreover 
just  after  OB  is  described,  w^  is  the  value,  and  just  before  OA  is  described, 
Wi  is  the  value ;  hence  the  intermediate  loops,  which  have  affected  w-^, 
must  be  even  in  number.  Let  OG  be  the  first  after  OB  which  affects  Wj, 
and  let  the  symbol  of  OG  be  Ir.  Then  beginning  OG  with  Wi,  the  value 
Wi  must  be  restored  by  a  complete  circuit  of  all  the  loops,  that  is,  it 
must  be  restored  by  OB;  and  therefore  the  value  must  be  w^  when 
beginning  OA,  or  Wi  must  be  restored  before  OA.  Let  OH  be  the  first 
loop  after  OG  to  restore  Wj ;  then,  by  proceeding  as  above,  we  can  deform 
all  the  loops  between  OG  and  OH  over  OG,  with  the  result  that  no  such 
deformed  loop  affects  Wi  and  that  OG  and  OH  are  both  Ir.  Hence  all  the 
loops  affecting  lUi  can  be  arranged  in  pairs  having  the  same  symbol. 

Since  OG  and  OH  are  a  pair  with  the  same  symbol,  every  loop  between 
OB  and  OG  can  be  passed  unchanged  over  OG  and  OH  together.  When 
this  is  done,  pass  OG  over  OB  so  that  it  becomes  2?-,  and  then  OH  over 
OB  so  that  it  also  is  2r.  Thus  these  deformed  loops  OG,  OH  are  a  pair 
2r;  and  therefore  OA  can,  without  change,  be  deformed  over  both  so  as 
to  be  next  to  OB.  Let  this  be  done  with  all  the  pairs ;  then,  finally,  we 
have 

(a)    loops  not  affecting  Wj,  {b)   a  pair  with  the  symbol  12, 

(c)    pairs  affecting  Wq  and  not  w^,      (d)    loops  not  affecting  Wj. 

We  thus  have  a  pair  12  and  loops  not  affecting  w^,  so  that  such  a  change 
has  been  effected  as  to  make  all  the  loops  affecting  tUi  possess  the  symbol  12. 

For  the  second  case,  when  OG  is  the  first  loop  to  restore  w,,,  the 
value  with  which  the  loop  OB  whose  symbol  is  12  began  to  be  described,  we 
treat  the  loops  between  OB  and  00  in  a  manner  similar  to  that  adopted  in 
the  former  case  for  loops  between  OA  and  OB;  so  that,  remembering  that 
now  W2  instead  of  the  former  Wi  is  the  value  dealt  with  in  the  recurrence,  we 
can  deform  these  loops  into  ' 

(a)  loops  behind  OB  which  change  Wj  but  not  W2, 

(b)  OB  and  OG,  the  symbol  of  each  of  which  is  12. 

Now  OB  was  next  to  OA  ;  hence  the  set  (a)  are  now  next  to  OA.  Each  of 
them  when  passed  over  OA  drops  the  number  1  from  its  symbol,  and  so  the 
whole  system  now  consists  of 

(tt)   loops  behind  OA  not  affecting  lu^,     (b)    OA,  OB,  OG  each  of  which 
is  12,     (c)    other  loops. 

Begin  again  with  the  value  w^  before  OA.  Before  00'  the  value  is  Wj ; 
and  the  whole  circuit  of  the  loops  must  restore  Wj,  which  must  therefore 
occur  before  OA.     Let  OD  be  the  first  loop  by  which  'ii\  is  restored.     Then 


187.]  ON   CONFIGURATION    OF   LOOPS  411 

treating  the  loops  between  OC  and  OD,  as  formerly  those  between  the  initial 
positions  of  OA  and  OB  were  treated,  we  shall  have 

(a)    loops  behind  OA  not  affecting  w-,,     (b)    OA,  OB  each  being  12, 
(c)    loops  between  OB  and  00  not  affecting  Wj,     {d)    OG,  OD  each 
being  12,     (e)   other  loops. 
Except  that  fewer  loops  affecting  Wj  have  to  be  reckoned  with,  the  con- 
figuration is  now  in  the  same  condition  as  at  the  end  of   the  first  stage. 
Proceeding  therefore  as  before,  we  can  arrange  that  all  the  loops  affecting  Wj 
occur  in  pairs  with  the  symbol  12.     Moreover,  each  of  the  loops  in  the  set 
(c)  can  be  passed  unchanged  over  OA  and  OB ;  so  that,  finally,  we  have 

(a)  pairs  of  loops  with  the  symbol  12,     (6')    loops  not  affecting  w^. 
We  keep  (a)  in  pairs,  so  that  any  desired  deformation  of  loops  in  (b')  over 
them  can  be  made  without  causing  any  change  ;  and  we  treat  the  set  (b')  in 
the  same  manner  as  before,  with  the  result  that  the  set  (b')  is  replaced  by 

(b)  pairs  of  loops  with  the  symbol  23,  (c')  loops  not  affecting  ui^  or  w... 
And  so  on,  with  the  ultimate  result  that  the  loops  can  be  made  in  pairs  in 
which  each  symbol  is  of  the  form  {m,  m  +  1)  for  vi  =  l,  ... ,  n  —  l. 

188.  We  now  come  to  Clebsch's  Theorem  that  the  loops  thus  made  can 
be  so  deformed  that  there  is  only  one  pair  12,  only  one  pair  23,  and  so  on, 
until  the  last  symbol  (n—  1,  n),  which  is  the  common  symbol  of  p  +  1  pairs. 

This  can  be  easily  proved  after  the  establishment  of  the  lemma  that,  if 
there  be  two  pairs  12  and  one  pair  23,  the  loops  can  be  deformed  into  one  pair 
12  and  ttvo  pairs  23. 

The  actual  deformation  leading  to  the  lemma  is  shewn  in  the  accom- 
panying scheme  :    the  deformations   implied  by 

the  continuous  lines  are  those  of  a  loop  from  the  

left  to  the  right  of  the  respective  lines,  and  those     12      12      12 23      13      23 

implied  by  the  dotted  lines  are  those  of  a  loop     -^2      12      23      13      13      23 

from  the  right  to  the  left  of  the  respective  lines. 

•  •  1''       12       13       13       23       23 

It  is   interes.ting  to   draw  figures,  representing 

the  loops  in  the  various  configurations.  12      23      12      13      23      23 

By  the  continued  use  of  this  lemma  we  can  12  23  23  12  23  23 
change  all  but  one  of  the  pairs  12  into  pairs  23,  -^^  ^^  gg  23  23  23 
all  but  one  of  the  pairs  23  into  pairs  34,  and  so 

on,  the  final  configuration  being  that  there  are  one  pair  12,  one  pair  23,  ... 
and  p-M  pairs  {n-l,  n).     Thus  Clebsch's  theorem  is  proved. 

189.  We  now  proceed  to  the  construction  of  the  Riemann's  surface. 
Each  loop  is  associated  with  a  branch-point,  and  the  order  of  interchange 

for  passage  round  the  branch-point,  by  means  of  the  loop,  is  given  by  the 
numbers  in  the  symbol  of  the  loop. 


412  CONSTRUCTION    OF  [189. 

Hence,  in  the  configuration  which  has  been  obtained,  there  are  two  branch- 
points 12 :  we  therefore  connect  them  (as  in  §  176)  by  a  line,  not  necessarily 
along  the  direction  of  the  two  loops  12  but  necessarily  such  that  it  can, 
without  passing  over  any  branch-point,  be  deformed  into  the  lines  of  the 
two  loops ;  and  we  make  this  the  branch-line  between  the  first  and  the 
second  sheets.  There  are  two  branch-points  23 :  we  connect  them  by  a  line 
not  meeting  the  former  branch-line,  and  we  make  it  the  branch-line  between 
the  second  and  the  third  sheets.  And  so  on,  until  we  come  to  the  last  two 
sheets.  There  are  2p  4-  2  branch -points  n  —  1,  n:  we  connect  these  in  pairs 
(as  in  §  176)  hy  p  +  1  lines,  not  meeting  one  another  or  any  of  the  former 
lines,  and  we  make  them  the  p  +  1  branch-lines  between  the  last  two  sheets. 

It  thus  appears  that,  when  the  winding-points  of  a  Rienianns  surface  with 
n  sheets  of  connectivity  2p  +  1  are  all  simple,  the  surface  can  be  taken  in  such 
a  form  that  there  is  a  single  hranch-line  hetiueen  consecutive  sheets  except  for  the 
last  two  sheets :  and  hetiueen  the  last  two  sheets  there  are  p+1  branch-lines. 
This  form  of  Riemann's  surface  may  be  regarded  as  the  canonical  form  for  a 
surface,  all  the  branch-points  of  which  are  simple. 

Further,  let  AB  be  a  branch-line  such  as  12.  Let  two  points  P  and  Q 
be  taken  in  the  first  sheet  on  opposite  sides  of  AB,  so  that  PQ  in  space  is 
infinitesimal ;  and  let  P'  be  the  point  in  the  second  sheet  determined  by  the 
same  value  of  z  as  P,  so  that  P'Q  in  the  sheet  is  infinitesimal.  Then  the 
value  Wi  at  P  is  changed  by  a  loop  round  A  (or  round  B)  into  a  value  at  Q 
differing  only  infinitesimally  from  iu.2,  which  is  the  value  at  P' :  that  is,  the 
change  in  the  function  from  Q  to  P'  is  infinitesimal.  Hence  the  value  of  the 
function  is  continuous  across  a  line  of  passage  from  one  sheet  to  another. 

190.  The  genus  of  the  foregoing  surface  is  jo ;  and  it  was  remarked,  in 
§  170,  that  a  convenient  surface  of  reference  of  the  same  genus  is  that  of  a 
solid  sphere  with  p  holes  bored  through  it.  It  is,  therefore,  proper  to  in- 
dicate the  geometrical  deformation  of  a  Riemann's  surface  of  this  canonical 
form  into  a  ^-holed  sphere. 

The  Riemann's  surface  consists  of  n  sheets  connected  chainw'ise  each  with 
a  single  branch-line  to  the  sheet  on  either  side  of  it,  except  that  the  first  is 
connected  only  with  the  second  and  that  the  last  two  have  p  +  1  branch- 
lines.  We  may  also  consider  the  whole  surface  as  spherical  and  the  sequence 
of  the  sheets  from  the  inside  outwards :  and  the  outmost  sheet  can  be  con- 
sidered as  bounded. 

Let  the  branch-line  between  the  first  and  the  second  sheets  be  made  to 
lie  along  part  of  a  great  circle.  Let  the  first  sheet  of  the  Riemann's  surface 
be  reflected  in  the  plane  of  this  great  circle :  the  line  becomes  a  long- 
narrow  hole  along  the  great  circle,  and  the  reflected  sheet  becomes  a  large 
indentation  in  the  second  sheet.     Reversing  the  process  of  §  169,  we  can 


190.]  CANONICAL  SURFACE  413 

change  the  new  form  of  the  second  sheet,  so  that  it  is  spherical  again :  it  is 
now  the  inmost  of  the  n  —  1  sheets  of  the  surface,  the  connectivity  and  the 
ramification  of  which  are  unaltered  by  the  operation. 

Let  this  process  be  applied  to  each  surviving  inner  sheet  in  succession. 
Then,  after  n  —  2  operations,  there  will  be  left  a  two-sheeted  surface ;  the 
outer  sheet  is  bounded  and  the  two  sheets  are  joined  by  j9  +  1  branch- 
lines;  so  that  the  connectivity  is  still  2p  +  1.  Let  these  branch-lines  be 
made  to  lie  along  a  greab  circle :  and  let  the  inner  surface  be  reflected 
in  the  plane  of  this  circle.  Then,  after  the  reflexion,  each  of  the  branch-lines 
becomes  a  long  narrow  hole  along  the  great  circle ;  and  there  are  two 
spherical  surfaces  which  pass  continuously  into  one  another  at  these  holes, 
the  outer  of  the  surfaces  being  bounded.  By  stretching  one  of  the  holes 
and  flattening  the  two  surfaces,  the  new  form  is  that  of  a  bifacial  flat 
surface :  each  of  the  p.  holes  then  becomes  a  hole  through  the  body 
bounded  by  that  surface ;  the  stretched  hole  gives  the  extreme  geo- 
metrical limits  of  the  extension  of  the  surface,  and  the  original  boundary  of 
the  outer  surface  becomes  a  boundary  hole  existing  in  only  one  face.  The 
body  can  now  be  distended  until  it  takes  the  form  of  a  sphere,  and  the  final 
form  is  that  of  the  surface  of  a  solid  sphere  with  p  holes  bored  through  it 
and  having  a  single  boundary. 

This  is  the  normal  surface  of  reference  (§  170)  of  connectivity  2p  +  1. 

As  a  last  ground  of  comparison  between  the  Riemann's  surface  in  its 
canonical  form  and  the  surface  of  the  bored  sphere,  we  may  consider  the 
system  of  cross-cuts  necessary  to  transform  each  of  them  into  a  simply 
connected  surface. 

We  begin  with  the  spherical  surface.  The  simplest  irreducible  circuits 
are  of  two  classes,  (i)  those  which  go  round  a  hole,  (ii)  those  which  go  through 
a  hole ;  the  cross-cuts,  2p  in  number,  which  make  the  surface  simply  con- 
nected, must  be  such  as  to  prevent  these  irreducible  circuits. 

Round  each  of  the  holes  we  make  a  cut  a,  the  first  of  them  beginning 
and  ending  in  the  boundary :  these  cuts  prevent  circuits  through  the  holes. 
Through  each  hole  we  make  a  cut  b,  beginning  and  ending  at  a  point  in  the 
corresponding  cut  a :  we  then  make  from  the  first  b  a  cut  Ci  to  the  second  a, 
from  the  second  b  a  cut  Ca  to  the  third  a,  and  so  on.  The  surface  is  then 
simply  connected :  a-i  is  a  cross-cut,  bi  is  a  cross-cut,  Ci  +  a^  is  a  cross-cut,. 
62  is  a  cross-cut,  c^  +  a^  is  a  cross-cut,  and  so  on.  The  total  number  is 
evidently  2p,  the  number  proper  for  the  reduction ;  and  it  is  easy  to  verify 
that  there  is  a  single  boundary. 

To  compare  this  dissection  with  the  resolution  of  a  Riemann's  surface  by 
cross-cuts,  say  of  a  two-sheeted  surface  (the  ?i-sheeted  surface  was  trans- 
formed into  a  two-sheeted  surface),  it  must  be  borne  in  mind  that  only  p  of 
the  p  +  1  branch-lines  were  changed  into  holes  and  the  remaining  one,  which. 


414 


DEFORMATION 


[190. 


after  the  partial  deformation,  was  a  hole  of  the  Rieraann's  surface,  was 
stretched  out  so  as  to  give  the  boundary. 

It  thus  appears  that  the  direction  of  a  cut  a  round  a  hole  in  the  normal 
surface  of  reference  is  a  cut  round  a  branch-line  in  one  sheet,  that  is,  it  is  a 
cut  a  as  in  the  resolution  (§  180)  of  the  Riemann's  surface  into  one  that  is 
simply  connected. 

Again,  a  cut  6  is  a  cut  from  a  point  in  the  boundary  across  a  cut  a  and 
through  the  hole  back  to  the  initial  point ;  hence,  in  the  Riemann's-  surface, 
it  is  a  cut  from  some  one  assigned  branch-line  across  a  cut  a,.,  meeting  the 
branch-line  surrounded  by  a^,  passing  into  the  second  sheet  and,  without 
meeting  any  other  cut  or  branch-line  in  that  surface,  returning  to  the  initial 
point  on  the  assigned  branch-line.  It  is  a  cut  h  as  in  the  resolution  of  the 
Riemann's  surface. 

Lastly,  a  cut  c  is  made  from  a  cut  6  to  a  cut  a.  It  is  the  same  as  in  the 
resolution  of  the  Riemann's  surface,  and  the  purpose  of  each  of  these  cuts  is 
to  change  each  of  the  loop-cuts  a  (after  the  first)  into  cross-cuts. 

A  simple  illustration  arises  in  the  case  of  a  two-sheeted  Eiemann's  surface,  of  genus  ^  =  2. 
The  various  forms  are  : — 

(i)  the  surface  of  a  two-holed  sphere,  with  the  directions  of  cross-cuts  that  resolve  it 
into  a  simply  connected  surface  ;  as  in  (i),  fig.  65,  B,  K  being  at  opposite  edges 
of  the  cut  Cj  where  it  meets  a2 :  H^  G  at  opposite  edges  where  it  meets  hi :  and 
so  on ; 

(ii)  the  spherical  surface,  resolved  into  a  simply  connected  surface,  bent,  stretched, 
and  flattened  out  ;   as  in  (ii),  fig.  65  ; 

(iii)   the  plane  Riemann's  surface,  resolved  by  the  cross-cuts  ;   as  in  fig.  63,  p.  402. 


Fig.  65. 


Numerous  illustrations  of  transformations  of  Eiemann's  surfaces  are  given  by 
Hofmann,  Methodik  der  stetigen  Deformation  von  zweihlattrigen  Riemann' schen  Flachen, 
(Halle  a.  S.,  Nebert,  1888). 


191.]  OF  RiEM Ann's  surfaces  415 

191.  We  have  seen  that  a  bifacial  surface  with  a  single  boundary  can  be 
deformed,  at  least  geometrically,  into  any  other  bifacial  surface  with  a  single 
boundary,  provided  the  two  surfaces  have  the  same  connectivity;  and  the 
result  is  otherwise  independent  of  the  constitution  of  the  surface,  in  regard 
to  sheets  and  to  form  or  position  of  branch-lines.  Further,  in  all  the  geo- 
metrical deformations  adopted,  the  characteristic  property  is  the  uniform 
correspondence  of  points  on  the  surfaces. 

Now  with  every  Riemann's  surface,  in  its  initial  form,  an  algebraical 
equation  f(iu,z)  =  0  is  associated;  but  when  deformations  of  the  surface 
are  made,  the  relations  that  establish  uniform  correspondence  between 
different  forms,  practically  by  means  of  conformal  representation,  are  often 
of  a  transcendental  character  (Chap.  XX.).  Hence,  when  two  surfaces  are 
thus  equivalent  to  one  another,  and  when  points  on  the  surfaces  are 
determined  solely  by  the  variables  in  the  respective  algebraical  equations, 
no  relations  other  than  algebraical  being  taken  into  consideration,  the 
uniform  correspondence  of  points  can  only  be  secured  by  assigning  a  new 
condition  that  there  be  uniform  transformation  between  the  variables  w  and 
2  of  one  surface  and  the  variables  tu'  and  /  of  the  other  surface.  And,  when 
this  condition  is  satisfied,  the  equations  are  such  that  the  deficiencies  of  the 
two  (generalised)  curves  represented  by  the  equations  are  the  same,  because 
they  are  equal  to  the  common  connectivity.  It  may  therefore  be  expected 
that,  when  the  variables  in  an  equation  are  subjected  to  uniform  transfor- 
mation, the  genus  of  the  equation  is  unaltered  ;  or  in  other  words,  that  the 
deficiency  of  a  curve  is  an  invariant  for  uniform  transformation. 

This  inference  is  correct :  the  actual  proof  is  directly  connected  with 
geometry  and  the  theory  of  Abelian  functions,  and  is  given  in  treatises 
on  those  subjects*.  We  shall  return  to  the  theorem  in  connection  with 
birational  transformation,  which  will  be  discussed  later :  merely  remarking 
now  that  the  result  is  of  importance  here,  because  it  justifies  the  adoption 
of  a  simple  normal  surface  of  the  same  genus  as  the  surface  of  reference. 

*  Clebsch's  Vorlesungen  iiber  Geometrie,  t.  i,  p.  459,  where  other  references  are  given ;  Salmon's 
Higher  Plane  Curves,  pp.  93,  319  ;  Clebsch  und  Gordan,  Theorie  der  AheVsehen  Functionen, 
Section  3;    Brill,  Math.  Ann.,  t.  vi,  pp.  33—65. 


CHAPTER   XYI. 

Algebraic  Functions  and  their  Integrals. 

192.  In  the  preceding  chapter  sufficient  indications  have  been  given  as 
to  the  character  of  the  Riemann's  surface  on  which  the  ?? -branched  function 
w,  determined  by  the  equation 

f{iu,z)  =  Q, 

can  be  represented  as  a  uniform  function  of  the  position  of  the  variable. 
It  is  unnecessary  to  consider  algebraically  multiform  functions  of  position 
on  the  surface,  for  such  multiformity  would  merely  lead  to  another  surface 
of  the  same  kind,  on  which  the  algebraically  multiform  functions  would 
be  uniform  functions  of  position ;  transcendentally  multiform  functions  of 
position  will  arise  later,  through  the  integrals  of  algebraic  functions.  It 
therefore  remains,  at  the  present  stage,  only  to  consider  the  most  general 
uniform  function  of  position  on  the  Riemann's  surface. 

On  the  other  hand,  it  is  evident  that  a  Riemann's  surface  of  any  number 
of  sheets  can  be  constructed,  with  arbitrary  branch-points  and  assigned 
sequence  of  junction  ;  the  elements  of  the  surface  being  subject  merely  to 
general  laws,  which  give  a  necessary  relation  between  the  number  of  sheets, 
the  ramification  and  the  connectivity,  and  which  require  the  restoration  of 
any  value  of  the  function  after  the  description  of  some  properly  chosen 
irreducible  circuit.  The  essential  elements  of  the  arbitrary  surface,  and  the 
merely  general  laws  indicated,  are  independent  of  any  previous  knowledge 
of  an  algebraical  equation  associated  with  the  surface ;  and  a  question  arises 
whether,  when  a  Riemann's  surface  is  given,  an  associated  algebraical  equa- 
tion necessarily  exists. 

Two  distinct  subjects  of  investigation,  therefore,  arise.  The  first  is  the 
most  general  uniform  function  of  position  on  a  surface  associated  with  a  given 
algebraical  equation,  and  its  integral ;  the  second  is  the  discussion  of  the 
existence  of  functions  of  position  on  a  surface  that  is  given  independently 


192.]  FUNCTIONS   OF   POSITION  417 

of  an  algebraical  equation.  Both  of  them  lead,  as  a  matter  of  fact,  to  the 
theory  of  transcendental  (that  is,  non-algebraical)  functions  of  the  most 
general  type,  commonly  called  Abelian  transcendents.  But  the  first  is, 
naturally,  the  more  direct,  in  that  the  algebraical  equation  is  initially 
given :  whereas,  in  the  second,  the  prime  necessity  is  the  establishment 
of  the  so-called  Existence-Theorem — that  such  functions,  algebraical  and 
■  transcendental,  exist. 

193.  Taking  the  subjects  of  investigation  in  the  assigned  order,  we 
suppose  the  fundamental  equation  to  be  irreducible,  and  rational  as 
regards  both  the  dependent  and  the  independent  variable ;  the  general  form 
is  therefore 

W'^  Go  (Z)  +  W"-1  G,(z)+...+  wGn-,  (z)  +  Gn  (z)  =  0, 

the  coefficients  Go{z),  Gi(z),  ...,  Gn{z)  being  rational  integral  functions  of 
the  variable  z. 

The  infinities  of  w  are,  by  §  95,  the  zeros  of  Go  (z)  and,  possibly,  z  =  qo  . 
But,  for  our  present  purpose,  no  special  interest  attaches  to  the  infinity  of  a 
function,  as  such;  we  therefore  take  wGo(z)  as  a  new  dependent  variable, 
and  the  equation  then  is 

f{w,  z)  =  lu''  +  tu''-^g, {z)+  ...  +  ivgn-i (z)  +  gn (z)  =  0, 

in  which  the  functions  g{z)  are  rational  integral  functions  of  z. 

The  distribution  of  the  branches  for  a  value  of  z  which  is  an  ordinary 
point,  and  the  determination  of  the  branch-points  together  with  the  cyclical 
grouping  of  the  branches  round  a  branch-point,  may  be  supposed  known. 
When  the  corresponding  7i-sheeted  Riemann's  surface  (say  of  connectivity 
2p  + 1)  is  constructed,  then  w  is  a  uniform  function  of  position  on  the 
surface. 

Now  not  merely  w,  but  every  rational  function  of  w  and  z,  is  a  uniform 
function  of  position  on  the  surface ;  and  its  branch-points  (though  not 
necessarily  its  infinities)  are  the  same  as  that  of  the  function  w. 

Conversely,  every  uniform  function  of  position  on  the  Riemann's  surface, 
having  accidental  singularities  and  infinities  only  of  finite  order,  is  a  rational 
function  of  w  and  z.  The  proof*  of  this  proposition,  to  which  we  now 
proceed,  leads  to  the  canonical  expression  for  the  most  general  uniform 
function  of  position  on  the  surface,  an  expression  which  is  used  in  Abel's 
Theorem  in  transcendental  integrals. 

Let  w'  denote  the  general  uniform  function,  and  let  w/,  Wz,  ...,  w^  denote 
the  branches  of  this  function  for  the  points  on  the  n  sheets  determined  by 

*  The  proof  adopted  follows  Prym,  Crelle,  t.  Ixxxiii,  (1877),  pp.  251—261 ;   see  also  Klein, 
XJeher  Riemann's  Theorie  der  algebraischen  Functionen  und  Hirer  Integrale,  p.  57. 

F.  F.  27 


418 


UNIFORM   FUNCTIONS   OF   POSITION 


[193. 


the  arithmetical  magnitude  z\  and  let  w-^,  w^,  ...,  Wn  be  the  corresponding 
branches  of  w  for  the  magnitude  2.     Then  the  quantity 

where  s  is  any  positive  integer,  is  a  symmetric  function  of  the  possible  values 
of  w^w' ;  it  has  the  same  value  in  whatever  sheet  2  may  lie  and  by  whatever 
path  2  may  have  attained  its  position  in  that  sheet ;  the  said  quantity  is 
therefore  a  uniform  function  of  z.  Moreover,  all  its  singularities  are  accidental 
in  character,  by  the  initial  hypothesis  as  to  w'  and  the  known  properties  of 
w ;  they  are  finite  in  number ;  and  therefore  the  uniform  function  of  2  is 
rational.  Let  it  be  denoted  by  hs(2),  which  is  an  integral  function  only 
when  all  the  singularities  are  for  infinite  values  of  2;   then 

an  equation  which  is  valid  for  any  positive  integer  s,  there  being  of  course 
the  suitable  changes  among  the  rational  integral  functions  h  {2)  for  changes 
in  s.  It  is  unnecessary  to  take  s'^n,  when  the  equations  for  the  values 
0,  1,  ...,  w—  1  of  5  are  retained:  for  the  equations  corresponding  to  values 
of  s^n  can  be  derived,  from  the  n  equations  that  are  retained,  by  using 
the  fundamental  equation  determining  w. 

Solving  the  equations 

W-^W-^  ^W^W^ +...-{■  WnWn  =hi{2), 


to  determine  w/,  we  have 


w. 


1,         1,  ...,  1 

Wi,  tU2,    ••■,  Wn 

W^~,  Wi,    ...,  Wn' 

-,  Wn' 


ho(2),  1,  ...,    1 

}h(2),  ^2,  ...,    Wn 

h2{2),  Wi,  ...,    lUri 

hn-i(2),  W2^-^  ...,    lUn 


The  right-hand  side  is  evidently  divisible  by  the  product  of  the  differences 
of  Wa,  Ws,  ...,  Wn',  and  this  product  is  a  factor  of  the  coefficient  of  ty/. 
Then,  if 

n 
(W  -  W.2)  (W  -  W3)  . . .  {lU  -  tUn)  =    S    krW''~'; 

r=l 

where  k^  is  unity,  we  have,  on  removing  the  common  factor, 

,       knK  {Z)  +  kn-ik  (^)  +  . .  •  +  hhn-2  (^)  +  /<n-i  (2) 
^'  ~  (Wj  -  Wa)  (Wi  -Ws)...  (Wi  -  Wn)  ' 


193.]  ON  A  riemann's  surface  419 

But  f{w,  z)  =  {w  —  Wi)  {W  —  W^...{W—  IVn), 

so  that  ko  =  Wi  +  gi  (z), 

ks  =  tui"  +  iv^g^  {z)  +  g^  {z), 


K  =  Wi^-i  +  Wi'^-^^'i  {z)+...+  gn-i  (z). 
When  these  expressions  for  k  are   substituted   in   the   numerator   of  the 
expression  for  Wi,  it  takes  the  form  of  a  rational  integral  function  of  w 
of  degree  n  —  1  and  of  z,  say 

h,  (z)  Wi^-i  +  ^1  (z)  ^i"-^  +  . . .  +  Rn-2  (2)  Wi  +  Hn-i  (z). 

The    denominator    is    evidently    df/dwi,    when    iv    is    replaced    by   w^    after 
differentiation,  so  that  we  now  have 


tu,  = 


df/dtv-^ 

The  corresponding  form  holds  for  each  of  the  branches  of  w' :  and  therefore 
we  have 

,  ^  ho  (z)  w^-^  +  H,  (z)  w^-'  +  . . .  +  Hn-,  jz) 
^  dfldw 

^  Aq  jz)  w^-^  +  H,  jz)  w'^--  +  ■  ■ .  +  Hn-,  {z) 
^^»i-i  _|.  (^ _  1 ) w'^-'^gi {z)  +  ...+  gn-x  {z) ' 

so  that  w  is  a  rational  function  of  w  and  z.  The  proposition  is  therefore 
proved. 

By  eliminating  w  between  f{w,  2')  =  0  and  the  equation  which  expresses 
w'  in  terms  of  w  and  z,  or  by  the  use  of  §  99,  it  follows  that  w'  satisfies  an 
algebraical  equation 

/,(..»  =  0, 

where  /i  is  of  order  n  in  w' .  As  will  be  seen  later  (§  245),  fx  {w ,  z)  either  is 
irreducible,  or  is  an  exact  power  of  an  irreducible  polynomial.  When/j  {w',  z) 
is  irreducible,  the  equations  /(w,  z)  =  0  and  /i  (w',  z)  =  0  have  the  same 
Riemann's  surface  associated  with  them*. 

194.  It  thus  appears  that  there  are  uniform  functions  of  position  on 
the  Riemann's  surface  just  as  there  are  uniform  functions  of  position  in 
a  plane.  The  preceding  proposition  is  limited  to  the  case  in  which  the 
infinities,  whether  at  branch-points  or  not,  are  merely  accidental ;  had  the 
function  possessed  essential  singularities,  the  general  argument  would  still 
be  valid,  but  the  forms  of  the  uniform  functions  h  (z)  would  no  longer  be 
polynomial.  In  fact,  taking  account  of  the  difference  in  the  form  of  the 
surface  on  which  the  independent  variable  is  represented,  we  can  extend 
to  multiform  functions,  which  are  uniform  on  a  Riemann's  surface,  those 
propositions  for  uniform  functions  which  relate  to  expansion  near  an  ordinary 

*  Functions  related  to   one   another,   as   lu  and   iv'  then   are,   are   called  gleichverzweigt, 
Eiemann,  p.  93. 

27—2 


420  RATIONAL   FUNCTIONS  [194. 

point  or  a  singularity  or,  by  using  the  substitution  of  §  93,  a  branch 
singularity,  those  which  relate  to  continuation  of  functions,  and  so  on ; 
and  their  validity  is  not  limited,  as  in  Cor.  VI.,  §  90,  to  a  portion  of  the 
surface  in  which  there  are  no  branch-points. 

Thus  we  have  the  theorem  that  a  uniform  rational  function  of  position  on 
the  Riemanns  surface  has  as  many  zeros  as  it  has  infinities. 

The  number  is  called  *  the  degree  of  the  function. 

This  theorem  may  be  proved  as  follows. 

The  function  is  a  rational  function  of  w  and  z.  If  it  be  also  integral,  let  it  be 
w'=  U{zv,  z),  where  U  is  integral. 

Then  the  niimber  of  the  zeros  of  w'  on  the  surface  is  the  number  of  simultaneous  roots 
common  to  the  two  equations  U{w,  z)  =  0,  f{w,  z)  =  0.  If  u^  and  /^  denote  the  aggregates 
of  the  terms  of  highest  dimensions  in  these  equations — say  of  dimensions  X  and  /^  respec- 
tively— then  \fjL  is  the  number  of  common  roots,  that  is,  the  number  of  zeros  of  w. 

The  number  of  points,  where  w'  assumes  a  value  A,  is  the  number  of  simultaneous 
roots  common  to  the  equations  U{w,  z)  =  A,  f{w,  z)  —  0,  that  is,  it  is  X/x  as  before.  Hence 
there  are  as  many  points  where  w'  assumes  a  given  value  as  there  are  zeros  of  w' ;  and 
therefore  the  number  of  the  infinities  is  the  same  as  the  number  of  zeros.  The  number 
of  infinities  can  also  be  obtained  by  considering  them  as  simultaneous  roots  common  to 

U(w  z) 
If  the  function  be  not  integral,  it  can  (§  193)  be  expressed  in  the  form  w'=  y)       \i 

where  U  and  V  are  rational  integral  functions.  The  zeros  of  w'  are  the  zeros  of  U  and 
the  infinities  of  V,  the  numbers  of  which,  by  what  precedes,  are  respectively  the  same  as 
the  infinities  of  U  and  the  zeros  of  V.  The  latter  are  the  infinities  of  ^y' ;  and  therefore  w' 
has  as  many  zeros  as  it  has  infinities. 

Note  1.  When  the  numerator  and  the  denominator  of  a  uniform  fractional 
function  of  z  have  a  common  zero,  we  divide  both  of  them  by  their  greatest 
common  measure ;  and  the  point  is  no  longer  a  common  zero  of  their  new 
forms.  But  when  the  numerator  U(w,  z)  and  the  denominator  V(w,  z)  of  a 
uniform  function  of  position  on  a  Riemann's  surface  have  a  common  zero,  so 
that  there  are  simultaneous  values  of  w  and  z  for  which  both  vanish,  U  and  V 
do  not  necessarily  possess  a  rational  common  factor ;  and  then  the  common 
zero  cannot  be  removed. 

It  is  not  difficult  to  shew  that  this  possibility  does  not  affect  the  preceding  theorem. 

Note  2.  In  estimating  the  degree  of  a  function  through  (say)  its  zeros,  it 
is  necessary  to  have  a  clear  mode  of  estimating  the  multiplicity  of  a  zero ; 
likewise  of  course  for  its  infinities,  and  for  its  level  points. 

Let  w  =  a,  z  =  a,  be  a  zero  of  a  uniform  rational  function  U {w,  z)  of 
position  on  a  Riemann's  surface :  the  multiplicity  of  the  zero  is  estimated  by 
the  expression  of  U  in  the  immediate  vicinity  of  the  position.  For  this 
purpose,  let 

w  =  oi-r  y,     z  =  a  +  X. 

*  Sometimes  it  is  called  the  order  of  the  function. 


194.]  ON   A   RIEM ANN'S   SURFACE  421 

(If  a  be  infinite,  we  take  w  =  -;  if  ^  be  infinite,  we  take  z  =  -;  if  both  be 

y  oc 

infinite,  we  use  both  these  substitutions.) 

First,  let  a,  a  be  an  ordinary  point  on  the  Riemann's  surface,  that  is,  not  a 
branch -point;  then  the  equation 

/(a  +  y,a+x)  =  0  =f(a,  a) 

determines  3/  as  a  uniform  function  of  x  in  the  immediate  vicinity  of  a,  a,  so 
that  we  have 

y  =  \x  +  X^X'  +  .... 

Now  {/(lu,  z)=  U(a  +  y,  a  + x) 

dU        dU 

substitute  for  y  on  the  right-hand  side  the  above  value,  and  suppose  that 
then 

U(w,  z)  =  kox^  +  kiX^+'^  +  ..., 

in  the  immediate  vicinity  of  the  point  on  the  surface.  We  then  say  that  the 
zero  a,  a  of  U{w,  z)  is  of  multiplicity  I. 

Next,  let  the  point  a,  a  be  a  branch-point  on  the  Riemann's  surface,  so 
that  a  number  of  sheets  wind  into  one  another  at  the  point.  Then,  by  §  97, 
there  exists  a  variable  ^  such  that 

^  _  a  =  ^9, 

w-a.  =  t;p{vi-\-v^^-\-  ...), 

in  the  immediate  vicinity  of  a,  a :  the  positive  integers  p  and  q  having  no 
common  factor.  If  2^  =  00  give  rise  to  a  branch-point,  the  new  variable  would 
be  given  by  z  =  t~i;  if  a  were  infinite,  the  expression  for  w  would  be  of  the 
form  lu  =  ^~P  (vi  -f  ^2^+  ...)  ;  and  similarly  if  both  a  and  a  were  infinite.  As 
a,  a  is  a  zero  of  U,  we  have 

U  (iv,z)  =  (w  -  a)  —  +  {z  -  c)  ^  +.... 

Substitute  for  both  w  —  a.  and  z  —  c  on  the  right-hand  side :  and  suppose  that 
then 

valid  in  the  immediate  vicinity  of  the  point.  We  then  say*  that  ^  is 
an  infinitesimal  of  the  first  order,  and  therefore  that  the  zero  a,  a  is  of 
multiplicity  \. 

*  Eiemann,  Ges.  Werke,  p.  96.     The  justification  for  regarding  f  as  a  small  quantity  of  the 
first  order,  that  is,  the  same  as  x  when  the  point  is  not  a  branch-point,  is  that  the  value  of 

dz 


2iri  J  z-  i 


taken  round  a  simple  closed  curve  enclosing  a  is  q,  because  the  curve  passes  round  a  in  each  of 
the  q  sheets.     Thus  f  2  counts  as  having  q  zeros,  and  therefore  f  is  of  the  first  order. 


422  PATHS   OF   INTEGKATION  [195. 

195.  In  the  case  of  uniform  functions  it  was  seen  that,  as  soon  as 
their  integrals  were  considered,  deviations  from  uniformity  occurred.  Special 
investigations  indicated  the  character  of  the  deviations  and  the  limitations 
to  their  extent.  Incidentally,  special  classes  of  functions  were  introduced, 
such  as  many- valued  functions,  the  values  differing  by  multiples  of  a  constant; 
and  thence,  by  inversion,  simply-periodic  functions  were  deduced. 

So,  too,  when  multiform  functions  defined  by  an  algebraical  equation  are 
considered,  it  is  necessary  to  take  into  special  account  the  deviations  from 
uniformity  of  value  on  the  Riemann's  surface  which  may  be  introduced  by 
processes  of  integration.  It  is,  of  course,  in  connection  with  the  branch- 
points that  difficulties  arise ;  but,  as  the  present  method  of  representing  the 
variation  of  the  variable  is  distinct  from  that  adopted  in  the  case  of  uniform 
functions,  it  is  desirable  to  indicate  how  we  deal,  not  merely  with  branch- 
points, but  also  with  singularities  of  functions  when  the  integrals  of  such 
functions  are  under  consideration.  In  order  to  render  the  ideas  familiar 
and  to  avoid  prolixity  in  the  explanations  relating  to  general  integrals,  we 
shall,  after  one  or  two  propositions,  discuss  again  some  of  the  instances 
given  in  Chapter  IX.,  taking  the  opportunity  of  stating  general  results  as 
occasion  may  arise. 

One  or  two  propositions  already  proved  must  be  restated:  the  difference 
from  the  earlier  forms  is  solely  in  the  mode  of  statement,  and  therefore  the 
reasoning  which  led  to  their  establishment  need  not  be  repeated. 

I.  The  path  of  integration  betiueen  any  two  points  on  a  Riemann's  surface 
can,  without  affecting  the  value  of  the  integral,  be  deformed  in  any  possible 
continuous  manner  that  does  not  make  the  path  pass  over  any  discontinuity  of 
the  subject  of  integration. 

This  proposition  is  established  in  §  100. 

II.  A  simple  closed  curve  on  a  Riemann's  surface,  which  is  a  path  of 
integration,  can,  tuithout  affecting  the  value  of  the  integral,  be  deformed  in 
any  possible  continuous  manner  that  does  not  rnake  the  curve  pass  over  any 
discontinuity  of  the  subject  of  integration. 

Since  the  curve  on  the  surface  is  closed,  the  initial  and  the  final  points 
are  the  same ;  the  initial  branch  of  the  function  is  therefore  restored  after 
the  description  of  the  curve.  This  proposition  is  established  in  Corollary  II., 
§  100. 

III.  If  the  path  of  integration  be  a  curve  betiueen  tiuo  points  on  different 
sheets,  determined  by  the  same  algebraical  value  of  z,  the  curve  is  not  a  closed 
curve ;  it  must  be  regarded  as  a  path  between  the  two  points;  its  deformation 
is  subject  to  Proposition  I. 

No  restatement,  from  Chapter  IX.,  of  the  value  of  an  integral,  along 
a  path  which  encloses  a  branch -point,  is  necessary.     The  method  of  dealing 


195.]  ON  A  riemann's  surface  423 

with  the  point  when  that  value  is  infinite  will  be  the  same  as  the  method  of 
dealing  with  other  infinities  of  the  function. 

196.  We  have  already  obtained  some  instances  of  multiple-valued 
functions,  in  the  few  particular  integrals  in  Chapter  IX.;  the  differences  in 
the  values  of  the  functions,  arising  as  integrals,  consist  solely  of  multiples  of 
constants.  The  way  in  which  these  constants  enter  in  Riemann's  method  is 
as  follows. 

When  the  surface  is  simply  connected,  there  is  no  substantial  difference 
from  the  previous  theory  for  uniform  functions;  we  therefore  proceed  to  the 
consideration  of  multiply  connected  surfaces. 

On  a  general  surface,  of  any  connectivity,  take  any  two  points  Zq  and  z. 
As  the  surface  is  one  of  multiple  connection,  there  will  be  at  least  two 
essentially  distinct  paths  between  z^  and  z,  that  is,  paths  which  cannot  be 
reduced  to  one  another ;  one  of  these  paths  can  be  deformed  so  as  to  be 
made  equivalent  to  a  combination  of  the  other  with  some  irreducible  circuit. 
Let  ^1  denote  the  extremity  of  the  first  path,  and  let  z^  denote  the  same  point 
when  regarded  as  the  extremity  of  the  second ;  then  the  difference  of  the 
two  paths  is  an  irreducible  circuit  passing  from  ^i  to  Zr,.  When  this  circuit 
is  made  impossible  by  a  cross-cut  G  passing  through  the  point  z,  then  z-^ 
and  Z.2  may  be  regarded  as  points  on  the  opposite  edges  of  the  cross-cut :  and 
the  irreducible  circuit  on  the  unresolved  surface  becomes  a  path  on  the 
partially  resolved  surface  passing  from  one  edge  of  the  cross-cut  to  the  other. 

When  the  surface  is  resolved  by  means  of  the  proper  number  of  cross-cuts 
into  a  simply  connected  surface,  there  is  still  a  path  in  the  surface  from 
2^1  to  ^2  on  opposite  edges  of  the  cross-cut  C:  and  all  paths  between  z^  and  z^ 
in  the  resolved  surface  are  reconcileable  with  one  another.  One  such  path 
will  be  taken  as  the  canonical  path  from  z^  to  z^ ;  it  evidently  does  not  meet 
any  of  the  cross-cuts,  so  that  we  consider  only  those  paths  which  do  not 
intersect  any  cross-cut. 

If  then  Z  be  the  function  of  position  on  the  surface  to  be  integrated,  the 
value  of  the  integral  for  the  first  path  from  z^  to  z^  is 


Zdz; 

and  for  the  second  path  it  is  1     Zdz, 

or,  by  the  assigned  deformation  of  the  second  path,  it  is 

Zdz+  j'^Zdz, 


the  second  integral  being  taken  along  the  canonical  path  from  z^  to  Zz  in  the 
surface,  that  is,  along  the  irreducible  circuit  of  canonical  form,  which  would  be 
possible  in  the  otherwise  resolved  surface  were  the  cross-cut  C  obliterated. 


424  CONSTANT   OF   INTEGRATION  [196. 

The  difference  of  the  values  of  the  integral  is  evidently 

Zdz, 
which   is  therefore  the  change  made  in  the  value  of  the  integral   1    Zdz, 

J  Zo 

when  the  upper  limit  passes  from  one  edge  of  the  cross-cut  to  the  other ;  let 
it  be  denoted  by  I.  As  the  curve  is,  in  general,  an  irreducible  circuit,  this 
integral  /  may  not,  in  general,  be  supposed  zero. 

We  can  arbitrarily  assign  the  positive  and  the  negative  edges  of  some  one 
cross-cut,  say  A.  The  edges  of  a  cross-cut  B  that  meets  A  are  defined  to  be 
positive  and  negative  as  follows:  when  a  point  moves  from  one  edge  of  B  to 
the  other,  by  describing  the  positive  edge  of  J.  in  a  direction  that  is  to  the 
right  of  the  negative  edge  of  A,  the  edge  of  B  on  which  the  point  initially 
lies  is  called  its  jiositive  edge,  and  the  edge  of  B  on  which  the  point  finally 
lies  is  called  its  negative  edge.     And  so  on  with  the  cross-cuts  in  succession. 

The  lower  limit  of  the  integral  determining  the  modulus  for  a  cross-cut 
is  taken  to  lie  on  the  negative  edge,  and  the  upper  on  the  positive  edge. 

Regarding  a  point  ^  on  the  cross-cut  as  defining  two  points  z-^^  and  Z2  on 
opposite  edges  which  geometrically  are  coincident,  we  now  prove  that/o?'  all 
'points  on  the  cross-cut  which  can  be  reached  from  ^  without  passing  over  any 
other  cross-cut,  when  the  surface  is  resolved  into  one  that  is  simjdy  connected, 
the  integral  I  is  a  constant.  For,  if  ^'  be  such  a  point,  defining  z^'  and  Z2  on 
opposite  edges,  then  z^^z^z^z^'zi  is  a  circuit  on  the  simply  connected  surface, 
which  can  be  made  evanescent ;  and  it  will  be  assumed  that  no  infinities  of  Z 
lie  in  the  surface  within  the  circuit,  an  assumption  which  will  be  taken  into 
account  in  §§  197,  199.  Therefore  the  integral  of  ^,  taken  round  the  circuit, 
is  zero.     Hence 

r  Zdz  +  r  Zdz  4-  I ''  Zdz  +  I ''  Zdz  =  0, 

fZi  fZi  fZ/  fZi' 

that  is,  Zdz  -        Zdz  =        Zdz  -       Zdz. 

•I  Z,  J  2/  •    2i  •'  Zn 

Along  the  direction  of  the  cross-cut,  the  function  Z  is  uniform :  and 
therefore  Zdz  is  the  same  for  each  element  of  the  two  edges,  so  long  as  the 
cross-cut  is  not  met  by  any  other.  Hence  the  sums  of  the  elements  on  the 
two  edges  are  the  same  for  all  points  on  the  cross-cut  that  can  be  reached 
from  ^  without  meeting  a  new  cross-cut.  The  two  integrals  on  the  right- 
hand  side  of  the  foregoing  equation  are  equal  to  one  another,  and  therefore 
also  those  on  the  left-hand  side,  that  is, 

''Zdz=  r  Zdz, 


which  shews  that  the  integral  I  is  constant  for  different  points  on  a  portion  of 
cross-cut  that  is  not  met  by  any  other  cross-cut. 


196.] 


AT   A   CROSS-CUT 


425 


If  however  the  cross-cut  be  met  by  another  cross-cut  C ,  two  cases  arise 
according  as  C  has  only  one  extremity  on  G,  or  has  both  extremities  on  C, 

First,  let  C  have  only  one  extremity  0  on  G.  By  what  precedes,  the 
integral  is  constant  along  OP,  and  it  is  constant 
along  OQ ;  but  we  cannot  infer  that  it  is  the  same 
constant  for  the  two  parts.  The  preceding  proof 
fails  in  this  case ;  the  distance  2-2^2'  in  the  resolved 
surface  is  not  infinitesimal,  and  therefore  there  is 
no  element  Zdz  for  z^^  to  be  the  same  as  the  ® 
element  for  z-^^z-^.  Let  I^  be  the  constant  for  OP, 
Ii  that  for  QO ;  and  let  QP  be  the  negative  edge.     Then 

I,=  rZdz,       I,=  l     Zdz. 


Q 


z'^. 


z\0  z, 
Fig.  66. 


Let  /'  be  the  constant  value  for  the  cross-cut   OR,  and  let  OR  be    the 
negative  edge ;  then 

I'=       Zdz. 

In  the  completely  resolved  surface,  a  possible  path  from  z^  to  ^2'  is  ^2  to  z^, 
Zi  to  Zi,  z-[  to  z^  ;  it  therefore  is  the  canonical  path,  so  that 


J'  = 


Zdz^ 

=  -/2  +  /l  + 


Zdz^ 
Zdz. 


Zdz 


But        Zdz  is  an  integral  of  a  uniform  finite  function  along  an  infinitesimal 

arc  z-JJzi,  and  it  is  zero  in  the  limit  when  we  take  ^1  and  Zi  as  coincident. 
Thus 

or  the  constant  for  the  cross-cut  OR  is  the  excess  of  the  constant  for  the  part  of 
PQ  at  the  positive  edge  of  OR  over  the  constant  for  the  part  of  PQ  at  the 
negative  edge. 

Secondly,  let  G'  have  both  extremities  on  G,  close  to  one  another  so  that 
they  may  be  brought  together,  as  in  the  figure :  it 
is  effectively  the  case  of  the  directions  of  two  cross- 
cuts intersecting  one  another,  say  at  0.  Let  Ii,  I2, 
/3,/4be  the  constants  for  the  portions  QO,  OP,  OR, 
SO  of  the  cross-cuts  respectively,  and  let  Z3Z2  be  q 
the  positive  edge  of  QOP  ;  then  z^z.,  is  the  positive 
edge  of  SOR.     Then  if  @  (z)  denote  the  value  of 


the  integral  /    Zdz  at  0,  which  is  definite  because 

^0 


s 

Fig.  67. 


426 


MODULUS   OF   PERIODICITY 


[196. 


the  surface  is  simply  connected  and  no  discontinuities  of  Z  lie  within  the 
paths  of  integration,  we  have 


and 


so  that 


/,=       Zdz  =  %{z^-%{z,)\ 


h  =  \"'  Zdz  =  0  (^3)  -  ©  (^2),    I, = r  Zdz  =  ®  (^4)  -  ®  (^1) ; 

J   Zi  J  Si 


or  the  excess  of  the  constant  for  the  portion  of  a  cross-cut  on  the  positive  edge, 
over  the  constant  for  the  portion  on  the  negative  edge,  of  another  cross-cut  is 
equal  to  the  excess,  similarly  estimated,  for  that  other  cross-cut. 

Ex:  Consider  the  constants  for  the  various  portions  of  the  cross-cuts  in  the  canonical 
resolution  (§§  180,  181)  of  a  Riemann's  surface.  Let  the  constants  for  the  two  portions 
of  a^  be  Ay-i  -4/;  and  the  constants  for  the  two  portions  of  6,.  be  B^,  Bj! ;  and  let  the 
constant  for  c^  be  C,.. 

Then,  at  the  junction  of  c^  and  a^  +  \,  we  have 

at  the  junction  of  c^  and  6^,  we  have 

and,  at  the  crossing  of  a,,  and  &^,  we  have 

Now,  because  hi  is  the  only  cross-cut  which  meets  a^ , 
we  have  Ai=Ai  ;  hence  Bi  =  Bi',  and  therefore  Ci  =  0. 
Hence  ^2  =  ^42';  therefore  B'2,=B2,  and  therefore  also 
(72  =  0.     And  so  on. 

Hence  the  constant  for  each  of  the  portions  of  a  cross-cut  a  is  the  same;  the  constant  for 
each  of  the  portions  of  a  cross-cut  b  is  the  same  ;  and  the  constant  for  each  cross-cut  c  is  zero. 
A  single  constant  may  thus  be  associated  with  each  cross-cut  a,  and  a  single  constant  with 
each  cross-cut  6,  in  connection  with  the  integral  of  a  given  uniform  function  of  position  on 
the  Riemann's  surface.  It  has  not  been  proved — and  it  is  not  necessarily  the  fact — that 
any  one  of  these  constants  is  different  from  zero ;  but  it  is  sufficiently  evident  that,  if  all 
the  constants  be  zero,  the  integral  is  a  uniform  function  of  position  on  the  surface,  that  is, 
a  uniform  function  of  w  and  z. 

197.  Hence  the  values  of  the  integral  at  points  on  opposite  edges  of  a 
cross-cut  differ  by  a  constant. 

Suppose  now  that  the  cross-cut  is  obliterated :  the  two  paths  to  the  point 
z  will  be  the  same  as  in  the  case  just  considered  and  will  furnish  the  same 
values  respectively,  say  JJ  and  TJ-\-I.  But  the  irreducible  circuit  which 
contributes  the  value  /  can  be  described  any  number  of  times ;  and 
therefore,  taking  account  solely  of  this  irreducible  circuit  and  of  the  cross-cuts 


Fig.  68. 


197.]  FOR  A   CROSS-CUT  427 

which  render  other  circuits  impossible  on  the  resolved  surface,  the  general 
value  of  the  integral  at  the  point  z  is 

where  h  is  an  integer  and  JJ  is  the  value  for  some  prescribed  path. 

The  constant  /  is  called*  a  modulus  of  periodicity. 

It  is  important  that  every  modulus  of  periodicity  should  be  finite ;  the  path 
which  determines  the  modulus  can  therefore  pass  through  a  point  c  where 
Z  =  00 ,  or  be  deformed  across  it  without  change  in  the  modulus,  only  if  the 
limit  of  (z  —  c)Z  he  a  uniform  zero  at  the  point.  If,  however,  the  limit  of 
(z  —  c)  Z  at  the  point  be  a  constant,  implying  a  logarithmic  infinity  for  the 
integral,  or  if  it  be  an  infinity  of  finite  order  (the  order  not  being  necessarily 
an  integer),  implying  an  algebraic  infinity  for  the  integral,  we  surround 
the  point  c  by  a  simple  small  curve  and  exclude  the  internal  area  from  the 
range  of  variation  of  the  independent  variable  f.  This  exclusion  is  secured 
by  making  a  small  loop-cut  in  the  surface  round  the  point ;  it  increases 
by  unity  the  connectivity  of  the  surface  on  which  the  variable  is  represented. 

When  the  limit  of  {z  —  c)Z  is  a  uniform  zero  at  c,  no  such  exclusion 
is  necessary :  the  order  of  the  infinity  for  Z  is  easily  seen  to  be  a  proper 
fraction  and  the  point  to  be  a  branch-poiut. 

Similarly,  if  the  limit  of  zZ  for  ^r  =  oo  be  not  zero  and  the  path  which 
determines  a  modulus  can  be  deformed  so  as  to  become  infinitely  large,  it  is 
convenient  to  exclude  the  part  of  the  surface  at  infinity  from  the  range  of 
variation  of  the  vaiiable,  proper  account  being  taken  of  the  exclusion.  The 
reason  is  that  the  value  of  the  integral  for  a  path  entirely  at  infinity  (or 
for  a  point-path  on  Neumann's  sphere)  is  not  zero ;  z=  (X)  is  either  a 
logarithmic  or  an  algebraic  infinity  of  the  function.  But,  if  the  limit  of  zZ 
be  zero  for  z=  oo  ,  the  exclusion  of  the  part  of  the  surface  at  infinity  is 
unnecessary. 

198.  When,  then,  the  region  of  variation  of  the  variable  is  properly 
bounded,  and  the  resolution  of  the  surface  into  one  that  is  simply  connected 
has  been  made,  each  cross-cut  or  each  portion  of  cross-cut,  that  is  marked  off 
either  by  the  natural  boundary  or  by  termination  in  another  cross-cut, 
determines  a  modulus  of  periodicity.  The  various  moduli,  for  a  given 
resolution,  are  therefore  equal,  in  number,  to  the  various  portions  of  the 
cross-cuts.  Again,  a  system  of  cross-cuts  is  susceptible  of  great  variation, 
not  merely  as  to  the  form  of  individual  members  of  the  system  (which  does 
not  affect  the  value  of  the  modulus),  but  in  their  relations  to  one  another. 
The  total  number  of  cross-cuts,  by  which  the  surface  can  be  resolved  into  one 
that  is  simply  connected,  is  a  constant  for  the  surface  and  is  independent  of 

*  Sometimes  the  modulus  for  the  cross-cut. 

t  This  is  the  reason  for  the  assumption  made  on  p.  424. 


428  THE    NUMBER   OF    INDEPENDENT   MODULI  [198. 

their  configuration :  but  the  number  of  distinct  pieces,  defined  as  above,  is 
not  independent  of  the  configuration.  Now  each  piece  of  cross-cut  furnishes 
a  modulus  of  periodicity ;  a  question  therefore  arises  as  to  the  number  of 
independent  moduli  of  periodicity. 

Let  the  connectivity  of  the  surface  be  iV^+  1,  due  regard  being  had  to  the 
exclusions,  if  any,  of  individual  points  in  the  surface :  in  order  that  account 
may  be  taken  of  infinite  values  of  the  variable,  the  surface  will  be  assumed 
spherical.  The  number  of  cross-cuts  necessary  to  resolve  it  into  a  surface 
that  is  simply  connected  is  iV;  whatever  be  the  number  of  portions  of  the 
cross-cuts,  .the  number  of  these  portions  is  not  less  than  N'. 

When  a  cross-cut  terminates  in  another,  the  modulus  for  the  former  and 
the  moduli  for  the  two  portions  of  the  latter  are  connected  by  a  relation 

CO  —  COi'^  Qi2, 

SO  that  the  modulus  for  any  portion  can  be  expressed  linearly  in  terms  of 
the  modulus  for  the  earlier  portion  and  of  the  modulus  for  the  dividing 
cross-cut. 

Similarly,  when  the  directions  of  two  cross-cuts  intersect,  the  moduli  of 
the  four  portions  are  connected  by  a  relation 

a)i~  &)/  =  (02~  0)2  ; 
and  by  passing  along  one  or  other  of  the  cross-cuts,  some  relation  is  obtainable 
between  co^  and  &)/  or  between  co^  and  co^',  so  that,  again,  the  modulus  of  any 
portion  can  be  expressed  linearly  in  terms  of  the  modulus  for  the  earlier 
portion  and  of  moduli  independent  of  the  intersection. 

Hence  it  appears  that  a  single  constant  must  be  associated  with  each 
cross-cut  as  an  independent  modular  constant ;  and  then  the  constants 
for  the  various  portions  can  be  linearly  expressed  in  terms  of  these  inde- 
pendent constants.  There  are  therefore  N  linearly  independent  moduli  of 
periodicity:  but  no  system  of  moduli  is  unique,  and  any  system  can  be 
modified  partially  or  wholly,  if  any  number  of  the  moduli  of  the  system  be 
replaced  by  the  same  number  of  independent  linear  combinations  of  members 
of  the  system.  These  results  are  the  analytical  equivalent  of  geometrical 
results,  which  have  already  been  proved,  viz.,  that  the  number  of  independent 
simple  irreducible  circuits  in  a  complete  system  is  N,  that  no  complete 
system  of  circuits  is  unique,  and  that  the  circuits  can  be  replaced  by 
independent  combinations  reconcileable  with  them. 

199.  If,  then,  the  moduli  of  periodicity  of  a  function  U  at  the  cross-cuts 
in  a  resolved  surface  be  /j,  I^,  ••-,  In,  all  the  values  of  the  function  at 
any  point  on  the  unresolved  surface  are  included  in  the  form 

Z7  -h  mi/i  -f  mj/a  +  . . .  +  niNljY, 

where  mj ,  TOo  ,  . . . ,  nij^/-  are  integers. 


199.]  EXAMPLES    •  429 

Some  special  examples,  treated  by  the  present  method,  will  be  useful  in  leading  up  to 
the  consideration  of  integrals  of  the  most  general  functions  of  position  on  a  Eiemann's 
surface. 

[dz 
Ex.  1.     Consider  the  integral    I  — . 

The  subject  of  integration  is  uniform,  so  that  the  surface  is  one-sheeted.  The  origin 
is  an  accidental  singularity  and  gives  a  logarithmic  infinity  for  the  integral ;  it  is  therefore 
excluded  by  a  small  circle  round  it.  Moreover,  the  value  of  the  integral  round  a  circle 
of  infinitely  large  radius  is  not  zero:  and  therefore  3=oo  is  excluded  from  the  range  of 
variation.  The  boundary  of  the  single  spherical  sheet  can  be  taken  to  be  the  point  2=00  ; 
and  the  bounded  sheet  is  of  connectivity  2,  owing  to  the  small  circle  at  the  origin.  The 
surface  can  be  resolved  into  one  that  is  simply  connected  by  a  single  cross-cut  drawn 
from  the  boundary  at  2=co  to  the  circumference  of  the  small  circle. 

If  a  plane  surface  be  used,  this  cross-cut  is,  in  effect,  a  section  (§  103)  of  the  plane 
made  from  the  origin  to  the  point  s  =  oc . 

/■J 
There  is  only  one  modulus  of  periodicity  :  its  value  is  evidently  /  — ,  taken  round  the 

origin,  that  is,  the  modulus  is  2iri.     Hence  whenever  the  6 

path  of  variation  from  a  given  point  to  a  point  z  passes      " "'" "^  w  ^ 

from  A  to  B.  the  value  of  the  integral  increases  by  ^iri;  but 

if  the  path  pass  from  B  to  A,  the  value  of  the  integral  °' 

decreases  by  27rt.     Thus  A  is  the  negative  edge,  and  B  the  positive  edge  of  the  cross-cut. 

If,  then,  any  one  value  of  I      —  be  denoted  by  w,  all  values  at  the  point  in  the 

J  So    ^ 

unresolved  surface  are  of  the  form  w  +  2inTvi,  where  m  is  an  integer;  when  z  is  regarded 
as  a  function  of  w,  it  is  a  simply-periodic  function,  having  27rz  for  its  period. 

Ex.  2.     Consider    i  -^ ^  •     The  subject  of  integration  is  uniform,  so  that  the  surface 

consists  of  a  single  sheet.  There  are  two  infinities  +a,  each  of  the  first  order,  because 
{z  +  a)Z  is  finite  at  these  two  points  :  they  must  be  excluded  by  small  circles.  The  limit, 
when  2=00,  of  z/{z^  —  a^)  is  zero,  so  that  the  point  2=00  does  not  need  to  be  excluded. 
We  can  thus  regard  one  of  the  small  circles  as  the  boundary  of  the  surface,  which  is  then 
doubly  connected :  a  single  cross-cut  from  the  other  circle  to  the  boundary,  that  is,  in 
effect,  a  cross-cut  joining  the  two  points  a  and  —a,  resolves  the  surface  into  one  that  is 
simply  connected. 


TTl 


It  is  easy  to  see  that  the  modulus  of  periodicity  is  ^  :  that  A  is  the  negative  edge  and 


a 


B  the  positive  edge  of  the  cross-cut :  and  that,  if  w  be 

a  value  of  the  integral  in  the  unresolved  surface  at  any  _  ^  ^ B 

point,  all  the  values  at  that  point  are  included  in  the  A 

form  Fig.  70. 


w  +  n  — 


where  n  is  an  integer. 


Ex.  3.  Consider  j{a^  -  2^)  -  4  dz.  The  subject  of  integration  is  two-valued,  so  that  the 
surface  is  two-sheeted.  The  branch-points  are  ±  a,  and  co  is  not  a  branch-point,  so  that 
the  single  branch-line  between  the  sheets  may  be  taken  as  the  straight  line  joining  a 
and  -  a.  The  infinities  are  ±a  ;  but  as  (z  +  a)  {a^-z^)~2  vanishes  at  the  points,  they  do 
not  need  to  be  excluded.  As  the  limit  of  z(a'^-z^}~2,  for  2  =  co,  is  not  zero,  we  exclude 
2  =  00  by  small  curves  in  each  of  the  sheets. 


430 


EXAMPLES 


[199. 


a  circuit 


Taking  the  surface  in  the  spherical  form,  we  assign  as  the  boundary  the  small  curve 
round  the  point  2=00  in  one  of  the  sheets.  The  connectivity  of  the  surface,  through  its 
dependence  on  branch-lines  and  branch-points,  is  unity :  owing  to  the  exclusion  of  the 
point  s  =  00  by  the  small  curve  in  the  other  sheet,  the  connectivity  is  increased  by  one 
unit :  the  surface  is  therefore  doubly  connected.  A  single  cross-cut  will  resolve  the  surface 
into  one  that  is  simply  connected :  and  this  cross-cut  must  pass  from  the  boundary  at 
2  =  00  which  is  in  one  sheet  to  the  excluded  point  2=co. 

Since  the  (single)  modulus  of  periodicity  is  the  value  of  the  integral  along 
in  the  resolved  surface  from  one  edge  of  the  cross-cut 
to  the  other,  this  circuit  can  be  taken  so  that  in  the  ,,-.--''-- 

unresolved  surface  it  includes  the  two  branch-points;       ^ ^ __ 

and  then,  by  II.  of  §  195,  the  circuit  can  be  deformed      (^0^ 
until  it  is  practically  a  double  straight  line  in  the  upper 
sheet  on  either  side  of  the  branch-line,  together  with  two 
small  circles  round  a  and  -  a  respectively.    Let  P  be  the 
origin,  practically  the  middle  point  of  these  straight  lines. 


Consider  the  branch  (a^  -  2^)  -  2  belonging  to  the  upper 
sheet.     Its  integral  from  P  to  a  is 


^;i 


o 

Fig.  71- 


I     {a?  —  2^)    2  dz. 
J  0 


From  a  to  —a  the  branch  is  -{a^-z^)~^;  the  point  R  is  contiguous  in  the  surface, 
not  to  P,  but  (as  in  §  189)  to  the  point  in  the  second  sheet  beneath  P  at  which  the  branch 
is  _  (a2-22)~4,  the  other  branch  having  been  adopted  for  the  upper  sheet.  Hence,  from  a 
to  -  a  by  R,  the  integral  is 


_r«2-22\- 


■{a 

From  -ato  Q,  the  branch  is  +{a^-z^)~i,  the  same  branch  as  at  P:  hence  from  -atoQ, 
the  integral  is 

I       {a^  —  2^)  ~  a  dz. 
J  -a 

The  integral,  along  the  small  arcs  round  a  and  round  a'  respectively,  vanishes  for  each. 
Hence  the  modulus  of  periodicity  is 


{a^-z^)-idz+  -(a'--z^)-^dz+         {a^-z^)-hdz, 


that  is,  it  is  Stt. 

This  value  can  be  obtained  otherwise  thus.     The  modulus  is  the  same  for  all  points 
OB  the  cross-cut ;  hence  its  value,  taken  at  0'  where  2=  ao ,  is 

j{d^-z^)-hdz, 

passing  from  one  edge  of  the  cross-cut  at  0'  to  the  other,  that  is,  round  a  curve  in  the 
plane  everywhere  at  infinity.     This  gives 

9   / 
27^^■  Lt  2(a2-2''^)~i  =  =^  =  27r, 

the  same  value  as  before. 

The  latter  curve  round  0',  from  edge  to  edge,  can  easily  be  deformed  into  the  former 
curve  round  a  and  —  a  from  edge  to  edge  of  the  cross-cut. 


199.] 


OF   MODULI   OF   PERIODICITY 


431 


Again,  let  Wi  be  a  value  of  the  integral  for  a  point  2j^  in  one  sheet  and  W2  be  a  value 
for  a  point  ^2  in  the  other  sheet  with  the  same  algebraical  value  as  Sj :  take  zero  as  the 
common  lower  limit  of  the  integral,  being  the  same  zero 
for  the  two  integrals.  As  this  zero  may  be  taken  in 
either  sheet,  let  it  be  in  that  in  which  z^  lies  :  and  then       y^  O  i  -^ 


Wj  =1      (a^  —  z^)    2  dz. 


Fig.  72 


To  pass  from  0  to  Zo  for  W2,  any  path  can  be  justifiably  deformed  into  the  following: 
(i)  a  path  round  either  branch-point,  say  a,  so  as  to  return  to  the  point  under  0  in  the 
second  sheet,  say  to  O2,  (ii)  any  number  m  of  irreducible  circuits  round  a  and  —a,  always 
returning  to  O9  in  the  second  sheet,  (iii)  a  path  from  0^  to  Z2  lying  exactly  under  the  path 
from  0  to  Zi  for  Wi.  The  parts  contributed  by  these  paths  respectively  to  the  integral  W2 
are  seen  to  be 

fa  1  /"O 

(1)     a  quantity  +  TT,  arising  from  I     (a^-z^)-idz+       -  (a^ - z^)- i dz,  for  resisons 

similar  to  those  above ; 

(ii)    a  quantity  m27r,  where  m  is  an  integer  positive  or  negative ; 


(iii)    a  quantity  /   ' —{a^  —  z^)    "sdz. 

In  the  last  quantity  the  minus  sign  is  prefixed,  because  the  subject  of  integration  is 
everywhere  in  the  second  sheet.     Now  ^2=21,  and  therefore  the  quantity  in  (iii)  is 


-f; 


{a^-z'-)-^dz, 


that  is,  it  is  —  Wj ;  hence 


W2  =  (2?^^  + 1 )  TT  —  Wi . 
fz  _i 

If  then  we  take  w=  j     (a-  -  z'^)    2  dz,  the  integral  extending  along  some  defined  curve  from 

an  assigned  origin,  say  along  a  straight  line,  the  values  of  w  belonging  to  the  same 
algebraical  value  of  z  are  27nr  +  w  or  [2m  +  l)n  —  w,  and  the  inversion  of  the  functional 
relation  gives 

(f)  {iv)=z=(j)  (2n7r  +  w) 


where  m  and  n  are  any  integers. 

/"  dz 

Ex.  4.     Consider 


=  <^  {{2111+1)  n-io], 


-  ,  assuming  |  c  I  >  |  a  |.     The  surface  is  two-sheeted, 
(2  -  c)  (a2  -  s2)2 

with  branch-points  at  +a  but  not  at  00  :   hence  the  line  joining  a  and  —a  is  the  sole 

branch-line.     The  infinities  of  the  subject  of  integration  are  a,  —a,  and  c.     Of  these  a 

and  —a  need  not  be  excluded,  for  the  same  reason  that 

their  exclusion  was  not  required  in  the   last   example. 

But  c  must  be  excluded ;  and  it  must  be  excluded  in  both 

sheets,  because  z  =  c  makes   the   subject   of  integration 

infinite  in  both  sheets.     There  are  thus  two  points  of 

accidental  singularity  of  the  subject  of  integration ;   in 

the  vicinity  of  these   points,  the  two  branches  of  the 

subject  of  integration  are 


J__(a2_,2)-i+...^     _J_(a2_,2)-i_ 


Fig.  73. 


432 


MODULI    OF   PERIODICITY 


[199. 


the  relation  between  the  coefficients  of  {z-c)~'^  in  them  being  a  special  case  of  a  more 
general  proposition  (§  210).  And  since  zl{{z-e){a^  —  z'^)'^]  when  s=ao  is  zero,  oo  does  not 
need  to  be  excluded. 

The  surface  taken  plane  is  doubly  connected,  as  in  the  last  example,  one  of  the  curves 
surrounding  c,  say  that  in  the  upper  sheet,  being  taken  as  the  boundary  of  the  surface. 
A  single  cross-cut  will  suffice  to  make  it  simply  connected :  the  direction  of  the  cross-cut 
must  pass  from  the  c-curve  in  the  lower  sheet  to  the  branch-line  and  thence  to  the 
boundary  in  the  upper  sheet. 

There  is  only  a  single  modulus  of  periodicity,  being  the  constant  for  the  single  cross-cut. 
This  moduhis  can  be  obtained  by  means  of  the  curve  AB  in  the  first  sheet;  and,  on 
contraction  of  the  curve  (by  II.  §  195)  so  as  to  be  infinitesimally  near  c,  it  is  easily  seen  to 
be  27^^■(a^-c2)~i,  or  say  'iTr(<fi-a^)~i.  But  the  modulus  can  be  obtained  also  by  means 
of  the  curve  CD;  and  when  the  curve  is  contracted,  as  in  the  previous  example,  so  as 
practically  to  be  a  loop  round  a  and  a  loop  round  —a,  the  value  of  the  integral  is 

,[«  — ± — ^, 

J  -a  (z  —  c)  {a^  —  z^)2 

which  is  easily  proved  to  be  2Tr  {c^  —  a^)" ^. 

As  in  Ex.  3,  a  curve  in  the  upper  sheet,  which  encloses  the  branch-points  and  the 
branch-lines,  can  be  deformed  into  the  curve  AB. 

Ex.  5.     Consider  w  =  \{Az'^—g2Z  —  g^~^dz=\udz. 

The  subject  of  integration  is  two-valued,  and  therefore  the  Riemann's  surface  is 
two-sheeted.     The  branch-points  are  3=oo,  ej,  gg,  ^3,  where  ei,  62,  e^  are  the  roots  of 

Az^~goj-gz  =  0; 

and  no  one  of  them  needs  to  be  excluded  from  the  range  of  variation  of  the  variable. 

The  connectivity  of  the  surface  is  3,  so  that  two  cross-cuts  are  necessary  to  resolve 
the  surface  into  one  that  is  simply  connected.     The  configurations  of  the  branch-lines  and 


Fig.  74. 


of  the  cross-cuts  admit  of  some  variety ;   two  illustrations  of  branch-lines  are  given  in 
fig.  74,  and  a  point  on  Qi  in  each  diagram  is  taken  as  boundary. 

The  modulus  for  the  cross-cut  ^j— say  from  the  inside  to  the  outside — can  be  obtained 
in  two  different  ways.  First,  from  P,  a  point  on  Qi,  draw  a  line  to  63  in  the  first  sheet, 
then  across  the  branch-hne,  then  in  the  second  sheet  to  63  and  across  the  branch-line, 


199.] 


OF   INTEGRALS 


483 


then  in  the  first  sheet  round  63  and  back  to  P :  the  circuit  is  represented  by  the  double 
line  between  62  aiid  ^s-     The  value  of  the  integral  is 


\    udz  +1     ( —  XI)  dz,  that  is,  2  I     xidz. 
J  e^  J  €3  J  62 


Again,  it  can  be  obtained  by  a  line  from  F',  another  point  on  §1,  to  qo  ,  round  the  branch- 
point there  and  across  the  branch-line,  then  in  the  second  sheet  to  e^  and  round  ej,  then 
across  the  branch-line  and  back  to  F' :  the  value  of  the 
integral  is 

£1  =  2  \    udz. 


But  the  modulus  is  the  same  for  F  as  for  F' :  hence 
1:1  =  2  I    udz=2  I  \idz. 


roc 

:  (    udz=i 
J  ei 


This  relation  can  be  expressed  in  a  different  form.  The 
path  from  e^  to  63  can  be  stretched  into  another  form 
towards  z  =  cc  in  the  first  sheet,  and  similarly  for  the 
path  in  the  second  sheet,  without  affecting  the  value  of 
the  integral.  Moreover  as  the  integral  is  zero  for  2=cc, 
we  can,  without  affecting  the  -value,  add  the  small  part 
necessary  to  complete  the  circuits  from  62  to  cc  and  from  e^  to  co 
these  circuits  being  given  by  the  arrows,  we  have 


Fig.  75. 
The  directions  of 


^f; 


7idz  =  2  I     udz  +  2  I     udz, 


E. 


i  ex 


Ev 


J  e 


udz  =  E2  —  E^, 


or,  if 

for  X  =  l,  2,  3,  we  have* 

J  e.2 

say  E.2  =  Ei-\-Ez; 

and  El  is  the  modulus  of  periodicity  for  the  cross-cut  Q-^. 

In  the  same  way,  the  modulus  of  periodicity  for  Q2  is  found  to  be 


E,  =  2 


udz  and  to  be  2  I     udz, 
63  J  62 


the  equivalence  of  which  can  be  established  as  before. 

Hence  it  appears  that,  if  w  be  the  value  of  the  integral  at  any  point  in  the  surface, 
the  general  value  is  of  the  form  w  +  mEi  +  nE^,  where  m  and  ?i  are  integers.  As  the 
integral  is  zero  at  infinity  (and  for  other  reasons  which  have  already  appeared),  it  is 
convenient  to  take  the  fixed  limit  Zq  so  as  to  define  w  by  the  relation 

w=  I     icdz. 


Now  corresponding  to  a  given  arithmetical  value  of  z,  there  are  two  points  in  the 
surface  and  two  values  of  w:  it  is  important  to  know  the  relation  to  one  another  of  these 
two  values.  Let  z'  denote  the  value  in  the  lower  sheet :  then  the  path  from  z'  to  oo  can 
be  made  up  of 

(i)  a  path  from  z'  to  00 ' ;  (ii)  any  number  of  irreducible  circuits  from  00 '  to  qo  ' ; 
and  (iii)  across  the  branch -line  and  round  its  extremity  to  co . 


See  Ex.  7,  §  104. 


F.  F. 


28 


434  EXAMPLES  [199. 

These  parts  respectively  contribute  to  the  integral 

raa>  roo 

(i)  a  quantity  I     {  —  u)dz,  that  is,  —  I    udz,  or,  -w;   (ii)  a  quantity  inEi  +  nE^, 

where  m  and  n  are  integers ;  (iii)  a  quantity  zero,  since  the  integi'al  vanishes 
at  infinity  :   so  that 

If  now  we  regard  s  as  a  function  of  w,  say  2  =  ^(w),  we  have 

^(w)  =  5  =  ^(m£'i  +  »jE'3  +  w),       (^{w')=z'. 

But  /  =  2  arithmetically,  so  that  we  have 

z=(^{w)  =  f  {mEy^  +  nj&3  ±  w) 

as  the  function  expressing  z  in  terms  of  w. 

Similarly  it  can  be  proved  that 

f'{w)=±^o'{mEy  +  nEs±w), 

the  upper  and  the  lower  signs  being  taken  together.  Now  ^  {iv),  by  itself,  determines  a 
value  of  z,  that  is,  it  determines  two  points  on  the  sui-face :  and  ^'  (vj)  has  different  values 
for  these  two  points.     Hence  a  point  on  the  surface  is  uniquely  determined  hy  ^  i^w)  and 

Ex.  6.     Consider  %o=  /    {{\-z^){\-Bz'')]~\dz=  judz.     The  subject  of  integration  is 

two-valued,  so  that  the  surface  is  two-sheeted.  The  branch-points  are  ±1,  ±y,  but 
not  00  ;   no  one  of  the  branch-points  need  be  excluded,  nor  need  infinity. 

The  connectivity  is  3,  so  that  two  cross-cuts  will  render  the  surface  simply  connected : 
let  the  branch-lines  and  the  cross-cuts  be  taken  as  in  the  figure  (fig.  76). 

The  details  of  the  argument  follow  the  same  course  as  in  the  previous  case. 

The   modulus   of    periodicity   for    Q^   is   2 1      udz  =  4  /    zidz  =  4A'',    in    the    ordinary 

notation. 

1 

udz  =  2iK',  as  before. 

Hence,  if  m;  be  a  value  of  the  integral  for  a  point  z  in  the  first  sheet,  a  more  general 

value  for  that  point  is 

w-{:m4:K+n2iK'. 

Let  w'  be  a  value  of  the  integral  for  a  point  z'  in  the  second  sheet,   where  z'  is 
arithmetically  equal  to  z — the  point  in  the 
first  sheet  at  which  the  value  of  the  integral  Qi 

is  w ;  then  /^^^  y^-^ 

_I (C-  a    — « 

w'  =  2K+m4:K+n2iA' —  w,  k  \\  -1  v:^^^^'/ 

.  so  that,  if  we  invert  the  functional  relation 
and  take  z  =  sn  w,  we  have  ^^" 

su  w  =  2  =  sn  {w  +  4:mK +2niK') 

=  sn  {(4ot  +  2)  K+2niK'  -  w}. 


199.] 


OF   MODULI   OF   PERIODICITY 


435 


Esc.  7.     Consider  the  integral  w=  I  -. r-  ,  where  u={{l-z^)  (1  -kh^)}h. 


1 


As  in  the  last  case,  the  surface  is  two-sheeted :  the  branch-points  are  ±  1,  +j  ;  no  one 

of  them  need  be  excluded,  nor  need  s=qo  .  But  the  point  z  =  c  must  be  excluded  in  both 
sheets ;  for  expanding  the  subject  of  integration  for  points  in  the  first  sheet  in  the  vicinity 
of  s  =  c,  we  have 

^{(l-c2)(l-FO}-U..., 
and  for  points  in  the  second  sheet  in  the  vicinity  of  2  =  c,  we  have 

in  each  case  giving  rise  to  a  logarithmic  infinity  for  z  =  c. 

We  take  the  small  curves  excluding  z=c  in  both  sheets  as  the  boundaries  of  the 
surface.     Then,  by  Ex.  4,  §  178,  (or  because  one  of  these  curves  may  be  regarded  as  a 


Fig.  77. 


boundary  of  the  surface  in  the  last  example,  and  the  curve  excluding  the  infinity  in  the 
other  sheet  is  the  equivalent  of  a  loop-cut  which  (§  161)  increases  the  connectivity  by 
unity),  the  connectivity  is  4.  The  cross-cuts  necessary  to  make  the  surface  simply 
connected  are  three.     They  may  be  taken  as  in  the  figure ;  Qi  is  drawn  from  the  boundary 

in  one  sheet  to  a  branch-line  and  thence  round  t  to  the  boundary  in  the  other  sheet : 

^2  beginning  and  ending  at  a  point  in  Qi ,  and  Q3  beginning  and  ending  at  a  point  in  Q2  ■ 

The  moduli  of  periodicity  are : — 

for  Qi,  the  quantity  (Ql  =  )27^^{(l  -c^){l-  k"c^)}~i,  obtained  by  taking  a  small  curve 
round  c  in  the  upper  sheet : 


Q2,  the  quantity  (122  =  )  2 


dz 


_i  {z  —  c)u 


,    obtained   bj  taking   a  circuit   round   1 


and  T,  passing  from  one  edge  of  Q^  to  the  other  at  F: 


f    k        ciz 

§3,  the  quantity  (03=)  2  I        — ,  obtained  by  taking  a  circuit  round  -1 

and  —  -7 ,  passing  from  one  edge  of  Qz  to  the  other  at  G : 
so  that,  if  any  value  of  the  integral  at  a  point  be  ^<;,  the  general  value  at  the  point  is 

where  mj,  m2,  m^  are  integers. 

Conversely,  z  is  a  triply-periodic  function  of  w ;  but  the  function  of  w  is  not  uniform 
.(§  108). 

28—2 


436  MODULI   OF   PERIODICITY  [199. 

Ex.  8.     As  a  last  illustration  for  the  present,  con^der 

The  surface  is  two-sheeted ;  its  connectivity  is  3,  the  branch-points  being  ±1,  ±t  ,  but  not 


of  them.     To  consider  the  integral  at  infinity,  we  substitute  s  =  ;j,  and  then 


s=oo  .     No  one  of  the  branch-points  need  be  excluded,  for  the  integral  is  finite  round  each 

it  infinity 


^2 


{  dz'  (.     k'^  ,„ 

k     k'^  , 

giving  for  the  function  at  infinity  an  accidental  singularity  of  the  first  order  in  each 
sheet. 

The  point  z  =  cc  must  therefore  be  excluded  from  each  sheet:  but  the  form  of  w,  for 
infinitely  large  values  of  z,  shews  that  the  modulus  for  the  cross-cut,  which  passes  from 
one  of  the  points  (regarded  as  a  boundary)  to  the  other,  is  zero. 

The  figure  in  Ex.  6  can  be  used  to  determine  the  remaining  moduli.  The  modulus, 
for  §2  is 

^1  —  k'^x^\^ 


\—x^ 


dx 


=  4 dx 

with  the  notation  of  Jacobian  elliptic  functions.     The  modulus  for  §i  is 


dx 


jo{(l-/)(l-/&V)}* 
on  transforming  by  the  relation  ^^^^  ^  ^'2^2  _  j  .  the  last  expression  can  at  once  be  changed 
into  the  form  2^■  (K'  —  E'\  with  the  same  notation  as  before. 

If  then  w  be  any  value  of  the  integral  at  a  point  on  the  surface,  the  general  value 

there  is 

w  +  AmE+2ni{K'-E'), 

where  m  and  n  are  integers. 

200.  After  these  illustrations  in  connection  with  simple  cases,  we  may 
proceed  with  the  consideration  of  the  integral  of  the  most  general  uniform 
rational  function  lu  of  position  on  a  Riemann's  surface,  constructed  in  connec- 
tion with  the  algebraical  equation 

f{w,  z)  =  w''  +  w^'-^g^ {z)+...+  wgn-i (^)  +  9n (^)  =  0, 


200.]  INTEGRAL  OF   ALGEBRAIC   FUNCTION  437 

where  the  fanctions  g  {z)  are  rational  and  integral.  Subsidiary  explanations, 
which  are  merely  generalised  from  those  inserted  in  the  preceding  particular 
discussions,  will  now  be  taken  for  granted. 

Taking  w'  in  the  form  of  §  193,  we  have 

w=-K{z)  + ^ =  ~Kiz)  +  —^, 

dw  dw 

so  that  in  taking  the  integi'al  of  w  we  shall  have  a  term  -  |  Iiq  (z)  dz,  where 

ho  (z)  is  a  rational  function.  This  kind  of  integral  has  been  discussed  in 
Chapter  II. ;  as  it  has  no  essential  importance  for  the  present  investigation, 
it  will  be  omitted,  so  that,  without  loss  of  generality  merely  for  the  present 
purpose  *,  we  may  assume  h^  (z)  to  vanish ;  and  then  the  numerator  of  w' 
is  of  degree  not  higher  than  n  —  2  in  tv. 

The  value  of  z  is  insufficient  to  specify  a  point  on  the  surface :  the  values 
of  w  and  z  must  be  given  for  this  purpose,  a  requisite  that  was  unnecessary 
in  the  preceding  examples  because  the  point  z  was  spoken  of  as  being  in  the 
upper  or  the  lower  of  the  two  sheets  of  the  various  surfaces.  Corresponding 
to  a  value  a  of  z,  there  will  be  n  points :  they  may  be  taken  in  the  form 
(tti,  fli),  (tta,  a.^,  ...,  {an,  oLn),  whero  aj,  ...,  a„  are  each  arithmetically  equal  to 
a,  and  Wj ,  . . . ,  a„  are  the  appropriately  arranged  roots  of  the  equation 

f{w,  a)  =  0. 

U  (w  z) 
The  fimction  w    to  be  integi^ated  is  of  the  form       ^'    ' ,  where  TJ  is 

polynomial  of  degree  7i  —  2  in  w,  but  though  rational  in  z  it  is  not  necessarily 
integral  in  z. 

An  ordinary  point  of  w' ,  which  is  neither  an  infinity  nor  a  branch-point, 
is  evidently  an  ordinary  point  of  the  integral. 

The  infinities  of  the  subject  of  integration  are  of  prime  importance. 
They  are : — 

(i)      the  infinities  of  the  numerator, 

(ii)     the  zeros  of  the  denominator. 

The  former  are  constituted  by  (a),  the  poles  of  the  coefficients  of  powers  of  w 
in  Uiiv,  z),  and  (/S),  2^  =  00:  this  value  is  included,  because  the  only  infinities 
of  w,  as  determined  by  the  fundamental  equation,  arise  for  infinite  values  of 
z,  and  infinite  values  of  w  and  of  z  may  make  the  numerator  U  (w,  z) 
infinite. 

*  See  §  207,  where  ho  {z)  is  retained. 


438  INTEGRALS  [200. 

So   far   as   concerns  the  infinities  of  w    which  arise  when  2;  =  oo  (and 

therefore  w  =  00  ),  it  is  not  proposed  to  investigate  the  general  conditions 

that  the  integral  should  vanish  there.     The  test  is  of  course  that  the  limit, 

zTjiuu  z\ 
for  ^  =  00  ,  of )rr — -  should  vanish  for  each  of  the  n  values  of  w. 

dw  ' 

But  the  establishment  of  the  general  conditions  is  hardly  worth  the 
labour  involved;  it  can  easily  be  made  in  special  cases,  and  it  will  be 
rendered  unnecessary  for  the  general  case  by  subsequent  investigations. 

201.  The  simplest  of  the  instances,  less  special  than  the  examples 
already  discussed,  are  two. 

The  first,  which  is  really  that  of  most  frequent  occurrence  and  is  of  very 
great  functional  importance,  is  that  in  which  f{w,  z)  =  0  has  the  form 

w''-S{z)  =  {), 

where  S  {z)  is  of  order  2m  —  1  or  2m  and  all  its  roots  are  simple :    then 

-J-  =  2w  =  2^/8  (z).     In  order  that  the  limit  of  rrr —  may  be  zero  when 

dw 
z  =  ao ,  we  see  (bearing  in  mind  that  U,  in  the  present  case,  is  independent  of 
w)  that  the  excess  of  the  degree  of  the  numerator  of  U  over  its  denominator 
may  not  be  greater  than  m  —  2.     In  particular,  if  U  be  an  integral  function 
of  z,  a  form  of  U  which  would  leave  Jw'dz  zero  at  ^^  =  00  is 

U  ^  CqZ       '  +  CiZ  +  . . .  +  CyfisZ  +  Cjjj_2. 

As  regards  the  other  infinities  of  Ul\/S(z),  they  are  merely  the  roots  of 
8  {z)  =  0   or   they  are   the   branch-points,  each  of  the    first  order,  of  the 

equation 

10^ -8  (z)  =  0. 

By  the  results  of  §  101,  the  integral  vanishes  round  each  of  these  points ;  and 
each  of  the  points  is  a  branch-point  of  the  integral  function.  The  integral  is 
finite  everywhere  on  the  surface :  and  the  total  number  of  such  iyitegrals, 
essentially  different  from  one  another,  is  the  number  of  arbitrary  coefficients 
in  U,  that  is,  it  is  m—1,  the  same  as  the  genus  of  the  Riemanns  surface 
associated  with  the  equation. 

202.  The  other  important  instance  is  that  in  which  the  fundamental 
equation  is,  so  to  speak,  a  generalised  equation  of  a  plane  curve,  so  that 
gg  {z)  is  a  polynomial  function  of  z  of  degree  s :   then  it  is  easy  to  see  that, 

at  5;  =00,  each  branch  wcxzz,  so  that  ■^<xz''^-^:  hence  U(w,  z)  can  vary  only 

as  z'^~^,  in  order  that  the  condition  may  be  satisfied.  If  then  U{w,  z)  be  an 
integral  function  of  z,  it  is  evident  that  it  can  at  most  take  a  form  which 


202.]  OF  ALGEBRAIC  FUNCTIONS  439 

makes  U=0  the  generalised  equation  of  a  curve  of  degree  n  —  S;  while,  if  it  be 

V(w  z) 

—     '      ,  then  V  {w,  z),  supposed  integral  in  z,  can  at  most  take  a  form  which 

makes  V=0  the  generalised  equation  of  a  curve  of  degree  w  —  2. 

Other  forms  are  easily  obtainable  for  accidental  singularities  of  coefficients 
of  tu  in  U  (w,  z)  that  are  of  other  orders. 

As  regards  the  other  possible  infinities  of  the  integral,  let  c  be  an  acci- 
dental singularity  of  a  coefficient  of  some  power  of  lu  in  U{w,  z) ;  it  may  be 

assumed  not  to  be  a  zero  of  ^ .     Denote  the  n  points  on  the  surface  by 

(Ci,  h)>  (c2,  h),  ■•,  (Cn,  kn),  where  c^,  Co,  ...,  Cn  are  arithmetically  equal  to  c. 
In  the  vicinity  of  each  of  these  points  let  w'  be  expanded :  then,  near  (c^,  kr), 
we  have  a  set  of  terms  of  the  type 

+  ^.        .  \m-l  +■■■+  /T— -^  +  -—     +P(Z-  Cr), 


where  P{z  —  Cr)  is  a  converging  series  of  positive  integral  powers  of  z  —  Cr. 
A  corresponding  expansion  exists  for  every  one  of  the  n  points. 

The  integral  of  lu'  will  therefore  have  a  logarithmic  infinity  at  (c^,  kt), 
unless  Ai^r  is  zero;  and  it  will  have  an  algebraic  infinity,  unless  all  the 
coefficients  A^^^, ,  A^^r  are  zero. 

The  simplest  cases  are 

(i)     that   in    which    the    integral    has    a    logarithmic  infinity  but  no 
algebraic  infinity ;   and 

(ii)    that  in  which  the  integral  has  no  logarithmic  infinity, 

W  (in    f\ 

For  the  former,  w'  is  of  the  form '-rr^  ,  and  therefore  in  the  vicinity  of  c^ 

{z-c)?- 

we  have  tv'  —  — ^^^  +  P  {z  —  Cr), 

the  value  of  A^^r  being  ^Z   ^  ,  and  W  is  an  integral  function  of  kr,  of 

dkr 
degree  not  higher  than  n  —  2.     Hence 

^  »    W  (kr,  Cr) 

^     -^l,r=    ^     ^7 

r=\  r=l  ^ 

"bkf 
_    -      W{kr,6) 

dkr 


440  INFINITIES   OF   THE   INTEGRAL  [202. 

since  c  is  the  commQn  arithmetical  value  of  the  quantities  Ci,  Ca,  ...,  c„.  Now 
^1,  k^,  ...,  kn  are  the  roots  of 

/(^,c)  =  0, 

an  equation  of  degree  n,  while  W  is  of  degree  not  higher  than  n  —  2 ;  hence, 
by  a  known  theorem*, 

I      W{kr,c)        Q^ 

dkr 

n 

SO  that  S   ^1,,-  =  0- 

The  validity  of  the  result  is  not  affected  if  some  of  the  coefficients  A  vanish. 
But  it  is  evident  that  a  single  coefficient  A  cannot  be  the  only  non-vanishing 
coefficient ;  and  that,  if  all  but  two  vanish,  those  two  are  equal  and  opposite. 

This  result  applies  to  all  those  accidental  singularities  of  coefficients  of 
powers  of  iv  in  the  numerator  of  w'  which,  being  of  the  first  order,  give  rise 
solely  to  logarithmic  infinities  in  the  integral  of  w.  It  is  of  great  importance 
in  regard  to  moduli  of  periodicity  of  the  integral. 

(ii)  The  other  simple  case  is  that  in  which  each  of  the  coefficients 
Ai^r  vanishes,  so  that  the  integral  of  w'  has  only  an  algebraic  infinity  at 
the  point  Cr,  which  is  then  an  accidental  singularity  of  order  less  by  unity 
than  its  order  for  w'. 

In  particular,  if  in  the  vicinity  of  Cr,  the  form  of  w  be 

the  integral  has  an  accidental  singularity  of  the  first  order. 
It  is  easy  to  prove  that 

n 

so  that  a  single  coefficient  A  cannot  be  the  only  non-vanishing  coefficient ; 
but  the  result  is  of  less  importance  than  in  the  preceding  case,  for  all  the 
moduli  of  periodicity  of  the  integral  at  the  cross-cuts  for  these  points  vanish. 
And  it  must  be  remembered  that,  in  order  to  obtain  the  subject  of  integration 
in  this  form,  some  terras  have  been  removed  in  §  200,  the  integral  of  which 
would  give  rise  to  infinities  for  either  finite  or  infinite  values  of  z. 

It  may  happen  that  all  the  coefficients  of  powers  of  tv  in  the  numerator 
of  w'  are  integral  functions  of  z.  Then  ^  =  x  is  their  only  accidental 
singularity;   this  value  has  already  been  taken  into  account. 

*  Burnside  and  Panton,  Theory  of  Equations,   (7th  ed  )  vol.  i,  p.  172. 


203.]  OF   AN    ALGEBRAIC   FUNCTION  441 

203.  The  remaining  source  of  infinities  of  w\  as  giving  rise  to  possible 
infinities  of  the  integral,  is  constituted  by  the  aggregate  of  the  zeros  of 

^  =  0.     Such  points  are  the  simultaneous  roots  of  the  equations 

In  addition  to  the  assumption  already  made  that  /=  0  is  the  equation  of  a 
generalised  curve  of  the  ?ith  order,  we  shall  make  the  further  assumptions 
that  all  the  singular  points  on  it  are  simple,  that  is,  such  that  there  are  only 
two  tangents  at  the  point,  either  distinct  or  coincident,  and  that  all  the 
branch-points  are  simple. 

The  results  of  §  98  may  now  be  used.  The  total  number  of  the  points 
given  as  simultaneous  roots  is  n  (n  —  1) :  the  form  of  the  integral  in  the 
immediate  vicinity  of  each  of  the  points  must  be  investigated. 

Let  (c,  7)  be  one  of  these  points  on  the  Riemann's  surface,  and  let 
(c  +  ^,  7  +  f )  be  any  point  in  its  immediate  vicinity. 

I.  If  -^  ^  do  not  vanish  at  the  point,  then  (c,  7)  is  a  branch-point 
for  the  function  w.     We  then  have 

f(tu,  z)  =  A'^+  B'v^  +  quantities  of  higher  dimensions, 

for  points  in  the  vicinity  of  (c,  7),  so  that  u  cc  ^^  when  |  ^|  is  sufficiently  small. 
Then 

^—  =  25'l/  +  quantities  of  higher  dimensions 

when  j^j  is  sufficiently  small.  Hence,  for  such  values,  the  subject  of  integra- 
tion is  a  constant  multiple  of 

U  (7,  c)  -f-  positive  integral  powers  of  v  and  ^ 
^^  +  powers  of  ^  with  index  >  -|- 
that  is,  of  ^~^,  when  |  ^1  is  sufficiently  small.     The  integral  is  therefore  a 
constant  multiple  of  ^'^,  when  |  ^j  is  sufficiently  small;    and  its   value  is 
therefore   zero    round  the  point,   which   is  a  branch-point  for  the  function 
represented  by  the  integral. 

II.  If  ^ — ^  '  vanish  at  the  point,  we  have  (with  the  assumptions 
of  §  98), 

/ {w,  z)  =  A^^  +  2B^u  +  Gv"'  -h  terms  of  the  third  and  higher  degrees  ; 
and  there  are  two  cases. 

(i)     If  B^^  AC,  the  point  is  not  a  branch-point,  and  we  have 
Cv^B^=^ (B'  -AC)i+  integral  powers  ^^  ^^,.... 


442  INFINITIES   OF   ALGEBEAIC   FUNCTION  [203. 

as  the  relation  between  v  and  ^  deduced  froni/=  0.     Then 

^  =  2  (B^  +  Cv)  +  terms  of  second  and  higher  degrees 

=  X^+  higher  powers  of  ^. 

In  the  vicinity  of  (c,  7),  the  subject  of  integration  is 

U  (y,  c)  +  Dv  +  E^  +  positive  integral  powers 
A,^  +  higher  powers  of  ^ 

Hence  w^hen  it  is  integrated,  the  first  term  is -^-^  log  ^,  and  the  remain- 

A, 

ing  terms  are   positive  integral   powers   of  ^:    that    is,   such  a  point   is   a 
logarithmic  infinity  for  the  integral,  unless  U  (7,  c)  vanish. 

If,  then,  we  seek  integrals  which  have  not  the  point  for  a  logarithmic 
infinity  and  we  begin  with  U  as  the  most  general  function  possible,  we  can 
prevent  the  point  from  being  a  logarithmic  infinity  by  choosing  among  the 
arbitrary  constants  in  fT^  a  relation  such  that 

U{y,c)  =  Q. 

There  are  h  such  points  (§  98) ;   and  therefore  S  relations  among  the  , 
constants  in  the  coefficients  of  U  must  be  chosen,  in  order  to  prevent  the 
integral 


dz 


dw 

from  having  a  logarithmic  infinity  at  these  points.     When  these  are  chosen, 
the  points  become  ordinary  points  of  the  integral. 

(ii)     If  B^  =  AC,  the  point  is  a  branch-point ;  we  have 

B^+Cv  =  ^L^Km^'  +  N^^+... 

as  the  relation  between  ^  and  v  deduced  fromy=  0.     In  that  case, 

rif 

^  =  2  (B^+  Cv)  +  terms  of  the  second  and  higher  degrees 

s. 
=  Z^2  _|_  powers  of  ^  having  indices  >  |. 

In  the  vicinity  of  (c,  7),  the  subject  of  integration  is 

U  (7,  c)  +  Du  +  E^+  higher  powers 

L^  +  higher  powers  of  ^ 

Hence  when  it  is  integrated,  the  first  term  is  —  2  — \p-2  ^-f   and  it  can  be 

proved  that  there  is  no  logarithmic  term  ;   the  point  is  an  infinity  for  the 
integral,  unless  U(y,  c)  vanish. 


203.]  TO   BE   INTEGRATED  443 

If,  however,  among  the  arbitrary  constants  in  U  we  choose  a  relation  such 
that 

then  the  numerator  of  the  subject  of  integration 

=  Dv  +  E^-{-  higher  positive  powers 

=  A,'^+  yu,'^  +  higher  powers  of  ^, 

on  substituting  from  the  relation  between  v  and  ^  derived  from  the  funda- 
mental equation.     The  subject  of  integration  then  is 


that  is, 

the  integral  of  which  is 


2  -^  ^-  +  positive  powers. 


The  integral  therefore  vanishes  at  the  point :  and  the  point  is  a  branch-point 
for  the  integral.  It  therefore  follows  that  we  can  prevent  the  point  from 
being  an  infinity  for  the  function  by  choosing  among  the  arbitrary  constants 
in  t/"  a  relation  such  that 

r7(7,c)  =  o. 

There  are  k  such  points  (§  98) :  and  therefore  k  relations  among  the 
constants  in  the  coefficients  of  U  are  chosen  in  order  to  prevent  the  integral 
from  becoming  infinite  at  these  points.  Each  of  the  points  is  a  branch-point 
of  the  integral. 

204.  All  the  possible  sources  of  infinite  values  of  the  subject  of  integra- 
tion iv,  =  — ~— i ,  have  now  been  considered.     A  summary  of  the  preceding 

dw 
results  leads  to  the  following  conclusions  relative  to  fw'dz: — 

(i)     an  ordinary  point  of  w'  is  an  ordinary  point  of  the  integral : 

(ii)  for  infinite  values  of  z,  the  integral  vanishes  if  we  assign  proper 
limitations  to  the  form  of  U{w,  z) : 

(iii)  accidental  singularities  of  the  coefficients  of  powers  of  w  in 
U{w,z)  are  infinities,  either  algebraic  or  logarithmic  or  both 
algebraic  and  logarithmic,  of  the  integral : 

(iv)  if  the  coefficients  of  powers  of  w  in  U(w,z)  have  no  accidental 
singularities  except  for  z^cc ,  then  the  integral  is  finite  for 
infinite  values  of  z  (and  of  w)  when  U('w,z)  is  the  most  general 


444  INTEGEALS  [204. 

rational  integral  function  of  w  and  z  of  degree  n  —  3 ;  but,  if 
the  coefficients  of  powers  of  w  in  U{w,z)  have  an  accidental 
singularity  of  order  //,,  then  the  integral  will  be  finite  for 
infinite  values  of  z  (and  of  w)  when  U(w,z)  is  the  most 
general  rational  integral  function  of  w  and  z,  the  degree  in  w 
being  not  greater  than  n  —  2  and  the  dimensions  in  w  and  z 
combined  being  not  greater  than  n  +  /j,  —  S: 

(v)  those  points,  at  which  df/dw  vanishes  and  which  are  not  branch- 
points of  the  function,  can  be  made  ordinary  points  of  the 
integral,  if  we  assign  proper  relations  among  the  constants 
occurring  in  U{w,z) : 

(vi)  those  points,  at  which  dfjdw  vanishes  and  which  are  branch- 
points of  the  function,  can,  if  necessary,  be  made  to  furnish 
zero  values  of  the  integral  by  assigning  limitations  to  the 
form  of  U{w,z)\  each  such  point  is  a  branch-point  of  the 
integral  in  any  case. 

These  conclusions  enable  us  to  select  the  simplest  and  most  important 
classes  of  integrals  of  uniform  functions  of  position  on  a  Riemann's  surface. 

205.  The  first  class  consists  of  those  integrals  which  do  not  acquire* 
an  infinite  value  at  any  point ;  they  are  called  integrals  of  the^?"s^  kind^. 

The  integrals,  considered  in  the  preceding  investigations,  can  give  rise  to 
integrals  of  the  first  kind,  if  the  numerator  U{w,z)  of  the  subject  of  integra- 
tion satisfy  various  conditions.  The  function  U(w,z)  must  be  a  polynomial 
function  of  dimensions  not  higher  than  n  —  S  in  w  and  z,  in  order  that  the 
integral  may  be  finite  for  infinite  values  of  z  and  for  all  finite  values  of  z 
not  specially  connected  with  the  equation  f(w,z)  =  0;  for  certain  points 
specially  connected  with  the  fundamental  equation,  being  8  -f  /c  in  number, 
the  value  of  U{w,z)  must  vanish,  so  that  there  must  be  S-f /c  relations 
among  its  coefficients.  But  when  these  conditions  are  satisfied,  then  the 
integral  function  is  everywhere  finite,  it  being  remembered  that  certain 
limitations  on  the  nature  off{w,  z)  =  0  have  been  made. 

Usually  these  conditions  do  not  determine  U  (w,  z)  uniquely  save  as  to  a 
constant  factor ;  and  therefore  in  the  most  general  integral  of  the  first  kind  a 
number  of  independent  arbitrary  constants  will  occur,  left  undetermined  by 
the  conditions  to  which  U  ib  subjected.  Each  of  these  constants  multiplies 
an  integral  which,  everywhere  finite,  is  different  from  the  other  integrals  so 
multiplied;  and  therefore  the  number  of  different  integi-als  of  the  first  kind 

*  They  will  be  seen  to  be  multiform  functions  even  on  the  multiply  connected  Kiemann's 
surface,  and  they  do  not  therefore  give  rise  to  any  violation  of  the  theorem  of  §  40. 

t  The  German  title  is  erster  Gattung ;  and  similarly  for  the  integrals  of  the  second  kind 
and  the  third  kind. 


205.]  OF   THE   FIRST   KIND  445 

is  equal  to  the  number  of  arbitrary  independent  constants,  left  undetermined 
in  U.  It  is  evident  that  any  linear  combination  of  these  integrals,  with 
constant  coefficients,  is  also  an  integral  of  the  first  kind ;  and  therefore  a 
certain  amount  of  modification  of  form  among  the  integrals,  after  they  have 
been  obtained,  is  possible. 

The  number  of  these  integrals,  linearly  independent  of  one  another,  is 
easily  found.  Because  ?7  is  a  polynomial  function  of  w  and  z  of  dimen- 
sions n  —  3,  it  contains,  ^  (w  —  1)  {n  —  2)  terms  in  its  most  general  form  ;  but 
its  coefficients  satisfy  S  +  /c  relations,  and  these  are  all  the  relations  that 
they  need  satisfy.  Hence  the  number  of  undetermined  and  independent 
constants  which  it.  contains  is 

i(w-l)(/?.-2)-S-/c, 

which,  by  §  182,  is  the  genus  p  of  the  Riemann's  surface ;  and  therefore,  for 
the  present  case,  the  number  of  integrals,  tuhich  are  finite  everywhere  on  the 
surface  and  are  linearly  independent  of  one  another,  is  equal  to  the  genus  of 
the  Riemann's  surface. 

Moreover,  the  integral  of  the  first  kind  has  the  same  branch-points  as  the 
function  w.  Though  the  integral  is  finite  everywhere  on  the  surface,  yet  its 
derivative  w'  is  not  so  :  the  infinities  of  tu'  are  the  branch-points. 

The  result  has  been  obtained  on  the  original  suppositions  of  §  98,  which 
were,  that  all  the  singular  points  of  the  generalised  curve  f{w,z)  —  0  are 
simple,  that  is,  only  two  tangents  (distinct  or  coincident)  to  the  curve  can 
be  drawn  at  each  such  point,  and  that  all  the  branch-points  are  simple. 
Other  special  cases  could  be  similarly  investigated.  But  it  is  superfluous  to 
carry  out  the  investigation  for  a  succession  of  cases,  because  the  result  just. 
obtained,  and  the  result  of  |  201,  are  merely  particular  instances  of  a  general 
theorem  which  will  be  proved  in  Chapter  XVIII.,  viz.,  that,  associated  with 
a  Riemann's  surface  of  connectivity  2p-+l,  there  are  p  linearly  independent 
integi^als  of  the  first  kind  luhich  are  finite  everywhere  on  the  surface. 

The  function  U{w,z),  which  occurs  in  the  subject  of  integration  in  an 
integral  of  the  first  kind,  is  often  called  an  adjoint  polynomial  of  order  9i  —  3 ;. 
and  the  generalised  curve 

•    •  U(w,z)  =  0 

is  called  an  adjoint  curve  of  order  ?i  —  3. 

206.  The  functions,  which  thus  arise  out  of  the  integral  of  an  algebraic- 
function  and  are  finite  everywhere,  are  not  uniform  functions  of  position  on 
the  unresolved  surface.  If  the  surface  be  resolved  by  2p  cross-cuts  into  one 
that  is  simply  connected,  then  the  function  is  finite,  continuous  and  uniform 
everywhere  in  that  resolved  surface,  which  is  limited  by  the  cross-cuts  as  a 
single  boundary.     But  at  any  point  on  a  cross-cut,  the  integral,  at  the  two 


446  INTEGRALS  [206 

points  on  opposite  edges,  has  values  that  differ  by  any  integer  multiple  of 
the  modulus  of  the  function  for  that  cross-cut  (and  possibly  also  by  integer 
multiples  of  the  moduli  of  the  function  for  the  other  cross-cuts). 

Let  the  cross-cuts  be  taken  as  in  §  181 ;  and  for  an  integral  of  the  first 
kind,  say  W,  let  the  moduli  of  periodicity  for  the  cross-cuts  be 

coi,  CO2 ,  ••■,  <Wm)  lor  cii,  CL2,  ••'}  cip, 

and  <»p+i,  «^+2,  ■••,  (^2p,  for  b^,  b^,  ...,  bp, 

respectively;  the  moduli  for  the  portions  of  cross-cuts  Co,  c^,  ...,  Cp  have  been 
proved  to  be  zero. 

Some  of  these  moduli  may  vanish ;  but  it  will  be  proved  later  (§231)  that 
all  the  moduli  for  the  cross-cuts  a,  or  all  the  moduli  for  the  cross-cuts  b,  cannot 
vanish  unless  the  integral  is  a  mere  constant.  In  the  general  case,  with  which 
we  are  concerned,  we  may  assume  that  they  do  not  vanish ;  and  so  it  follows 
that,  if  W  be  a  value  of  an  integral  of  the  first  kind  at  any  point  on  the 
Riemann's  surface,  all  its  values  at  that  point  are  of  the  form 

•ip 

W  +     ^  nirCOr, 
r  =  l 

where  the  coefficients  m  are  integers. 

The  foregoing  functions,  arising  through  integrals  that  are  finite  every- 
where on  the  surface,  will  be  found  the  most  important  from  the  point  of 
view  of  Abelian  transcendents :  but  other  classes  arise,  having  infinities  on 
the  surface,  and  it  is  important  to  indicate  their  general  nature  before  passing 
to  the  proof  of  the  Existence-Theorem. 

207.  First,  consider  an  integral  which  has  algebraic,  but  not  logarithmic, 
infinities.  Taking  the  subject  of  integration,  as  in  the  preceding  case,  to  be 
the  most  general  possible  so  that  arbitrary  coefficients  enter,  we  can,  by 
assigning  suitable   relations  among  these    coefiicients,   prevent    any  of  the 

rlf 

points,  given  as  zeros  of  ^  =  0,  from  being  infinities  of  the  integral.     It 

follows  that  then  the  only  infinities  of  the  integral  will  be  the  points  that  are 
accidental  singularities  of  coefficients  of  powers  of  w  in  the  numerator  of  the 
general  expression  for  lu'.  These  singularities  must  each  be  of  the  second 
order  at  least :  and,  in  the  expansion  of  w'  in  the  vicinity  of  each  of  them, 
there  must  be  no  term  of  index  —  1,  for  it  is  the  index  that  leads,  on  integration, 
to  a  logarithm. 

Such  integrals  are  called  integrals  of  the  second  kind. 

The  simplest  integral  of  the  second  kind  has  an  infinity  for  only  a  single 
point  on  the  surface,  and  the  infinity  is  of  the  first  order  only :  the  integral 
is  then  called  an  elementary  integral  of  the  second  kind.     After  what  has 


207.]  OF   THE   SECOND    KIND  447 

been  proved  in  §  202  (ii),  it  is  evident  that  an  elementary  integral  of  the 
second  kind  cannot  occur  in  connection  with  the  equation  f{tu,  z)  =  0,  unless 
the  term  ho  (z)  of  §  200  be  retained  in  the  expression  for  tv'. 

Ex.  1.     Adopting  the  subject  of  integration  obtained  in  §  200,  we  have 

,     1  ,    .  .      U  (w,  z) 

diu 

where  U  is  of  the  character  considered  in  the  preceding  sections,  viz.,  it  is  of  degree  n  — 2 
in  IV  ;  various  forms  of  iv'  lead  to  various  forms  of  h^  {z)  and  of  U  {w,  z). 

If  —  ho  (z)  = -^ ,  and  if  c  be  not  a  singularity  of  the  coefficient  of  any  power  of  w 

in  U,  it  is  then  evident  that 

{w'dz  = h  I  —rrr-^dz; 

■'  z-c       \       of 

J       '^w 

and  the  integral  on  the  right-hand  side  can  by  choice  among  the  constants  be  made  an 
integral  of  the  iirst  kind.  The  integral  is  not,  however,  an  elementary  integral  of  the 
second  kind,  because  z=c\s  an  infinity  in  each  sheet. 

Ex.  2.  A  special  integral  of  the  second  kind  occurs,  when  we  take  an  accidental 
singularity,  say  z  =  c,  of  the  coefficient  of  some  power  of  w  in  U  (tv,  z)  and  we  neglect  Jiq  (z)  ; 
so  that,  in  effect,  the  subject  of  integration  v/  is  limited  to  the  form 

U{iv,z) 
df_     ' 
dtv 

U  being  of  degree  not  higher  than  n  —  2\n  iv.  To  the  value  z  =  c,  there  correspond  n  points 
in  the  various  sheets  ;  if,  in  the  immediate  vicinity  of  any  one  of  the  points,  to'  be  of  the 
form 

in  that  vicinity  the  integral  is  of  the  form 

z  —  e,. 

For  such  an  integral  the  sum  of  the  coefficients  A^.  is  zero  :  the  simplest  case  arises 
when  all  but  two,  say  Ai  and  A^,  oi  these  vanish.     The  integral  is  then  of  the  form 

in  the  vicinity  of  c^ ,  and  of  the  form 

^—  +  ^2(^-^2) 
Z  —  C2 

in  the  vicinity  of  c^.     But  the  integral  is  not  an  elementary  integral  of  the  second  kind. 

208.  To  find  the  general  value  of  an  integral  of  the  second  kind, 
all  the  arithmetically  infinite  points  would  be  excluded  from  the  Riemann's 
surface  by  small  curves :  and  the  surface  would  be  resolved  into  one  that  is 
simply  connected.  The  cross-cuts  necessary  for  this  purpose  would  consist  of 
the  set  of  2p  cross-cuts,  necessary  to  resolve  the  surface  as  for  an  integral  of 


448  ELEMENTARY  INTEGRAL  [208. 

the  first  kind,  and  of  the  k  additional  cross-cuts  in  relation  with  the  curves 
excluding  the  algebraically  infinite  points. 

Let  the  moduli  for  the  former  cross-cuts  be 

ej,  €2,  ...,  ep,  for  the  cuts  a-^,  az,  ...,  cip, 
e^+i,  fp+o,  ...,  e»p  for  the  cuts  61,  62 >  •••>  ^jp>  respectively : 
the  moduli  for  the  cuts  c  are  zero.  It  is  evident  fi:'om  the  form  of  the 
integral  in  the  vicinity  of  any  infinite  point  that,  as  the  integral  has  only 
an  algebraic  infinity,  the  modulus  for  each  of  the  k  cross-cuts,  obtained  by  a 
curve  from  one  edge  to  the  other  round  the  point,  is  zero.  Hence  if  one 
value  of  the  integral  of  the  second  kind  at  a  point  on  the  surface  be  E{z), 
all  its  values  at  that  point  are  included  in  the  form 

E{z)-\-^    7lr€r, 
r=l 

where  nj,  n^,  ...,  n^  are  integers. 

The  importance  of  the  elementary  integral  of  the  second  kind,  inde- 
pendently of  its  simplicity,  is  that  it  is  determined  by  its  infinity,  save  as  to  an 
additive  integral  of  the  first  kind. 

Let  7^1  {z)  and  E^  (z)  be  two  elementary  integrals  of  the  second  kind, 
having  their  single  infinity  common,  and  let  a  be  the  value  of  z  at  this  point ; 
then  in  its  vicinity  we  have 

E,(z)==^^  +  F,(z-a),  E,(z)^-^+P,(z-a), 

Z  —  (X-  Z        Oj 

and  therefore  A-^E^{z)—  A^E^{z)  is  finite  at  z  =  a.  This  new  function  is 
therefore  finite  over  the  whole  Riemann's  surface :  hence  it  is  an  integral  of 
the  first  kind,  the  moduli  of  periodicity  of  which  depend  upon  those  of  E^  {z) 
and  Eo,{z). 

Ex.  It  may  similarly  be  proved  that  for  the  special  case  in  Ex.  2,  §  207,  when  the 
integral  of  the  second  kind  has  two  simple  infinities  for  the  same  arithmetical  value  of  z  in 
different  sheets,  the  integral  is  determinate  save  as  to  an  additive  integral  of  the  first  kind. 

Let  ai  and  a-i  be  the  two  points  for  the  arithmetical  value  a  of  2  ;  and  let  F{z)  and  G  {£) 
be  two  integrals  of  the  second  kind  above  indicated  having  simple  infinities  at  aj  and  ag 
and  nowhere  else. 

Then  in  the  vicinity  of  a^  we  have 

so  that  BF  {z)-  AG  (2)  is  finite  in  the  vicinity  of  %. 
Again,  in  the  vicinity  of  ag,  we  have,  by  §  202, 

F{z)=~^^F^kZ-(l^\  6-'(2)  =  ;^+$2(^-«2), 

so  that  BF(z)  —  AG  (2)  is  finite  in  the  vicinity  of  ^2  also.  Hence  BF{z)-AG{z)  is  finite 
over  the  whole  surface,  and  it  is  therefore  an  integral  of  the  first  kind  ;  which  proves  the 
statement. 


208.]  OF   THE   SECOND   KIND  449 

It  therefore  appears  that,  if  F  {z)  be  any  such  integral,  every  other  integral  of  the  same 
nature  at  those  points  is  of  the  form  F  (z)  +  W,  where  W  is  an  integral  of  the  first  kind. 
Now  there  are  p  linearly  independent  integrals  of  the  first  kind  :  it  therefore  follows  that 
there  are  p  +  1  linearly  independent  integrals  of  the  second  kind,  which  have  simple 
infinities  with  equal  and  opposite  residues  at  two  points,  (and  at  only  two  points),  deter- 
mined by  one  algebraical  value  of  z. 

From  the  property  that  an  elementary  integral  of  the  second  kind  is 
determined  by  its  infinity  save  as  to  an  additive  integral  of  the  first  kind,  we 
infer  that  there  are  p  +  1  linearly  independent  elementary  integrals  of  the 
second  kind  tuith  the  same  single  infinity  on  the  Miemann's  surface. 

This  result  can  be  established  in  connection  with  /(w,  z)  =  0  as  follows.  The  subject 
of  integration  is 

U  (w,  z) 

{z-afT 
^        '  CIV 

where  for  simplicity  it  is  assumed  that  a  is  neither  a  branch-point  of  the  function 
nor  a  singular  point  of  the  ciu-ve  f{w,z)  =  0,  and  in  the  present  case  (I  is  of  degree 
n  —  l  in  to.  To  ensure  that  the  integral  vanishes  for  s=co,  the  dimensions  of  U{w,z) 
may  not  be  greater  than  n-  1.  Hence  (J{w,  z),  in  its  most  general  form,  is  a  polynomial 
function  of  w  and  z  of  degree  n  —  l  ;  the  total  number  of  terms  is  therefore  \n{n+\), 
which  is  also  the  total  number  of  arbitrary  constants. 

In  order  that  the  integral  may  not  be  infinite  at  each  of  the  8  -f-  k  singularities  of  the 
curve  /(?<',  2)  =  0,  a  relation  U  {y,  c)  =  0  must  be  satisfied  at  each  of  them  ;  hence,  on  this 
score,  there  are  b  +  <  relations  among  the  arbitrary  constants. 

Let  the  points  on  the  surface  given  by  the  arithmetical  value  a  of  2  be  (aj,  ai),  {a^,  a<^, 
...,  (a„,  a„).     The  integral  is  to  be  infinite  at  only  one  of  them;  so  that  we  must  have 

for  r  =  2,  3,  ...,n  ;  and  71  — 1  is  the  greatest  number  of  such  points  for  which  U  can  vanish, 
unless  it  vanish  for  all,  and  then  there  would  be  no  algebraic  infinity.     Hence,  on  this 
score,  there  are  n  —  l  relations  among  the  arbitrary  constants  in  U. 
In  the  vicinity  of  2  =  a,  iv=a,  let 

z  =  a  +  ^,         ^v=a  +  v■, 

r\-f  7\-f 

then  we  have  0  =  v  ^  +  tJ-  + ..., 

da       da 

where  ~-  is  the  value  of  ^  ,  and  ^  that  of  J- ,  for  z=a  and  w  =  a.     For  suflficientlv  small 

ca  dw  da  oz  '' 

values  of  1 V  I  and  j  ^  | ,  we  may  take 

For  such  points  we  have 

U{w,z)=U{a,a)+vj^+C^  +  ... 

da 


and  ^^  =  1'+— 


^{f^l 


dw     da     df    d{a,a) 
d^ 

P.  F.  •  29 


450  INTEGRALS  [208. 

/40  ■ 


TiiGii  uniGss  ^^ ^       - — , . 

U{a,  a)  d  (a,  a)      vf    9  (a,  a) 

da 
7\  (  -f  Tl\ 
for  («! ,  ai),  and  3  (7^  =  *^ 

for  (ct2,  02)5  («3)  "3)'  •••)  {'^n,  «n)»  there  will  be  terms  in  -  in  the  expansion  of  the  subject  of 

integration  in  the  vicinity  of  the  respective  points,  and  consequently  there  will  be 
logarithmic  infinities  in  the  integral.  Such  infinities  are  to  be  excluded  ;  and  therefore 
their  coefficients,  being  the  residues,  must  vanish,  so  that,  on  this  score,  there  appear  to 
be  n  relations  among  the  arbitrary  constants  in  U.  But,  as  in  §  210,  the  sum  of  the 
residues  for  any  point  is  zero  :  and  therefore,  when  n  —  1  of  them  vanish,  the  remaining 
residue  also  vanishes.  Hence,  from  this  cause,  there  are  only  n  —  \  relations  among  the 
arbitrary  constants  in   U. 

The  tale   of  independent   arbitrary   constants   in    U  {w,  z),   remaining  after   all   the 
conditions  are  satisfied,  is 

|?i  (n  + 1)  -  (8  +  Ac)  -  (w  -  ] )  -  (?i  -  1) 
=p+l. 
As  each  constant  determines  an  integral,  the  inference  is  that  there  are  p  +  1  linearly 
independent  elementary  integrals  of  the  second  kind  with  a  common  infinity. 

209.  Next,  consider  integrals  which  have  logarithmic  infinities,  inde- 
pendently of  or  as  well  as  algebraic  infinities.  They  are  called  integrals  of 
the  third  kind.  As  in  the  case  of  integrals  of  the  first  kind  and  the  second 
kind,  we  take  the  subject  of  integration  to  be  as  general  as  possible  so  that  it 
contains  arbitrary  coefficients ;  and  we  assign  suitable  relations  a.mong  the 
coefficients  to  prevent  any  of  the  points,  given  as  zeros  of  df/dtv,  ft-om  becoming 
infinities  of  the  integral.  It  follows  that  the  only  infinities  of  the  integral 
are  accidental  singularities  of  coefficients  of  powers  of  w  in  the  numerator 
of  the  general  expression  for  w' ;  and  that,  when  w'  is  expanded  for  points  in 
the  immediate  vicinity  of  such  an  expression,  the  term  with  index  —  1  must 
occur. 

To  find  the  general  value  of  an  integral  of  the  third  kind,  we  should 
first  exclude  from  the  Riemann's  surface  all  the  infiinite  points,  say 

tj,  I2,  . . . ,  l^> 

by  small  curves ;  the  surface  would  then  have  to  be  resolved  into  one  that 
is  simply  connected.  The  cross-cuts  for  this  purpose  would  consist  of  the 
set  of  2p  cross-cuts,  necessary  to  resolve  the  surface  for  an  integral  of  the 
first  kind,  and  of  the  additional  cross-cuts,  /j,  in  number  and  drawn  from  the 
boundary  (taken  at  some  ordinary  point  of  the  integral)  to  the  small  curves 
that  surround  the  infinities  of  the  function. 

The  moduli  for  the  former  set  may  be  denoted  by 
-OTj,  OT2,  ...,  '^p  for  the  cuts  cii,  Uo,  ...,  Qp, 
and  -^p+i,  ■OT^+2;  •••,  '^2p  for  the  cuts  bi,h2,  ...,hp  respectively; 


209.]  OF    THE    THIRD    KIND  451 

they  are  zero  for  the  cuts  c.  Taking  the  integral  from  one  edge  to  the  other 
of  any  one  of  the  remaining  cross-cuts  ^i,  l^,  ...,  Iq,  (where  Iq  is  the  cross-cut 
drawn  from  the  curve  surrounding  Iq  to  the  boundary),  its  value  is  given  by 
the  value  of  the  integral  round  the  small  curve  and  therefore  it  is  27riXq, 
where  the  expansion  of  the  subject  of  integration  in  the  immediate  vicinity 
of  z  =  lq  is 

M^-A^^<^-«- 

Then,  if  11  be  any  value  of  the  integral  of  the  third  kind  at  a  point  on  the 
unresolved  Riemann's  surface,  all  its  values  at  the  point  are  included  in  the 
form 

2p  ft. 

n  +  S  mj-OTr  +  27^^  S  iiqXq, 

r=l  q=l 

where  the  coefficients  irii,  ...,  ni^p,  rii,  ...,  n^i  are  integers. 

210.     It  can  be  proved  that  the  quantities  Xq  are  subject  to  the  relation 

X^  +  X2+...+X^  =  0. 

Let  the  surface  be  resolved  by  the  complete  system  of  2p  +  fi  cross-cuts :  the 
resolved  surface  is  simply  connected  and  has  only  a  single  boundary.  The 
subject  of  integration,  tu',  is  uniform  and  continuous  over  this  resolved  surface  : 
it  has  no  infinities  in  the  surface,  for  its  infinities  have  been  excluded ;  hence 

Jw'dz  =  0, 

when  the  integral  is  taken  round  the  complete  boundary  of  the  resolved 
surface. 

This  boundary  consists  of  the  double  edges  of  the  cross-cuts  a,  h,  c,  L, 
and  the  small  curves  round  the  //.  points  I ;  the  two  edges  of  the  same  cross- 
cut being  described  in  opposite  directions  in  every  instance. 

Since  the  integral  is  zero  and  the  function  is  finite  everywhere  along  the 
boundary,  the  parts  contributed  by  the  portions  of  the  boundary  may  be  con-  , 
sidered  separately. 

First,  for  any  cross-cut,  say  a^:  let  0  be  the  point  where  it  is  crossed  by  hq, 

and  let  the  positive  direction  of  description  of  the  whole  boundary  be  indicated 

by  the  arrows  (fig.  82,  §  230).     Then,  for  the  portion  Ga...E,  the  part  of  the 

[^  . 

integral  is  I     w'dz,  or,  if  Ca...E  be  the  negative  edge  (as  in  §  196),  the  part 
J  c 

oi  the  integral  may  be  denoted  by 

E 

w'dz. 
c 

The  part  of  the  integral   for  the   portion   F...aD,  being  the  positive 

f^  .  rF 

edge  of  the  cross-cut,  is       w'dz,  which  may  be  denoted  by  —  /     w'dz.     The 

29—2 


452  ELEMENTAEY  INTEGEAL  [210. 

course  and  the  range  for  the  latter  part  are  the  same  as  those  for  the 
former,  and  w'  is  the  same  on  the  two  edges  of  the  cross-cut ;    hence  the 

sum  of  the  two  is 

rE 
=  I    {w'  —  w')  dz, 
J  c 

which  evidently  vanishes*.     Hence  the  part  contributed  to  jw'dz  by  the  two 

edges  of  the  cross-cut  ciq  is  zero. 

Similarly  for  each  of  the  other  cross-cuts  a,  and  for  each  of  the  cross-cuts 
h,  c,  L. 

The  part  contributed  to  the  integral  taken  along  the  small  curve  enclosing 
Iq  is  tiriXq,  for  §'=1,  2,  ...,  fx:  hence  the  sum  of  the  parts  contributed  to  the 
integral  by  all  these  small  curves  is 

27rl    2    \q. 

All  the  other  parts  vanish,  and  the  integral  itself  vanishes ;  hence 

establishing  the  result  enunciated. 

CoROLLAEY.  An  integral  of  the  third  kind,  that  is,  having  logarithmic 
infinities  on  a  Riemanns  surface,  must  have  at  least  two  logarithmic  infinities.. 

If  it  had  only  one  logarithmic  infinity,  the  result  just  proved  would 
require  that  Xj  should  vanish,  and  the  infinity  would  then  be  purely 
algebraic. 

211.  The  simplest  instance  is  that  in  which  there  are  only  two. 
logarithmic  infinities;    their  constants  are  connected  by  the  equation 

\^-{-\^  =  0. 

If,  in  addition,  the  infinities  be  purely  logarithmic,  so  that  there  are  na 
algebraically  infinite  terms  in  the  expansion  of  the  integral  in  the  vicinity 
of  either  of  the  points,  the  integral  is  then  called  an  elementary  integral 
of  the  third  kind.  If  two  points  Cj  and  Cg  on  the  surface  be  the  two  infini- 
ties, and  if  they  be  denoted  by  assigning  the  values  Cj  and  c^  to  z ;  and  if 
Xi  =  1  =  —  X2  (as  may  be  assumed,  for  the  assumption  only  implies  division 
of  the  integral  by  a  constant  factor),  the  expansion  of  the  subject  of  inte- 
gration for  points  in  the  vicinity  of  Ci  is 

*  It  vanishes  from  two  independent  causes,  first  through  the  factor  10'  -  w',  and  secondly 
because  Zj^=z^,  the  breadth  of  any  cross-cut  being  infinitesimal. 

The  same  result  holds  for  each  of  the  cross-cuts  a  and  6. 

For  each  of  the  cross-cuts  c  and  L,  the  sum  of  the  parts  contributed  by  opposite  edges 
vanishes  only  on  account  of  the  factor  w'  -  to' ;  in  these  cases  the  variable  z  is  not  the  same 
for  the  upper  and  the  lower  limit  of  the  integral. 


211,]  OF   THE   THIRD   KIND  453 

and  for  points  in  the  vicinity  of  Cg  the  expansion  is 

Z-Cz 

Such  an  integral  may  be  denoted  by  Ilia :  its  modulus,  consequent  on 
the  logarithmic  infinity,  is  27ri. 

Ex.  1.  Prove  that,  if  ni2,  1123,  Hsi  be  three  elementary  integrals  of  the  third  kind 
having  Cx,  C2;  c^,  c^;  C3,  c^  for  their  respective  pairs  of  points  of  logarithmic  discontinuity, 
then  1X12  + 1123  +  Hsi  is  either  an  integral  of  the  first  kind  or  a  constant. 

Clebsch  and  Gordan  pass  from  this  result  to  a  limit  in  which  the  points  c^  and  c^ 
coincide  and  obtain  an  expression  for  an  elementary  integral  of  the  second  kind  in  the 
form  of  the  derivative  of  His  with  regard  to  c^ .  Klein,  following  Riemann,  passes  from  an 
elementary  integral  of  the  second  kind  to  an  elementary  integral  of  the  third  kind  by 
integrating  the  former  with  regard  to  its  parametric  laoint*. 

Ex.  2.  Reverting  again  to  the  integrals  connected  with  the  algebraical  equation 
fiw,  z)=0,  when  it  can  be  interpreted  as  the  equation  of  a  generalised  curve,  an  integral 
of  the  third  kind  arises  when  the  subject  of  integration  is 

,      V{w,  z) 
''  =  -37' 
cw 

where  V{tv,  z)  is  of  degree  to  — 2  in  w.  If  V  {w,  z)  be  of  degree  in  z  not  higher  than  w-  2, 
the  integral  of  w'  is  not  infinite  for  infinite  values  oi  z;  so  that  F(w,  2)  is  a  general  integral 
function  of  w  of  degree  n  —  2. 

Corresponding  to  the  arithmetical  value  c  of  2,  there  ai'e  n  points  on  the  surface,  say 
(•^i)  '^i))  ('^2j  ^2))  •••)  (c»)  ^n) ;  ^'D'd  the  expansion  of  w'  in  the  vicinity  of  (c^,  h^)  is 

3/  Z-Cr 

dk,. 

the  coefi&cients  of  the  infinite  tenns  being  subject  to  the  relation 

because  V(w,  2)  is  only  of  degree  n-2  in  w.  The  integral  of  lo'  will  have  a  logarithmic 
infinity  at  each  point,  unless  the  corresponding  coefl&cient  vanish. 

Not  more  than  n  —  2  of  these  coefficients  can  be  made  to  vanish,  unless  they  all  vanish ; 
and  then  the  integral  has  no  logarithmic  infinity.     Let  71 -2  relations,  say 

for  r=3,  4, ...,  TO,  be  chosen;  and  let  the  S  +  k  relations  be  satisfied  which  secure  that  the 
integral  is  finite  at  the  singularities  of  the  curve  /{to,  s)  =  0.  Then  the  integral  is  an 
elementary  integral  of  the  third  kind,  having  (ci,  ^1)  and  {c^,  Jc^^  for  its  points  of 
logarithmic  discontinuity. 

Ex.  3.  Prove  that  there  are  ^  +  1  linearly  independent  elementary  integrals  of  the 
third  kind,  having  the  sarae  logarithmic  infinities  on  the  surface. 

*  Clebsch  und  Gordan,  (I.e.,  p.  408,  note),  pp.  28 — 33 ;  Klein-Fricke,  Vorlesungen  Uber  die 
Theorie  der  elUptischen  Modulfunctionen,  t.  i,  pp.  518 — 522  ;   Riemann,  p.  100. 


454  CLASSES   OF   FUNCTIONS  [211. 

Ex.  4.     Shew  that,  in  connection  with  the  fundamental  equation 

any  integral  of  the  first  kind  is  a  constant  multiple  of 

'dz 


j;2' 

that  an  integral  of  the  second  kind,  of  the  class  considered  in  Ex.  2,  §  207,  is  given  by 

l—w, 
dz ; 

and  that  an  elementary  integral  of  the  third  kind  is  given  by 


I  — r  ^^• 
]    zw^ 


Ex.  5.     An  elementary  (Jacobian)  elliptic  integral  of  the  third  kind  occurs  in  Ex.  7, 
p.  435 ;  and  a  (Jacobian)  elliptic  integral  of  the  second  kind  occurs  in  Ex.  8,  p.  436. 

Shew  that  an  elementary  (elliptic)  integral  of  the  second  kind,  associated  with  the 
equation 

io^  =  ^z^-g^z-go„ 

and  having  its  infinity  at  (ci,  yj),  is 

'yi(^^+yi)  +  (6gi'-ig'2)(g-ci) 


/ 


{z-c^)'^w  ^''' 


and  that  an  elementary  (elliptic)  integral  of  the  third  kind,  associated  with  the  same 
equation  and  having  its  two  infinities  at  (cj,  yj),  (c2,  72)5  is 

1   f  fw+yi      w  +  y2\  dz 


2  J  \z  —  Ci       z  —  c<i)w 

Ex.  6.     Construct  an  elementary  integral  of  the  second  kind,  which  is  infinite  of  the 
first  order  at  2=0,  zi'=  1,  the  equation  between  lo  and  z  being 

w5  +  (s-l)(s2  +  l)2,=  0. 

(Math.  Trip.,  Part  II.,  1897.) 

A  sufficient  number  of  particular  examples,  and  also  of  examples  with 
a  limited  generality,  have  been  adduced  to  indicate  some  of  the  properties 
of  functions  arising,  in  the  first  instance,  as  integrals  of  multiform  functions 
of  a  variable  z  (or  as  integrals  of  uniform  functions  of  position  on  a 
Riemann's  surface).  The  succeeding  investigation  establishes,  from  the  most 
general  point  of  view,  the  existence  of  such  functions  on  a  Riemann's 
surface :  they  will  no  longer  be  regarded  as  defined  solely  by  integrals  of 
multiform  functions. 


CHAPTER   XYII. 

ScHWARz's  Proof  of  the  Existence-Theorem. 


212.  The  investigations  in  the  preceding  chapter  were  based  on 
the  supposition  that  a  fundamental  equation  was  given,  the  appropriate 
Riemann's  surface  being  associated  with  it.  The  general  expression  of 
uniform  functions  of  position  on  the  surface  was  constructed,  and  the 
integrals  of  such  functions  were  considered.  These  integrals  in  general 
were  multiform  on  the  surface,  the  deviation  from  uniformity  consisting 
in  the  property  that  the  difference  between  any  two  of  the  infinite  number  of 
values  could  be  expressed  as  a  linear  combination  of  integral  multiples  of 
certain  constants  associated  Avith  the  function.  Infinities  of  the  functions 
defined  by  the  integrals,  and  the  classification  of  the  functions  according  to 
their  infinities,  were  also  considered. 

But  all  these  investigations  were  made  either  in  connection  with 
very  particular  forms  of  the  fundamental  equation,  or  with  a  form  of  not 
unlimited  generality :  and,  for  the  latter  case,  assumptions  were  made, 
justified  by  the  analysis  so  far  as  it  was  carried,  but  not  established 
generally. 

In  order  to  render  the  consideration  of  the  propositions  complete,  it  must 
be  made  without  any  limitations  upon  the  general  form  of  fundamental 
equation. 

Moreover,  the  second  question  of  §  192,  viz.,  the  existence  of  functions 
(both  uniform  and  multiform)  of  position  on  a  surface  given  independently  of 
any  algebraical  equation,  is  as  yet  unconsidered. 

The  two  questions,  in  their  generality,  can  be  treated  together.  In  the 
former  case,  with  the  fundamental  equation  there  is  associated  a  Riemann's 
surface,  the  branching  of  which  is  determined  by  that  fundamental  equation ; 
in  the  latter  case,  the  Riemann's  surface  with  assigned  branching  is  supposed 


456  INITIAL   SIMPLIFICATION  [212. 

given*.  We  shall  take  the  surface  as  having  one  boundary  and  being  other- 
wise closed ;  the  connectivity  is  therefore  an  uneven  integer,  and  it  will  be 
denoted  by  2p  +  1. 

213.  The  problem  can  be  limited  initially,  so  as  to  prevent  unnecessary 
complications.  All  the  functions  to  be  discussed,  whether  they  be  algebraic 
functions  or  integrals  of  algebraic  functions,  can  be  expressed  in  the  form 
u  +  iv,  where  u  and  v  are  two  real  functions  of  two  independent  real  variables 
x  and  y.  It  has  already  (§  10)  been  proved  that  both  u  and  v  satisfy  the 
equation 

and  that,  if  either  m  or  v  be  known,  the  other  can  be  derived  by  a  quadra- 
ture at  most,  and  is  determinate  save  as  to  an  additive  arbitrary  constant. 
Since  therefore  w  is  determined  by  u,  save  as  to  an  additive  constant,  we 
shall,  in  the  first  place,  consider  the  properties  of  the  real  function  u  only. 

The  result  is  valid  so  long  as  v  can  be  determined,  that  is,  so  long  as  the 
function  u  has  differential  coefficients.  It  will  appear,  in  the  course  of  the 
present  chapter,  that  no  conditions  are  attached  to  the  derivatives  of  u  along 
the  boundary  of  an  area,  so  that  the  determination  of  v  along  such  a  boundary 
seems  open  to  question. 

It  has  been  (§  36)  proved,  in  a  theorem  due  to  Schwarz,  that,  if  w  a 
function  of  z  be  defined  for  a  half-plane  and  if  it  have  real  finite  continuous 
values  along  any  portion  of  the  axis  of  x,  it  can  be  symmetrically  continued 
across  that  portion  of  the  axis.  The  continuation  is  therefore  possible  for  the 
real  part  u  of  the  function  lu;  and  the  values  of  u  are  the  real  finite  continuous 
values  of  w  along  that  portion  of  the  axis. 

It  will  be  seen,  in  Chapters  XIX.,  XX.  that,  by  changing  the  independent 
variables,  the  axis  of  x  can, be  changed  into  a  circle  or  other  analytical  line 
(I  221);  so  that  a  function  u,  defined  for  an  interior  and  having  real  finite 
continuous  values  along  any  portion  of  the  boundary,  can  be  continued  across 
that  portion  of  the  boundary,  which  is  therefore  not  the  limit  of  existence  f 

*  The  surface  is  supposed  given ;  we  are  not  concerned  with  the  quite  distinct  question  as 
to  how  far  a  Riemann's  surface  is  determinate  by  the  assignment  of  its  number  of  sheets,  its 
branch-points  (and  consequently  of  its  connectivity),  and  of  its  branch-hnes.  This  question  is 
discussed  by  Hurwitz,  Math.  Ann.,  t.  xxxix,  (1891),  pp.  1 — 61.  He  shews  that,  if  fl  denote  the 
ramification  (§  178)  of  the  surface  which,  necessarily  an  even  integer,  is  defined  as  the  sum  of 
the  orders  of  its  branch-points,  a  two-sheeted  surface  is  made  uniquely  determinate  by  assigned 
branch-points;  the  number  of  essentially  distinct  three-sheeted  surfaces,  made  determinate  by 
assigned  branch-points,  is  ^  (3"~^-  1) ;  and  so  on,  the  number  being  finite  for  finite  values  of  fi. 
It  is  easy  to  verify  that  the  number  of  distinct  three-sheeted  surfaces,  with  4  assigned  points 
as  simple  branch-points,  is  4 :  an  example  suggested  to  me  by  Prof.  W.  Burnside. 

t  The  continuation  indicated  will  be  carried  out  for  the  present  case  by  means  of  the 
combination  of  areas  (§  222),  and  without  further  reference  to  the  transformation  indicated 
or  to  Schwarz's  theorem  on  symmetrical  continuation. 


213.]  POTENTIAL   FUNCTION  457 

of  u.     The  derivatives  of  u  can  then  be  obtained  in  the  extended  space  and 
so  V  can  be  determined  for  the  boundary*. 

And,  what  is  more  important,  it  will  be  found  that,  under  conditions  to  be 
assigned,  the  number  of  functions  u  that  are  determined  is  double  the  number 
of  functions  w  that  are  determined ;  the  complete  set  of  functions  u  lead  to 
all  the  parts  u  and  v  of  the  functions  w  (§  234,  note). 

214.  The  infinities  of  u  at  any  point  are  given  by  the  real  parts  of  the 
terms  which  indicate  the  infinities  of  w.  Conversely,  when  the  infinities  of  u 
are  assigned  in  functional  form,  those  of  w  can  be  deduced,  the  form  of  the 
associated  infinities  of  v  first  being  constructed  by  quadratures. 

The  periods  of  tu,  being  the  moduli  at  the  cross-cuts,  lead  to  real  constants 
as  differences  oi  u  at  opposite  edges  of  cross-cuts,  or,  if  we  choose,  as  constant 
differences  of  values  of  u  at  points  on  definite  curves,  conveniently  taken  for 
reference  as  lines  of  possible  cross-cuts.  Conversely,  a  real  constant  modulus 
for  u  is  the  real  partf  of  the  corresponding  modulus  of  w. 

Hence  a  function,  w,  of  position  on  a  Riemann's  surface  is,  except  as  to  an 
additive  constant,  determined  by  a  real  function  u  of  x  and  y  (where  x  -I-  iy  is 
the  independent  variable  for  the  surface),  if  ii,  be  subject  to  the  conditions : — 

(i)  it  satisfies  the  equation  V~u  =  0  at  all  points  on  the  surface  where 
its  derivatives  are  not  infinite  : 

(ii)  if  it  be  multiform,  its  values  at  any  point  on  the  surface  differ  by 
linear  combinations  of  integral  multiples  of  real  constants : 
otherwise,  it  is  uniform : 

(iii)     it  may  have  specified  infinities,  of  given  form  in  the  vicinity  of 
assigned  points  on  the  surface. 

In  addition  to  these  general  conditions  imposed  upon  the  function  u,  it  is 
convenient  to  admit  as  a  further  possible  condition,  for  portions  of  the  surface, 
that  the  function  u  shall  assume,  along  a  closed  curve,  values  which  are 
always  finite.  But  it  must  be  understood  that  this  condition  is  used  only  for 
subsidiary  purposes :  it  will  be  seen  that  it  causes  no  limitation  on  the  final 
result,  all  that  is  essential  in  its  limitations  being  merged  in  the  three 
principal  conditions. 

The  questions  for  discussion  are  therefore  (i),  the  existence  of  functions ^: 
satisfying   the    above    conditions    in    connection    with    a    given    Riemann's 

*  See  Phragmen,  Acta  Math.,  t.  xi^,  (1890),  pp.  225 — 227,  for  some  remarks  upon  this 
question. 

+  The  imaginary  parts  of  the  moduli  of  w  are  determinate  with  the  imaginary  part  of  w  -. 
see  remark  at  end  of  §  213,  and  the  further  reference  there  given. 

X  The  functions  u  (and  also  v)  are  of  great  importance  in  mathematical  physics  for  two- 
dimensional  phenomena  in  branches  such  as  gravitational  attraction,  electricity,  hydrodynamics, 
and  heat.  In  all  of  them,  the  function  represents  a  poteutial;  and,  consequently,  in  the  general 
theory  of  functions,  it  is  often  called  a.  potential  function. 


458  METHODS   OF   SOLUTION  [214, 

surface,  the  connectivity  of  which  is  2j)  +  1  as  dependent  upon  its  branching 
and  the  number  of  its  sheets;  and  (ii),  assuming  that  the  functions  exist, 
their  determination  by  the  assigned  conditions. 

215.  There  are  many  methods  for  the  discussion  of  these  questions.  The 
potential  function,  both  for  two  and  for  three  dimensions  in  space,  first  arose 
in  investigations  connected  with  mathematical  physics  :  and,  so  far  as  concerns 
such  subjects,  its  theory  was  developed  by  Green,  Gauss,  Poisson,  Stokes,  Lord 
Kelvin,  Maxwell  and  others.  Their  investigations  have  reference  to  appli- 
cations to  mathematical  physics ;  and  they  do  not  tend  towards  the  solution  of 
the  questions  just  propounded  in  relation  to  the  general  theory  of  functions. 

Klein  uses  considerations  drawn  from  mathematical  and  experimental 
physics  to  establish  the  existence  of  potential  functions  under  the  assigned 
conditions.  The  proof  that  will  be  adopted  brings  the  stages  of  the  investi- 
gation into  closer  relations  with  the  preceding  and  the  succeeding  parts  of 
the  subject  than  is  possible  if  Klein's  method  be  followed*. 

To  establish  the  existence  of  the  functions  under  the  assigned  conditions.. 
Riemannf  uses  the  so-called  Dirichlet's  Principle  J ;  but  as  E-iemann's  proof 
of  the  principle  is  inadequate,  his  proof  of  the  existence-theorem  cannot  be 
considered  complete. 

There  are  two  other  principal,  and  independent,  methods  of  importance, 
each  of  which  effectively  establishes  the  existence  of  the  functions,  due  to 
Neumann  and  to  Schwarz  respectively;  each  of  them  avowedly  dispenses! 
with  the  use  of  Dirichlet's  Principle. 

The  courses  of  the  methods  have  considerable  similarly.  Both  begin 
with  the  construction  of  the  function  for  a  circular  area.  Neumann  uses 
what  is  commonly  called  the  method  of  the  arithmetic  mean,  for  gradual 
approximation  to  the  value  of  the  potential  function  for  a  region  bounded 
by  a  convex  curve :  Schwarz  uses  the  method  of  conformal  representation, 
to  deduce  from  results  previously  obtained,  the  potential  function  for  regions 
bounded  by  analytical  curves ;  and  both  authors  use  certain  methods  for 
combination  of  areas,  for  each  of  which  the  potential  function  has  been 
constructed  II . 

*  Klein's  proof  occurs  in  his  tract,  already  quoted,  Ueber  Riemann's  Theorie  der  algehraischen 
Functionen  und  ihrer  Integrale,  (Leipzig,  Teubner,  1882),  and  it  is  modified  in  his  memoir 
"  Neue  Beitrage  zur  Riemann'schen  Functionentheorie,"  Math.  Ann.,  t.  xxi,  (1883),  pp.  141 — 218, 
particularly  pp.  160 — 162. 

t  Ges.  Werke,  pp.  35—39,  pp.  96—98. 

X  Riemann  enunciates  it,  (I.e.),  pp.  34,  92. 

§  Neumann,  Vorlesungen  uber  Rieviann's  Theorie  der  AbePschen  Integrale,  (2nd  ed.,  1884), 
p.  238  ;    Schwarz,   Ges.  Werke,  ii,  p.  171. 

II  Neumann's  investigations  are  contained  in  various  memoirs,  Math.  Ann.,  t.  iii,  (1871), 
pp.  325—349;  ib.,  t.  xi,  (1877),  pp.  558—566;  ib.,  t.  xiii,  (1878),  pp.  255—300;  ib.,  t.  xvi, 
(1880),  pp.  409—431;   and  the  methods  are  developed  in  detail  and  amplified  in  his  treatise 


215.]  SUMMARY   OF   SCHWARZ'S   ARGUMENT  459 

What  follows  in  the  present  chapter  is  based  upon  Schwarz's  investi- 
gations :  the  next  chapter  is  based  upon  the  investigations  of  both  Schwarz 
and  Neumann,  and,  of  course,  upon  Riemann's  memoirs. 

The  foUowiDg  summary  of  the  general  argument  will  serve  to  indicate  the  main  line  of 
the  proof  of  the  establishment  of  potential  functions  satisfying  assigned  conditions. 

I.  A  potential  function  u  is  uniquely  determined  by  the  conditions  :  that  u,  as  well 

as  its  derivatives  '—,  ^,  5— g,  ^-^  (which  satisfy  the  equation  VfM  =  0),  shall  be  uniform, 

finite  and  continuous,  for  all  points  within  the  area  of  a  circle;  and  that,  along  the 
circumference  of  the  circle,  the  function  shall  assume  assigned  values  that  are  always 
finite,  uniform  and,  except  at  a  limited  number  of  isolated  points  where  there  is  a  sudden 
(finite)  change  of  value,  continuous.     (§§  216 — 220.) 

II.  By  using  the  principle  of  conformal  representation,  areas  bounded  by  curves  other 
than  circles — say  by  analytical  curves — are  obtained,  over  which  the  potential  function  is 
imiquely  determined  by  general  conditions  within  the  area  and  assigned  values  along  its 
boundary.     (§  221.) 

III.  The  method  of  combination  of  areas,  dependent  upon  an  alternating  process, 
leads  to  the  result  that  a  function  exists  for  a  given  region,  satisfying  the  general 
conditions  in  that  region  and  acquiring  assigned  finite  values  along  the  boundary  when 
the  region  can  be  obtained  by  combinations  of  areas  that  can  be  conformally  represented 
upon  the  area  of  a  circle.     (§  222.) 

IV.  The  theorem  is  still  valid  when  the  region  (supposed  simply  connected)  contains, 
a  branch-point :  the  winding-surface  is  transformed  by  a  relation 

into  a  single-sheeted  surface,  for  which  the  theorem  has  already  been  established. 

When  the  surface  is  multiply  connected,  we  resolve  it  by  cross-cuts  into  one  that  is 
simply  connected,  before  discussing  the  function.     (§  223.) 

Ueber  das  logarithmische  und  Neioton'sche  Potential  (Leipzig,  Teubner,  1877)  and  in  his  treatise 
quoted  in  the  preceding  note.  In  this  connection,  as  -well  as  in  relation  to  Schwarz's  investi- 
gations, and  also  in  view  of  some  independence  of  treatment,  Harnack's  treatise,  Die  Grundlagen 
der  Theorie  des  logarithmischen  Potentiates  und  der  eindeutigen  P otentialf unction  in  der  Ebene 
(Leipzig,  Teubner,  1887),  and  a  memoir  by  Harnack,  Math.  Ann.,  t.  xxxv,  (1890),  pp.  19 — 40, 
may  be  consulted. 

A  modification  of  Neumann's  proof,  due  to  Klein,  is  given  in  the  first  volume  (pp.  508 — 522) 
of  the  treatise  cited  on  p.  453,  note. 

Schwarz's  investigations  are  contained  in  various  memoirs  occurring  in  the  second  volume 
of  his  Gesammelte  Werke,  pp.  108—132,  133—148,  144—171,  175—210,  303—306  :  their  various 
dates  and  places  of  publication  are  there  stated.  A  simple  and  interesting  general  statement 
of  the  gist  of  his  results  will  be  found  in  a  critical  notice  of  the  two  volumes  of  his  collected 
works,  written  by  Henrici  in  Nature  (Feb.  5,  12,  1891,  pp.  321—323,  349—352).  There  is  a 
comprehensive  memoir  by  Ascoli,  based  upon  Schwarz's  method,  "Integration  der  Differential- 
gleichung  V^u  =  0  in  einer  beliebigen  Riemann'schen  Flache,"  {Bih.  t.  kongl.  Svenska  Vet.  Akad. 
Handl.,  bd.  xiii,  1887,  Afd.  1,  n.  2  ;  83  pp.) ;  a  thesis  by  Jules  Biemann,  Sur  le  prohleme  de 
Dirichlet,  (These,  Gauthier-'Villars,  Paris,  1888),  discusses  a  number  of  Schwarz's  theorems 
(see,  however,  Schwarz,  Ges.  Werke,  t.  ii,  pp.  356 — 358) ;  and  an  independent  memoir  by  Prym, 
Crelle,  t.  Ixxiii,  (1871),  pp.  340—364,  may  be  consulted. 

The  literature  of  this  part  of  the  subject  is  very  wide  in  extent :  many  other  references  are 
given  by  the  authors  already  quoted. 


460  PRELIMINARY   LEMMAS  [215. 

V.  Real  functions  exist  on  a  Riemann's  surface,  which  are  everywhere  finite  and 
uniquely  determinate  by  arbitrarily  assigned  real  moduli  of  periodicity  at  the  cross-cuts. 
(§§  224-227.) 

VI.  Functions  exist,  satisfying  the  conditions  in  (V.)  except  that  they  may  have  at 
isolated  points  on  the  surface,  infinities  of  an  assigned  form.     (§  229.) 

216.  We  shall,  in  the  first  place,  discuss  potential  functions  that  have 
no  infinities,  either  algebraic  or  logarithmic,  over  some  continuous  area  on 
the  surface  limited  by  a  simple  closed  boundary,  or  by  a  number  of  non-inter- 
secting simple  closed  curves  constituting  the  boundary ;  for  the  present,  the 
area  thus  enclosed  will  be  supposed  to  lie  in  one  and  the  same  sheet,  so  that 
we  may  regard  the  area  as  lying  in  a  simple  plane. 

At  all  points  within  the  area  and  on  its  boundary,  the  function  u  is  finite 
and  will  be  supposed  uniform  and  continuous ;  for  all  points  within  the*  area 
(but  not  necessarily  for  points  on  the  boundary),  the  derivatives 

du      du     d^u     dhi 
dec '    dy'    dx^ '    dy'^ 

are   uniform,  finite   and  continuous  and  they  satisfy  the   equation  V'^u  =  0. 
These  may  be  called  the  general  conditions. 

Two  cases  occur  according  as  the  character  of  the  derivatives  at  points  in 
the  area  is  or  is  not  assigned  for  points  on  the  boundary ;  if  the  character  be 
assigned,  there  will  then  be  what  may  be  called  boundary  conditions.  The 
two  cases  therefore  are  : — 

(A)  When  a  function  u  is  required  to  satisfy  the  general  conditions, 

and  its  derivatives  are  required  to  satisfy  the  boundary  con- 
ditions : 

(B)  When  the  only  requirement  is  that  the  function  shall  satisfy  the 

general  conditions. 
Before   proceeding    to    the    establishment    of  what    is   the    fundamental 
proposition  in  Schwarz's   method,  it  is   convenient  to  prove  three  lemmas 
and  to  deduce  some  inferences  that  will  be  useful. 

Lemma  I.     If  two  functions  u-^  and  Wg  satisfy  the  general  conditions  for  tiuo 

regions  I\  and  T^  respectively,  which  have  a  common  portion  T  that  is  more 

.  than  a  point  or  a  line,  and  if  u^  and  u^  he  the  same  for  the  common  portion  T, 

or  if  they  he  the  same  for  any  part  of  T  that  is  more  than  a  point  or  a  line, 

then  they  define  a  single  function  for  the  luhole  region  composed  of  T^  and  T^. 

This  proposition  can  be  made  to  depend  upon  the  continuation  of 
analytic  functions*,  whether  in  a  plane  (§  34)  or,  in  view  of  a  subsequent 
transformation  (§  223),  on  a  Riemann's  surface. 

The  real  function  u-^  defines  a  function  Wi  of  the  complex  variable  z,  for 
any  point  in  the  region  T^ ;  and  for  points  within  this  region,  the  function  w^ 

*  For  other  proofs,  see  Schwarz,  Ges.  Werke,  t.  ii,  pp.  201,  202,  and  references  there  given. 


216.]  FOR   SCHWARZ'S   PROOF  461 

is  uniquely  determined  by  means  of  its  own  value  and  the  values  of  its 
derivatives  at  any  point  within  T^,  obtained,  if  necessary,  by  a  succession  of 
elements  in  continuation.  Hence  the  value  of  w^  and  its  derivatives  at  any 
point  within  T  defines  a  function  existing  over  the  whole  of  T^ . 

Similarly  the  real  function  Wg  defines  a  function  W2  within  T^,  and  this 
function  is  uniquely  determined  over  the  whole  of  T^,  by  its  value  and  the 
values  of  its  derivatives  at  any  point  within  T. 

Now  the  values  of  Wj  and  u^  are  the  same  at  all  points  in  T,  and  therefore 
the  values  of  tfj  and  w^  are  the  same  at  all  points  in  T,  except  possibly  for  an 
additive  (imaginary)  constant,  say  ia,  so  that 

Hence  for  all  points  in  T,  (supposed  not  to  be  a  point,  so  that  we  may  have 
derivatives  in  every  direction  (§  8):  and  not  to  be  a  line,  so  that  we  may 
have  derivatives  in  all  directions  from  a  point  on  the  line),  the  derivatives 
of  Wj  agree  with  those  of  w^]  and  therefore  the  quantities  necessary  to  define 
the  continuation  of  w-^  from  T  over  T^  agree  with  the  quantities  necessary  to 
define  the  continuation  of  lu.,  from  T  over  T.2,  except  only  that  w^  and  w^ 
differ  by  an  imaginary  constant.  Hence,  having  regard  to  the  form  of  the 
elements,  Wj  and  w^  can  be  continued  over  the  region  composed  of  T^  and  To, 
and  their  values  differ  (possibly)  by  the  imaginary  constant.  When  we  take 
the  real  parts  of  the  functions,  we  have  Ui  and  Wg  defining  a  single  function 
existing  over  the  whole  region  occupied  by  the  combination  of  T^  and  T„. 

The  other  two  lemmas  relate  to  integrals  connected  with  potential 
functions. 

Lemma  II.  Let  u  he  a  function  required  to  satisfy  the  general  conditions^ 
and  let  its  derivatives  he  required  to  satisfy  the  houndary  conditions,  for  an 
area  S  hounded  hy  simple  non-intersecting  curves:  then 

du  , 
^ds=0: 

on 

where  the  integral  is  extended  round  the  ivhole  boundary  in  the  direction  that  is 
positive  with  regard  to  the  hounded  a7xa  S;  and  dn  is  an  element  of  the  normal 
to  a  houndary-line  drawn  towards  the  interior  of  the  space  enclosed  hy  that 
houndary -line  regarded  merely  as  a  simple  closed  curve*. 

Let  P  and  Q  be  any  two  functions,  which,  "as  well  as  their  first  and  second 
derivatives  with  regard  to  x  and  to  y,  are  uniform,  finite  and  continuous  for 

*  The  element  dn  of  the  normal  is,  by  this  definition,  measured  inwards  to,  or  outwards 
from,  the  area  S  according  as  the  particular  boundary-line  is  described  in  the  positive,  or  in  the 
negative,  trigonometrical  sense.  Thus,  if  S  be  the  space  between  two  concentric  circles,  the 
element  dn  at  each  circumference  is  drawn  towards  its  centre ;  the  directions  of  integration  are 
as  in  §  2. 


^462  PRELIMINARY   LEMMAS  .  [216. 

all  points  within  S  and  on  its  boundary.  Then,  proceeding  as  in  §  16  and 
taking  account  of  the  conditions  to  which  P  and  Q  are  subject,  we  have 

where  V^  denotes  J^  +  |^  ,  the  double  integrals  extend  over  the  area  of  S,  and 
the  single  integral  is  taken  round  the  whole  boundary  of  S  in  the  direction 
that  is  positive  for  the  bounded  area  8. 

Let  ds  be  an  element  PT  of  arc  of  the  boundary  at  a  point  {x,  y),  and  dn 
be  an  element  TQ  of  the  normal  at  T  drawn  to  the 
interior  of  the  space  included  by  the  boundary- 
line  regarded  as  a  simple  closed  curve ;  and  let  -^ 
be  the  inclination  of  the  tangent  at  T.     Then  in 

(i)  as  TO  is  drawn  to  the  interior  of  the  area  in-        ^  P 

^  .  .  Fia    78 

eluded  by  the  curve,  the  direction  of  integration 

being  indicated  by  the  arrow  (so  that  S  lies  within  the  curve),  we  have 

dx  =  ds  cos  ^fr  -  dn  sin  -v/r,     dy  =  ds  sin  i/r  -t-  dn  cos  \/f ; 
and  therefore  it  follows  that,  for  any  function  R, 

dR        dR  .     ,      dR        , 

TT—  =  —  -;:-  sm.Y  +  '^  cos  Tir. 

dn  ox  oy 

Now  for  variations  along  the  boundary  we  have  dn  =  0,  so  that 

dR  ,      dR  ,       dR  , 

—  ^:—  ds  =  ^:— dy  — -t:--  dx.    ' 
dn  dx    ""       dy 

And  in  (ii),  as  TQ  is  drawn  to  the  interior  of  the  area  included  by  the  curve, 
the  direction  of  integration  being  indicated  by  the  arrow  (so  that  S  lies 
without  the  curve),  we  have 

dx  =  {— ds)  cos  yjr -\- dn  sin  ^Ir,     dy  =  {-ds)  sin  yfr -dn  cos  ■xjr, 

11..  dR      dR  .     ,      dR        , 

and  therefore  d'  ^  ^  ^^^'^~  dv  ^°^ '^' 

so  that,  as  before,  for  variations  along  the  boundary, 

dR  ,      dR,       dR, 

dn  ox  dy 

Hence,  with  the  conventions  as  to  the  measurement  of  dn  and  ds,  we  have 

both  integrals  being  taken  round  the  whole  boundary  of  <S  in  a  direction  that 
is  positive  as  regards  >S'.     Therefore 


216.]  FOR   SCHWARZ'S   PROOF     '  453 

In  the  same  way,  we  obtain  the  equation 

and  therefore       [  [(PV^Q  -  QV^P)  dxdy  =^(^q'^-~- P  ^^\  ds, 

where  the  double  integral  extends  over  the  whole  of  >S^,  and  the  single 
integral  is  taken  round  the  whole  boundary  of  S  in  the  direction  that  is 
positive  for  the  bounded  area  S. 

Now  let  u  be  a  potential  function  defined  as  in  the  lemma;  then  u 
satisfies  all  the  conditions  imposed  on  P,  as  well  as  the  condition  V^z^  =  0 
throughout   the  area  and  on   the  boundary.     Let  Q  =  l;    so    that  V'-^Q  =  0, 

;^  =  0.     Each  element  of  the  left-hand  side  is  zero,  and  there  is  no  dis- 

on 

continuity  in  the  values  of  P  and  Q;  the  double  integral  therefore  vanishes, 
and  we  have 

J  on 
the  result  which  was  to  be  proved. 

Ex.  1.  Let  u  be  a  potential  function  as  in  the  lemma:  and  let  the  area  S  be  the 
interior  of  a  circle  of  radius  R.  Let  two  concentric  circles  of  radii  ri  and  ?"2  be  drawn 
such  that  R'^ri>r2^0:   then 

I      u  (ri  ,(f))d(f)=  j      u  (;-2 ,4>)d4), 
a  result  due  to  Schwarz. 

Take  any  concentric  circle  of  radius  r  such  that  R>r>0;  and  consider  the  space 
between  this  circle  and  S.  The  function  u  satisfies  the  general  conditions  over  this  space, 
and  its  derivatives  satisfy  the  boundary  conditions  for  the  whole  contoiu- ;  hence 

f  dtc  ,       f  c>u  ,      - 
7^  ds+     ;^as=0, 

/  5  on         J  on 

where  the  first  integral  is  taken  in  the  counterclockwise  direction  round  S,  and  the  second 
clockwise  round  the  circle  of  radius  r.  On  account  of  the  character  of  u  over  the  whole 
of  S,  and  the  character  of  its  derivatives  along  the  circumference  of  S,  we  have 


and  therefore 


/ 


^— as=0; 

sdn 


taken  round  the  circle  of  radius  r.     But  =^=^^  along  this  circumference,  and  ds=ro 

on     or 

hence,  dropping  the  factor  r,  we  have 

'du 


or     ^ 
Integration  with  respect  to  r  between  ri  and  7-2  leads  to  the  result  stated. 


464  POTENTIAL   FUNCTION  [216. 

Ex.  2.  Let  w  be  a  potential  function  as  in  the  lemma :  and  let  the  area  S  be  the  space 
lying  without  the  circumference  of  the  circle  of  radius  R.  Let  two  concentric  circles 
of  radii  r^  and  7-2  be  drawn,  such  that  r<i>i\'^R:  a  precisely  similar  proof  leads  to  the 
result 


But  if  the  derivatives  of  m  are  not  required  to  satisfy  the  boundary 
conditions,  the  equation  in  Lemma  II  may  not  be  inferred;  we  then  have 
the  following  proposition. 

Lemma-' III.  Let  u  he  a  function,  which  is  only  required  to  satisfy  the 
general  conditions  for  an  area  S;  and  let  u  he  any  other  function,  which 
is  required  to  satisfy  the  general  conditions  for  that  area  and  may  or  may 
not  he  required  to  satisfy  the  houndary  conditions.  Let  A  he  an  area  entirely 
enclosed  in  S  and  such  that  no  point  of  its  whole  houndary  lies  on  any  part  of 
the  whole  boundary  of  S;  then 

where  the  integral  is  taken  round  the  whole  houndary  of  A  in  a  direction 
which  is  positive  with  regard  to  the  hounded  area  A,  and  the  element  dn  of 
the  normal  to  a  houndary-line  is  drawn  towards  the  interior  of  the  space 
enclosed  hy  that  houndary-line,  regarded  merely  as  a  simple  closed  curve. 

The  area  A  is  one  over  which  the  functions  u  and  11  satisfy  the  general 
conditions.  The  derivatives  of  these  functions  satisfy  the  boundary  conditions 
for  A,  because  they  are  uniform,  finite  and  continuous  for  all  points  inside  8, 
and  the  boundary  of  A  is  limited  to  lie  entirely  within  S.     Hence 

1    {ii^^u'  —  uV'^u)dxdy  =  —  \iu^ u  ^jds, 

the  integrals  respectively  referring  to  the  area  of  A  and  its  boundary  in  a 
direction  positive  as  regards  A.  But,  for  every  point  of  the  area,  V^u  =  0, 
V^u'  =  0;  and  tt  and  It' are  finite.  Hence  the  double  integral  vanishes,  and 
therefore 

taken  round  the  whole  boundary  of  A  in  the  positive  direction. 

One  of  the  most  effective  modes  of  choosing  a  region  A  of  the  above 
character  is  as  follows.  Let  a  simple  curve  Cj  be  drawn  lying  entirely  within 
the  area  S,  so  that  it  does  not  meet  the  boundary  of  S;  and  let  another 
simple  curve  Cg  be  drawn  lying  entirely  within  Cj,  so  that  it  does  not  meet 
d,  and  that  the  space  between  Cj  and  Co  lies  in  S.  This  space  is  an  area  of 
the  character  of  A,  and  it  is  such  that  for  all  internal  points,  as  well  as  for 
all  points  on  the  whole  of  its  boundary  (which  is  constituted  by  Cj  and  C2), 


216.]  DETERMINED   FOR   A   CIRCLE  465 

the  conditions  of  the  preceding  lemma  apply.  The  curve  C^  in  the  above 
integration  is  described  positively  relative  to  the  area  which  it  includes :  the 
curve  Ci  is  described,  as  in  §  2,  negatively  relative  to  the  area  which  it 
includes.     Hence,   for  such  a  space,  the  above  equation  is 

dii        ,dit\  ,        if    du        ,dii\  ,        - 

if  the  integrals  be  now  extended  round  the  two  curves  in  a  direction  that  is 
positive  relative  to  the  area  enclosed  by  each,  and  if  in  each  case  the  normal 
element  dn  be  drawn  from  the  curve  towards  the  interior. 

217.  We  now  proceed  to  prove  that  a  function  u,  required  to  satisfy  the 
general  conditions  for  an  area  included  within  a  circle,  is  uniquely  determined 
hy  the  series  of  values  assigned  to  u  along  the  circumference  of  the  circle. 

Let  the  circle  8  be  of  radius  R  and  centre  the  origin.  Take  an  internal 
point  Zo  =  re^'  (so  that  r  <  R),  and  its  inverse  z^'^r'e'^^  (such  that  rr'  =  R'^), 
so  that  V  is  external  to  the  circle.     Then  the  curves  determined  by 

z-  Zq         r 

'    ~  n^^ 

z  —  Z(,  \       ri 

for  real  values  of  X,  are  circles  which  do  not  meet  one  another.  The  boundary 
of  S  is  determined  by  A,  =  1,  and  X=  0  gives  the  point  z^  as  a  limiting  circle  : 
and  the  whole  area  of  S  is  obtained  by  making  the  real  parameter  X 
change  continuously  from  0  to  1. 

Lemma  III.  may  be  applied.  We  choose,  as  the  ring-space,  the  area 
included  between  the  two  circles  determined  by  Xj  and  X^,  where 

1  >  Xi  >  Tu  >  0, 

and  the  positive  quantities  1— Xj,  X^  can  be  made  as  small  as  we  please. 
Then  we  have 

[/    du         ,  du\  ,         r/    du        ,  du\  , 

where  the  integrals  are  taken  round  the  two  circumferences  in  the  trigono- 
metrically  positive  direction  (dn  being  in  each  case  a  normal  element  drawn 
towards  the  centre  of  its  own  circle),  and  the  function  ii'  satisfies  the  general 
and  the  boundary  conditions  for  the  ring-area  considered.  Moreover,  the 
area  between  the  circles,  determined  by  Xj  and  Xg,  is  one  for  which  u  satisfies 
the  general  conditions,  and  its  derivatives  certainly  satisfy  the  boundary 
conditions :    hence 

'du 


jrn'^'^  =  ^' 


ds2  =  0. 

dn 


Now  the  function  u  is  at  our  disposal,  subject  to  the  general  conditions 
for  the  area  between  the  two  X-circles  and  the  boundary  conditions  for  each 

F.  F.  30 


466  INTEGRAL   EXPRESSION    FOR  [217. 

of  those  circles.     All  these  conditions  are  satisfied  by  taking  u'  as  the  real 
part  of  log  {    _   °,  j ,  that  is,  in  the  present  case, 

11  =  log 


z  —  z„ 


For  all  points  on  the  outer  circle,  u'  is  equal  to  the  constant  log  ( ^  >-i ) ,  so 
that 


U'  ;r-  dSi  =  0 

on 


and  similarly  for  all  points  on  the  inner  circle,  u'  is  equal  to  the  constant 
log  ( -^  Xg ) ,  so  that 


u'  ^  ds2  =  0. 


Again,  for  a  point  z  on  the  outer  circle,  whose  angular  coordinate  is  yjf, 

the  value  of  tt—  for  an  inward  drawn  normal  is  (^  11) 
on  " 

(E^  -  r^Xj'f . 

~\R  (i?2  -  r2)  {R^  -  2RrX^  cos  (i/r  -  </>)  +  r'X,^}  ' 

and  because  the  radius  of  that   outer  circle  is  \R  {R-  —  r-)/(R^  —  r-Xi^),  we 


have 


\R{R'-r^^) 


Denoting  by  /"(A-j,  \/r)  the  value  of  m  at  this  point  yjr  on  the  circle  determined 
by  A,i,  we  have 

J  ^  d^  ^'^  =  -  J  0  ^^''^'  ^^  R^-2RrX,oos{t-cj.)  +  r"^x;^  ^^^- 
Similarly  for  the  inner  circle,  the  normal  element  again  being  drawn  towards 
its  centre,  we  have 

ju~ds,  =  -j^  f  (X,,  t)  ^.  _  2^,^  eos  (t  -  c/,)  +  r^V  ^^^^ 
Combining  these  results,  we  have 

Jo  /(^-  ^)  iJ.  _  2RrX,  cos  (t-  <^)  +  r^V  "^"^ 

In  the  analysis  which  has  established  this  equation,  Xj  and  Xg  can  have  all 
values  between  1  and  0 :  the  limiting  value  0  is  excluded  because  then  u' 
is  not  finite,  and  the  limiting  value  1  is  excluded  because  no  supposition  has 


217.]  A   POTENTIAL   FUNCTION  467 

been  made  as  to  the  character  of  the  derivatives  of  u  at  the  circumference 

But  the  equation  which  has  been  obtained  involves  only  the  values  of  u 
and  not  the  values  of  its  derivatives.  On  account  of  the  general  conditions 
satisfied  by  u,  the  values  of  u,  represented  by /(A,  i/r),  are  finite  and  continuous 
within  and  on  the  circumference  of  the  circle :  they  therefore  are  finite  and 
continuous  for  all  values  of  A,  from  0  to  1,  including  both  X.  =  0  and  \  =  1. 
Hence  the  integral 

(since  r  <  R),  is  also  finite  and  continuous  for  all  these  values  of  X,  both  X  =  0 
and  X  =  1  inclusive.  The  preceding  equation  has  been  proved  true,  however 
small  the  positive  quantities  1  —  A.;  and  A^  may  be  taken  ;  we  now  infer  that 
it  is  valid  when  we  take  Xj  =  1 ,  X.,  =  0. 

When  X2  =  0,  the  corresponding  circle  collapses  to  the  point  Zq  :  the  value 
of  /(X2,  yjr)  is  then  the  value  of  ii  at  Zq,  say  u{r,  cf>);  and  the  integral 
connected  with  the  second  circle  is  2Tru  (r,  0). 

When  Xi  =  1,  the  corresponding  circle  is  the  circle  of  radius  R ;  the  value 
■of  /(Xi,  yfr)  is  then  the  assigned  value  of  u  at  the  point  yjr  on  the  circum- 
ference, say  the  function  f{;^)-     Substituting  these  values,  we  have 

u  (r,  (/>)  =  ^  fyW  R._^nrooI{l-<^)  +  r^  ^^' 
the  integral  being  taken  positively  round  the  circumference,  of  the  circle  >S^. 

It  therefore  appears  that  the  function  u,  subjected  to  the  general 
conditions  for  the  area  of  the  ciicle,  is  uniquely  determined  by  the  values 
assigned  to  it  along  the  circumference  of  the  circle. 

The  general  conditions  for  u  imply  certain  restrictions  on  the  boundary 
values.  These  values  must  be  finite,  continuous  and  uniform :  and  therefore 
f{^),  as  a  function  of  -y^r,  must  be  finite,  continuous,  uniform  and  periodic  in 
i/r  of  period  27r. 

218.  It  is  easy  to  verify  that,  when  the  boundary  values  fi'^)  are  not 
otherwise  restricted,  all  the  conditions  attaching  to  u  are  satisfied  by  the 
function  which  the  integral  represents. 

Since  the  real  part  of  (i^e*^  +  z)/{Re'^^  —  z)  is  the  fraction 

{B?  -  r-)/{E2  -  2Rr  cos  (->/--(/))  +  r% 

it  follows  that  u  is  the  real  part  of  the  function  F  {z),  defined  by  the  equation 


„,  ,       1    [Re'^^  +  z   nt  ,\j  , 


30—2 


468  INTEGRAL  EXPRESSION   FOR  [218. 

For  all  values  of  z  such  that  \z\<'R,  the  fraction  can  be  expanded  in  a  series 
of  positive  integral  powers  of  z,  which  converges  unconditionally  and  uni- 
formly; and  therefore  F  {z)  is  a  uniform,  continuous,  analytical  function^ 
everywhere  finite  for  such  values  of  z.  Hence  all  its  derivatives  are  uniform, 
continuous,  analytical  functions,  finite  for  those  values  of  z ;  and  these 
properties   are   possessed    by    the    real    and    the    imaginary   parts    of    such 

derivatives.     Now^  „  ^,  _  ^   is  the  real   part  of  i^ ^ — --^-^ ;    and  therefore,. 

for  all  integers  m  and  n  positive  or  zero,  it  is  a  uniform,  finite  and  continuous- 
function  for  points  such  that  |^|<jR,  that  is,  for  points  within  the  circle. 
Moreover,  since  u  is  the  real  part  of  a  function  of  z,  and  has  its  differential 
coefficients  uniform,  finite  and  continuous,  it  satisfies  the  differential  equation 

To  infer  the  continuity  of  approach  of  u  (r,  <p)  to  f((f>)  as  r  is  made  equal 
to  R,  we  change  the  integral  expression  for  u  (r,  0)  into 

Moreover  for  all  values  of  r  <R  (but  not  for  r  =  R),  we  have 

1    p— *  R^-r''  ,.      1  r^       A^^r  ^       , 

^s—  ^^ — ^7d a ■>  do  =  -    tan  M  ^ tan  ^ 

27rj_^     R^  —  2Rr  COS  6  +  r^  tt  |_  [R  —  r 

and  therefore 

I=u(r,4>)-f(<py 

1    r^"-*.'...        .       .....  R^-r^ 


277 -<i) 

=  1; 


IIZ  l/<^ +  *>-/'«  SOT 


COS  6  +  r^ 


de. 


Let  (H)  denote  the  subject  of  integration  in  the  last  integral.  Then,  as  r 
is  made  to  approach  indefinitely  near  to  R  in  value,  @  becomes  infinitesimal 
for  all  values  of  6  except  those  which  are  extremely  small,  say  for  values  of  6 
between  —h  and  +  S.  Dividing  the  integral  into  the  corresponding  parts,, 
we  have 

i=J-     %de  +  ^\      ede  +  —-      me. 

Let  i¥be  the  greatest  value  of  f(^)  for  points  along  the  circle.     Then  the 
first  integral  and  the  second  integral  are  less  than 

ct>-S  R^-r^  and^"-^-^2ilf  ^^-^ 

2vr  (R-ry  +  2Rr{l-cosS)  27r  {R  -  rf  +  2Rr  (1  -  cos  8) 

respectively ;    by  taking  r  indefinitely  near  to  R  in  value,  these  quantities 
can  be  made  as  small  as  we  please.     For  the  third  integral,  let  k  be  the 
greatest  value  of f  ((f)  +  6)  —/(cp)  for  values  of  0  between  8  and  —  8  :  then  the: 
third  integral  is  less  than 

k    /•«  R'-r'  ,. 

aff, 


27r/_.  R' -  2Rr  cos  6  +  r^ 


'218.] 


POTENTIAL   FUNCTION 


469 


"k            f\K-\-r\ 
that  is,  it  is  less  than  —  tan~^  (  ^  -^ 8  j ;  so  that,  when  r  is  made  nearly 

equal  to  R,  the  third  integral  is  less  than  k. 

If  then  k  be  infinitesimal,  as  is  the  case  when  f{<^)  is  everywhere  finite 
and  continuous,  the  quantity  /  can  be  diminished  indefinitely ;  hence  u  (r,  ^) 
continuously  changes  into  the  function  f(<j))  as  r  is  made  equal  to  R.  The 
verification  that  the  function,  defined  by  the  integral,  does  satisfy  the  general 
conditions  for  the  area  of  the  circle  and  assumes  the  assigned  values  along 
the  circumference  is  thus  complete. 

Ex.  1.  It  will  be  convenient  to  possess  an  upper  limit  for  \u{r,  <i>)-u  (0)  |  for  the 
circumference  of  a  circle  of  radius  r,  concentric  with  the  given  circle,  r  being  less  than  R : 
say  R  —  r'^p,  where  p  is  a  quantity  that  may  not  be  made  as  small  as  we  please. 


We  have 


and  clearly 


so  that 


^^'^^^  =  liy^'^^m-2Rr 


R^-r^ 


cos  {■\f^  —  (f>)  +  r^ 


df, 


uiO)  =  - 


fWd^; 


dy^. 


To  indicate  one  upper  limit  for  the  modulus  of  the  right-hand  side,  we  can  proceed  as 
follows.  Let  0  be  the  common  centre  of  the  two  circles ;  P  the  point  {R,  0)  on  the  outer 
circle,  Q  the  point  {r,  y\r)  on  the  inner  circle ;  and  let  x  denote 
the  angle  between  QP  and  OQ  produced.     Then 

R  cos  (•(//■  —  0)  —  r=PQ  cos  Xi 
so  that 

27r{«(r,  <^)-«(0)}  =  2rj^  f  {^) -^  d^. 

Let  M  be  the  maximum  value  of  \f{'^)\  along  the  circum- 
ference of  the  outer  circle :  then  an  upper  limit  for  the  modulus 
of  the  right-hand  side  will  be  given  by  taking/(-v|r)  =  J/"  when  cos;^ 
is  positive  and  /(■</')=  —  i/"when  cos;^  is  negative.    Writing  -^-(^  =  6,  we  divide  the  range 

T 

of  integration  into  two  parts;  viz.  6=—a  to  +a,  where  cosa  =  -n;  and  d  =  a  to  2Tr-a. 
Over  the  former,  we  take  f{y\t)  =  M;  over  the  latter  /(>//■)=  —M. 

Returning  now  to  the  initial  expressions,  with  these  values,  we  have 

R^-r^ 


2n{u{r,  (j))-u{0)}\<M 

+  M 
<2M 


R^-2Rrcos0  +  r^ 

'2Tr-a  /  ^2  —  ^2 


1  ]dd 


R"^ 


+  : 


R^-r'' 


2Rr  COB  6  +  r^ 
-l]dd 


1- 


R^-2Rrcos6-i-r^ 


+  l]dd 


d6. 


470  DISCONTINUOUS   VALUES  [218. 

Now 


J: 


^^-^^  ..;^  =  2tan-|5+-^an^] 


B? - ^Rr cos 6 +  r'^  (i?-r        2j  ' 

so  that 


2iT  {?*  (r,  (^)  -  M  (0)}  I  <  2M 


2  tan" 


^3^^V"" 


+  2Jf[,r-a-|.r-2tan-i(|±^)y^| 


<4Jf[2tan-i(|±-;^,)-< 

where  ?/2  =  tan^a,  and  for  the  inverse  function  we  take  the  smallest  positive  angle  with 
the  indicated  tangent.     Now 

2r  tan  - 
tan~i  ( "n tan  \a]  —  ^a  =  tan~i 


(i?-r)  +  (i?  +  r)tan2| 


r  sm  a 
:tan" 


^  - r  cos  a 


=tan""i  \ r^  =  sin~i  ^, 

I(i22_r2)ij  ^ 

and  therefore 

AM  r 

I  ti  (r,  <^)  —  %  (0)  I  < sin  ~  -^  -p . 

The  form*  obtained  is  valid  for  values  of  r  such  that  Q^r^R-p,  where  p  is  a  finite 
quantity  that  may  be  not  large  but  cannot  be  made  as  small  as  we  please. 

Ex.  2.  Shew  that,  if  M  denote  the  maximum  value  (supposed  positive)  of  /(>//•)  for 
points  along  the  circumference  of  the  circle  and  if  u  (0)  vanish,  then 

4  T 

tf  (r,  <^)<— iftan-i -p.  (Schwarz.) 

77  it 

219.  But  in  view  of  subsequent  investigations,  it  is  important  to  consider 
the  function  represented  by  the  integral  when  the  periodic  function  /(<^) 
which  occurs  therein  is  not  continuous,  though  still  finite,  for  all  points  on 
the  circumference.  The  contemplated  modification  in  the  continuity  is  that 
which  is  caused  by  a  sudden  change  in  value  of /(</>)  as  <^  passes  through  a 
value  a  :  we  shall  have 

f{a^e)-f{a-e)  =  A, 

when  €  is  ultimately  zero.     Then  the  following  proposition  holds  : — 

Let  a  function  /(</>)  he  periodic  in  Itt,  finite  everywhere  along  the  circle, 
and  continuous  save  at  an  assigned  point  a  where  it  undergoes  a  sudden 
increase  in  value :  a  function  u  can  be  obtained,  which  satisfies  the  general 
conditions  for  the  circle  except  at  such  a  point  of  discontinuity  in  the  value  of 
f  {(}>),  and  acquires  the  values  off(^)  along  the  circumference. 

*  It  is  due  to  Schwarz,   Ges.  Werke,  t.  ii,  p.  190. 


#    • 


219.]  ALONG   THE   CIRCUMFERENCE  471 

Let  p  be  a  quantity  <R:  then  along  the  circumference  of  a  circle  of  radius 
p,  the  general  conditions  are  everywhere  satisfied  for  the  function  u,  so  that,  if 
a  {p,  t/t)  be  the  value  at  any  point  of  its  circumference,  the  value  of  u  at  any 
internal  point  is  given  by 

''  ^^'  ^^  =  ^  i'J  ''  ^P'  ^>  p^-2prcL~i^-<f.)  +  r^  ^^- 

Now  p  can  be  gradually  increased  towards  R,  because  the  general  conditions 
are  satisfied ;  but,  when  p  is  actually  equal  to  R,  the  continuity  of  u  {p,  -yjr)  is 
affected  at  the  point  a.  We  choose  a  finite  arbitrary  quantity  e,  which  can 
be  made  as  small  as  we  please ;  and  we  divide  the  integral  into  three  parts, 
viz.,  0  to  a  —  e,  a  —  e  to  a  +  e,  and  a  +  e  to  'Itt,  Avhen  p  is  very  nearly  equal  to 
R.  For  the  first  and  the  third  of  these  parts,  p  can,  as  in  the  preceding 
investigation,  be  changed  continuously  into  R  without  affecting  the  value  of 
the  integral.     If  we  denote  by  p  the  integral 

where  the  range  of  integration  does  not  include  the  part  from  a  —  e  to  a  +  e, 
and  where  the  values  /(a  —  e),  /  (a  +  e)  are  assigned  to  u  (R,  a.  —  e),  u'  (R,  a.  +  e), 
respectively ;  the  sum  of  the  integrals  for  the  first  and  the  third  intervals  is 
p  +  A,  where  A  is  a  quantity  that  vanishes  with  R  —  p,  because  the  subject  of 
integration  is  everywhere  finite.  For  the  second  interval,  the  integral  is 
equal  to  q+  A',  where 

1    /•«+^  R^  —  r^ 

^  =  2^]  ._/^^>i?^-^/^7cos(^-<^)+r^^^' 

and  A'  is  a  quantity  vanishing  with  R  —  p  because  the  subject  of  integration 
is  everywhere  finite.  So  far  as  concerns  q,  let  M  be  the  greatest  value  of 
Ifi^jr)  j,  so"  that  M  is  finite  ,  then 

a  quantity  which,  because  of  the  mode  of  occurrence  of  the  arbitrary  quantity 
e,  can  be  made  less  than  any  finite  quantity,  however  small,  provided  r  is 
never  actually  equal  to  R.  If  then,  an  infinitesimal  arc  from  a  —  e  to  a  +  e 
be  drawn  so  as,  except  at  its  assigned  extremities,  to  lie  within  the  area  of 
the  circle,  the  last  proviso  is  satisfied  :  and  the  effect  is  practically  to  exclude 
the  point  a  from  the  region  of  variation  of  m  as  a  point  for  which  the  function 
is  not  precisely  defined.     With  this  convention,  we  therefore  have 

"  ('■■  ■^)  -  iijr'  ^^-  ^^  R'  -  ^bZh"^  -  ^) + ..  -^t = a + A' + ,, 

so  that,  by  making  p  ultimately  equal  to  R  and  e  as  small  as  we  please,  the 
difference  between  u  (r,  </>)  and  the  integral  defined  as  above  can  be  made  less 


472  DISCONTINUITY   IN   VALUE  [219. 

than  any  assigned  quantity,  however  small.  Hence  the  integral  is,  as  before, 
equal  to  the  function  u  (r,  (f>),  provided  that  the  point  a  be  excluded  from  the 
range  of  integration,  the  value  /(a  — e)  just  before  ■^  =  (x  and  the  value 
/(a  +  e)  just  after  ■ylr=a  being  assigned  to  u{R,ylr). 

It  therefore  appears  that  discontinuities  may  occur  in  the  boundary 
values  when  the  change  is  a  finite  change  at  a  point,  provided  that  all 
the  values  assigned  to  the  boundary  function  be  finite. 

Corollary.  The  boundary  value  may  have  any  limited  number  of  points 
of  discontinuity,  provided  that  no  value  of  the  function  be  infinite  and  that  at 
all  points  other  than  those  of  discontinuity  the  periodic  function  be  uniform, 
finite  and  continuous :  and  the  integral  will  then  represent  a  p)otential  function 
satisfying  the  general  conditions. 

The  above  analysis  indicates  why  discontinuities,  in  the  form  of  infinite 
values  at  the  boundary,  must  be  excluded :  for,  in  the  vicinity  of  such  a 
point,  the  quantity  M  can  have  an  infinite  value  and  the  corresponding 
integral  does  not  then  necessarily  vanish.  Hence,  for  example,  the  real 
part  of 


eRe"^^  -  Re"'' 


is  not  a  function  that,  under  the  assigned  conditions,  can  be  made  a  boundary 
value  for  the  function  u. 

There  is  however  a  different  method  of  taking  account  of  the  discon- 
tinuities ;  it  consists  in  associating  other  particular  functions,  each  having 
one  (and  only  one)  discontinuity,  taken  in  turn  to  be  the  assigned  discontinuities 
of  the  required  functions,  and  thus  modifying  the  boundary  conditions.  The 
method  might  not  prove  the  simplest  way  of  proceeding  in  any  special  case, 
because  of  the  substantial  modification  of  those  conditions ;  but  this  is  of 
relatively  less  importance  in  the  establishment  of  an  existence-theorem,  which 
is  the  present  quest. 

Two  cases  arise,  according  as  the  assigned  discontinuity  in  value  takes 
place  at  a  point  of  continuity  in.  the  curvature  of  the  boundary,  or  at  a 
point  of  discontinuity  in  the  curvature. 

In  the  former,  when  the  discontinuity  is  required  to  occur  at  a  point  of 
continuous  curvature,  we  know  (§  3)  that  the  argu- 
ment of  a  point  experiences  a  sudden  change  by  tt 
when  the  path  of  the  point  passes  through  the 
origin.  Let  a  point  P  on  a  circle  (fig.  79,  i)  be 
considered  relative  to  ^4  :  the  inclination  of  -4 P  to 

TT 

the  normal,  drawn  inwards  at  A,  is  -^  —  -g-  (a  —  0), 


219.] 


ALONG   THE    CIRCUMFERENCE 


473 


aad  o{  AQ  to  the  same  line  is  — 


_  1 


!(«-</>') 


,  so  that  there  is  a  sudden 


change  by  tt  in  that  inclination.     Now,  taking  a  function 


9i4>)=--  *an-^ 


tan  ||  -  1  («  -  </>)}    ' 


and  limiting  the  angle,  defined  by  the  inverse  function,  so  that  it  lies 
between  -  ^tt  and  +^77,  as  may  be  done  in  the  above  case  and  as  is 
justifiable  with  an  argument  determined  inversely  by  its  tangent,  the 
function  g{<^)  undergoes  a  sudden  change  J.  as  </>  increases  through  the 
value  a.  Moreover,  all  the  values  of  g  ((p)  are  finite :  hence  g  (0)  is  a 
function  which  can  be  made  a  boundary  value  for  the  function  u.  Let  the 
function  thence  determined  be  denoted  by  iia- 

By  means  of  the  functions  u^,  we  can  express  the  value  of  a  function  u 
whose  boundary  value  /{(p)  has  a  limited  number  of  permissible  discontinuities. 
Let  the  increases  in  value  be  A^, ... ,  A^  at  the  points  a^,  a2,  ...,  a.m  respect- 
ively :  then,  if  gn  {(f))  denote 


tan 

TT 


t&n\--^(an-(f)) 


we  have  gn  (an  +  e)  —  gn  (a«  —  e)  =  An,  when  e  is  infinitesimal.     Hence 

/(«,„  +  f )  -/ (««  -  e)  -  [gn  (an  +  e)  -  gn  (««  -  e)} 
has  no  discontinuity  at  a„,  that  is,  f((f>)  —  gni'P)  has  no  discontinuity  at  «„. 

m 

Hence  also  /(</>)—  2  gn{4>)   l^^s   no   discontinuity   at  a^,  ...,  a^,   and 

n  =  l 

therefore  it  is  uniform,  finite,  and  continuous  everywhere  along  the  circle ; 
and  it  is  periodic  in  27r.  By  §  218,  it  determines  a  function  U  which  satisfies 
the  general  conditions. 

Each    of  the    functions  gn{4>)  determines  a  function  Un  satisfying  the 
general  conditions :    hence,  as  u  is  determined  by  /(^),  we  have 

m 
It  -    2    Un  =   U, 
n  =  \ 

which  gives  an  expression  for  u  in  terms  of  the  simpler  functions  Un  and  of  a 
function  U  determined  by  simpler  conditions  as  in  §  218. 

Ex.     Shew  that,  if  /(\/^)  =  l  from  -Itt  to  +\iv  and  =0  from  +\-n-  to  f  tt,  then  u  is  the 

real  part  of  the  function 

1  ,      \+iz- 
-T-  log  -. ; 


and  obtain  a  corresponding  expression  for  a  function,  which  is  equal  to  1  from  -  a  to  +  a, 
and  is  equal  to  0  from  a  to  27r  —  a. 


474  GENERAL    _  [219. 

In  the  second  case,  when  a  discontinuity  (say  K  in  magnitude)  is  required 
to  occur  at  a  point  of  sudden  change  in 
curvature,  we  proceed   similarly.     Let  A 
be   such   a  point,  and  suppose  the  curve 
(fig.  79,  ii)  referred  to  0  as  pole :  as  in  the 
first  case,  we  consider  the  point  P  relative 
to  A.     Let   6   denote  the   internal   angle 
QAP,  so  that   e  is    not  equal   to   ir ;    let 
TPO  =  ct>,,  AQO=^(Po,  P00'=  d„  A00'=  a, 
QOO'  =  02-     Then  the  inclination  of  AP  to 
AO  is  (/)i  —  (a  —  ^i),  and  the  inclination  of 
AQ  to  AO  is  —  {tt  —  c])2-  {S.2  —  oi)},  so  that,  in  passing  from  the  direction  AP 
to  the  direction  A  Q,  there  is  a  sudden  change  of  </>!  —  ^2  +  tt,  which  is  e  in 
the  limit  when  P  and  Q  move  up  to  A.     Accordingly,  if  we  take  a  function 

—  tan-i  [tan  ((/>-«+  6)], 

where  6  is  the  polar  angle  and  ^  is  the  inclination  of  the  (backwards-drawn) 
tangent  to  the  radius  vector,  this  function  suddenly  increases  in  value  by  K 
as  the  point  P  passes  through  A  towards  Q. 

The  result  is  used  in  exactly  the  same  way  as  in  the  preceding  case :  and 
we  obtain  a  new  function,  assigned  as  the  succession  of  boundary  values,  the 
new  (boundary)  function  being  free  from  all  discontinuities. 

Ex.     Obtain  the  expression  of  such  a  function  in  the  case  of  two  equal  circles,  when 
the  boundary  curve  consists  of  the  two  arcs  each  external  to  the  other. 

220.  The  general  inference  from  the  investigation  therefore  is,  that  a 
function  of  two  real  variables  x  and  y  is  uniquely  determined  for  all  points 
within  a  circle  by  the  following  conditions : — 

(i)     at  all  points  within  the  circle,  the  function  u  and  its  derivatives 

du    du    d^u     d'U  ^    ,  -r  12    -^         j  ^-  ^ 

TT-  ,  TT-  ,  ^:r-„,  ;:r^   must   be  unitorm,  finite  and    continuous,  and 

ox    oy    dor    oy^ 

must  satisfy  the  equation  V-w  =  0  : 

(ii)  if /((^)  denote  a  function,  which  is  periodic  in  </>  of  period  27r,  is 
finite  everywhere  as  the  point  ^  moves  along  the  circumference, 
is  continuous  and  uniform  at  all  except  a  limited  number  of 
isolated  points  on  the  circle,  and  at  those  excepted  points 
undergoes  a  sudden  prescribed  (finite)  change  of  value,  then 
to  u  is  assigned  the  value /(</))  at  all  points  on  the  circumference 
except  at  the  limited  number  of  points  of  discontinuity  of  that 
boundary  function. 

And  an  analytical  expression  has  been  obtained,  the  function  represented  by 
which  has  been  verified  to  satisfy  the  above  conditions. 


220.]  PROPERTIES  475 

We  now  proceed  to  obtain  some  important  results  relating  to  a  function 
u,  defined  by  the  preceding  conditions. 

I.  The  value  of  u  at  the  centre  of  the  circle  is  the  arithmetic  mean  of  its 
values  along  the  circunference. 

For,  by  taking  r  =  0,  we  have 

the  right-hand  side  being  the  arithmetic  mean  along  the  circumference. 

II.  If  the  function  he  a  uniform  constant  along  the  circumference,  it  is 
equal  to  that  constant  everywhere  in  the  interior. 

For,  let  C  denote  the  uniform  constant ;  then 

^^^''^^  =  2^7^ Jo  R^-2Rroo,^^-^)  +  r^'^^ 

=  G, 

for  all  values  of  r  less  than  R,  that  is,  everywhere  in  the  interior. 

But  if  the  function,  though  not  varying  continuously  along  the  circum- 
ference, should  have  different  constant  values  in  different  finite  parts,  as,  for 
instance,  in  the  example  in  |  219,  then  the  inference  can  no  longer  be  drawn. 

III.  //  the  function  he  uniform,  finite  and  continuous  everywhere  in  the 
plane,  it  is  a  constant. 

Since  the  function  is  everywhere  uniform,  finite  and  continuous,  the 
radius  R  of  the  circle  of  definition  can  be  made  infinitely  large :  then,  as 
the  limit   of  the   fraction  {R^— r')/{R^— 2 Rr  cos  {-yfr  — (f))  + 7^^}   is   unity,  we 

have 

1    [^' 
U  (r,  c}))  =  ^r—  I       u(cc  ,  i/r)  d-ylr, 

277  J   Q 

the  integral  being  taken  round  a  circle  of  infinite  radius  whose  centre  is  the 
origin.  But,  by  (I.)  above,  the  right-hand  integral  is  u  (0),  the  value  at  the 
centre  of  the  circle  ;  so  that 

u  (r,  (J3)  =  u  (0), 
and  therefore  u  has  the  same  value  everywhere. 

This  is  practically  a  verification  of  the  proposition  in  §  40,  that  a  uniform, 
finite  and  continuous  function  w,  which  has  no  infinity  anywhere,  is  a  constant. 

IV.  A  uniform,  finite  and  continuous  function  u  cannot  have  a  n^aximum 
value  or  a  minimum,  value  at  any  point  in  the  interior  of  a  region  over  which, 
subject  to  the  general  conditions  as  to  the  differential  coefficients,  it  satisfies  the 
differential  equation  Vht  =  0. 


476  MAXIMUM   OR   MINIMUM   VALUES  [220. 

If  there  be  any  such  point  not  on  the  boundary,  it  can  be  surrounded  by 
an  infinitesimal  circle  for  the  interior  of  which,  as  well  as  for  the  circum- 
ference of  which,  u  satisfies  both  the  general  and  the  boundary  conditions ; 
hence 

the  integral  being  taken  round  the  circumference.     But  in  the  immediate 

vicinity  of  such   a  point,  ^    has    everywhere  the   same   sign,   so  that  the 

integral  cannot  vanish :   hence  there  is  no  such  point  in  the  interior. 

In  the  same  way,  it  may  be  proved  that  there  cannot  be  a  line  of 
maximum  value  or  a  line  of  minimum  value  within  the  surface :  and  that 
there  cannot  be  an  area  of  maximum  value  or  an  area  of  minimum  value 
within  the  surface. 

V.  It  therefore  follows  that  the  maximum  values  for  any  region  are  to  he 
found  on  its  boundary :  and  so  also  are  the  minimum  values. 

If  M  be  the  maximum  value,  and  if  m  be  the  minimum  value,  of  the 
function  for  points  along  the  boundary,  then  the  value  of  the  function  for  an 
interior  point  is  <  ilf  and  is  >  m  and  can  therefore  be  represented  in  the  form 
Mp  +  m(l  —  p),  where  p  is  'a  real  positive  proper  fraction,  varying  from  point 
to  point.  Also  p  never  vanishes,  except  for  the  minimum  on  the  boundary ; 
and  it  is  never  so  great  as  unity,  except  for  the  maximum  on  the  boundary. 

In  particular,  let  a  function  have  the  value  zero  for  a  part  of  the 
boundary  and  have  the  value  unity  for  the  rest,  the  points  (if  any)  where  the 
sudden  change  from  0  to  1  takes  place  being  cut  off  as  in  §  219.  Let  a  line 
be  drawn  from  any  point  of  the  boundary,  through  the  interior,  to  any  other 
point  of  the  boundary ;  and  at  each  extremity  let  it  cut  the  boundary  at  a 
finite  angle.  The  value,  which  the  function  has  for  points  along  the  line,  in 
the  interior  is  always  positive  and  has  an  upper  limit  q,  a  proper  fraction. 
But  q  will  vary  from  one  line  to  another.  If  the  region  be  a  circle  and  q  be 
the  proper  fraction  for  a  line  in  the  circle,  then  the  value  along  that  line  of  a 
function  u,  which  is  still  zero  over  the  former  part  of  the  boundary  but  has  a 
varying  positive  value  ^  fj.  along  the  remainder,  is  evidently  ^  q/x.  This 
fraction  q  may  be  called  the  fractional  factor  for  the  line  in  the  supposed 
distribution  of  boundary  values. 

Again,  let  a  function  have  a  value  zero  over  part  of  the  boundary,  and 
have  positive  values  over  the  rest  of  it,  the  greatest  of  them  being  unity:  and 
suppose  that  there  is  no  sudden  discontinuity  in  value.  When  a  line  is  drawn 
(as  above)  through  the  area,  both  of  its  extremities  being  at  zero  values 
on  the  boundary,  let  the  value  of  the  function  along  the  line  be  q',  where  q' 
varies  from  point  to  point ;  it  is  zero  at  the  extremities  of  the  line,  and  it  is 


220.]  FRACTIONAL   FACTORS  477 

never  so  great  as  1,  because  a  maximum  value  cannot  occur  in  the  interior; 
hence  O^q  ^Q,  where  Q  is  a  real  positive  quantity,  greater  than  0  and  less 
than  1. 

The  fraction  Q  is  the  fractional  factor  for  the  line. 

Lastly,  let  a  function  have  a  value  zero  over  part  of  the  boundary,  and 
have  values  over  the  rest  of  it,  some  positive,  others  negative ;  let  —  C  be  the 
minimum,  and  +  D  the  maximum,  where  C  > 0,  D  >0;  and  suppose  that 
there  is  no  sudden  discontinuity  in  value.  Let  a  line  be  drawn  (as  above) 
through  the  area,  beginning  at  one  point  of  zero  value  and  ending  at  another  ; 
the  value  of  the  function  along  Jbhis  line  is  —  C  +  q"  (D  +  C),  where  q"  varies 
from  point  to  point  along  the  line.  Now  q"  cannot  be  as  small  as  0,  for  the 
minimum  —  C  is  not  to  be  found  on  the  line ;  nor  can  it  be  as  large  as  1,  for 
the  maximum  D  is  not  to  be  found  on  the  line.     Hence 

0<R^q"^Q<l, 

where  E  is  a  real  positive  quantity  greater  than  0  and  less  than  Q,  and  Q  is 
a  real  positive  quantity  less  than  1. 

The  fractions  Q  and  R  may  be  called  the  major  and  the  minor  fractional 
factors  for  the  supposed  distribution  of  boundary  values  ;  they  are  not. 
necessarily  independent  of  C  and  D. 

VI.  It  may  be  noted  that  the  second  of  these  propositions  can  now 
be  deduced  for  any  simply  connected  surface.  For  when  a  function  is 
constant  along  the  boundary,  its  maximum  value  and  its  minimum  value 
are  the  same,  say  X :  then  its  value  at  any  point  in  the  interior  is 
X-p  +  X  (1  —  p),  that  is,  X,  the  same  as  at  the  boundary.  Consequently  if 
two  functions  u-^  and  U2  satisfy  the  general  conditions  over  any  region,  and 
if  they  have  the  same  value  at  all  points  along  the  boundary,  then  they 
are  the  same  for  all  points  of  the  region.  For  their  difference  satisfies 
the  general  conditions :  it  is  zero  everywhere  along  the  boundary :  hence 
it  is  zero  over  the  whole  of  the  bounded  region. 

If,  then,  a  function  u  satisfy  the  general  conditions  for  any  region,  it  is 
unique  for  assigned  boundary  values  that  are  everywhere  finite,  uniform,  and 
continuous  except  at  isolated  points. 

221.  The  explicit  expression  of  u  with  boundary  values,  that  are 
arbitrary  within  the  assigned  limits,  has  been  determined  for  the  area 
enclosed  by  a  circle :  the  determination  being  partially  dependent  upon  the 
form  assumed  in  §  217  for  the  subsidiary  function  u'.  The  assumption  of 
other  forms  for  ii,  leading  to  other  curves  dependent  upon  a  parametric 
constant,  would  lead  by  a  similar  process  to  the  determination  of  u  for  the 
area  limited  by  such  families  of  curves. 

But  without  entering  into  the  details  of  such  alternative  forms  for  il  ,  we 
can  determine   the   value   of  u,  under  corresponding  conditions,  for  curves 


478  EXISTENCE    FOR  [221. 

derivable  from  the  circle  by  the  principle  of  conformal  representation*. 
Suppose  that,  by  means  of  a  relation 

^  =  ^  (0  =  ^  (I  +  iv), 
or,  say  x  +  iy  =p  (f,  r))  +  iq  (^,  r]), 

where  p  and  q  are  real  functions  of  f  and  r],  the  area  contained  within  the 
circle  is  transformed,  point  by  point,  into  the  area  contained  within  another 
curve  which  is  the  transformation  of  the  circle :  then  the  function  u  {x,  y) 
becomes,  after  substitution  for  x  and  y  in  terms  of  ^  and  r],  a  function,  say  U, 
of  ^  and  7}. 

Owing  to  the  character  of  the  geometrical  transformation,  p  and  q  (and 
their  derivatives  with  regard  to  f  and  rf)  are  uniform,  finite  and  continuous 
within  corresponding  areas.     Hence 

U{^,r))  =  u{x,y); 

dU  _dudp      dudq        dU  _dudp      dudq 
9^       dxd^      dyd^'       dr]       dx  drj      dydrj' 

so  that  the  function  IT.  satisfies  the  general  conditions  for  the  new  area 
bounded  by  the  new  curve. 

Moreover,  u  has  assigned  values  along  the  circular  boundary  which  is 
transformed,  point  by  point,  into  the  new  boundary ;  hence  U  has  those 
assigned  values  at  the  corresponding  points  along  the  new  boundary.  Thus 
the  function  U  is  uniquely  determined  for  the  new  area  by  conditions  which 
are  exactly  similar  to  those  that  determine  u  for  a  circle :  and  therefore  the 
potential  function  is  uniquely  determined  for  any  area,  which  can  he  con- 
formally  represented  on  the  area  of  a  circle,  by  the  general  conditions  of 
§  216  and  the  assignment  of  values  that  are  finite  and,  except  at  a  limited 
number  of  isolated  points  where  they  may  suffer  sudden  (finite)  changes  of 
value,  uniform  and  continuous  at  all  points  along  the  boundary  of  the  area. 

One  or  two  examples  of  very  special  cases  are  given,  merely  by  way  of 
illustration.  The  general  theory  of  the  transformation  of  a  circle  or  an 
infinite  straight  line  into  an  analytical  curve  will  be  considered  in  Chapter 
XX.  But,  meanwhile,  it  is  sufiicient  to  indicate  that,  by  the  principle  of 
conformal  representation,  w^e  can  pass  from  the  circle  to  more  general  curves 
as  the  boundary  of  to  area  within  which  the  potential  function  is  defined  by 
conditions  similar  to  those  for  a  circle :  in  particular  that,  by  assuming  the 
result  of  §§  265,  266,  we  can  pass  from  the  circle  to  an  analytical  curve  as  the 
boundary  of  such  an  area. 

*  The  general  idea  of  the  principle,  and  some  illustrations  of  it,  as  expounded  in 
Chapters  XIX.  and  XX.,  will  be  assumed  known  in  the  argument  which  follows:  see  especially 
§§  265,  266. 


221.]  ANALYTICAL  CURVE  479 

Ex.  1.  A  function  u  satisfying  the  general  conditions  for  a  circle  of  radius  unity  and 
centre  the  origin,  and  having  assigned  values  /(^)  along  the  circumference,  is  determined 
at  any  internal  point  by  the  equation 


^  ^^'  *^)  =  ^  /r^^^^^  l-2rcol(^^-0)  +  r2^^- 


Now  the  circle  and  its  interior  are  transformed  by  the  equation 

2+1  =  — 

into  a  parabola  and  the  excluded  area  (§  257) :  so  that,  if  R,  6  be  polar  coordinates  of 
any  point  in  that  excluded  area,  we  have 

rcos0  =  2i2~4cos^^  — 1,         rsin(^=  —  %R~^sm\6. 

Corresponding  to  the  circle  r=l,  we  have  the  parabola 

if  e  determine  the  point  on  the  parabola,  which  corresponds  to  y\r  on  the  circle,  we  have 

cos  a//' =  2  cos^ -^  9  - 1 , 
or  ■v//-  =  e. 

Hence  the  function  U  {R,  e)  assumes  the  values  /(0)  along  the  boundary  of  the 
parabola. 

Also  l-r2=^(i2lcosi^-l), 

1  -  2r  cos  (\|^  -  0)  +  r2  =  --  [R  cos^  |e  -.2/2*  cos  |e  cos  ^  (e  +  (9)  + 1]  ; 

and  therefore  we  have  the  following  result : — 

A  function  which  satisfies  the  general  conditions  for  the  area  bounded  by  and  lying  on  the 
convex  side  of  the  parabola  R  cos'^  ^  9  =  1  and  is  required  to  assume  the  value  f  (9)  at  points 
along  the  parabola,  is  defined  uniqziely  for  a  point  {r,  6)  external  to  the  parabola  by  the 
integral 

"'V(9)  , rioosie-l ^ ^^ 

0  l-2r2cos^9cosi(9  +  ^)  +  rcos2|9 

The  function  /(9)  may  suffer  finite  discontinuities  in  value  at  isolated  points:  elsewhere 
it  must  be  finite,  continuous  and  uniform. 


27rj( 


Ex.  2.     Obtain  an  expression  for  u  at  points  within  the  area  of  the  same  parabola, 
by  using 

0  =  tan2(i7rC2) 

as  the  equation  of  transformation  of  areas  (§  257). 
Ex.  3.     When  the  equation 

i+C 
is  used,  then,  \i  z=x  +  iy  and  f=X+il^,  we  have 

_   _l_2'2-F2  +  z"2A" 

X  +  ty=        j^2  +  (l+l^)2^- 

If  the  point  (  describe  the  whole  length  of  the  axis  of  X  from   -co  to  +qo,  so  that 
^e  may  take  ^=A'=tan0  with  <^  increasing  from   -^tt  to  +^7r,  we  have  ^= cos  20, 


480  COMBINATION  [221. 

3/  =  sin  20;  and  z  describes  the  whole  circumference  of  a  circle,  centre  the  origin  and 
radius  unity,-  in  a  trigonometrically  positive  direction  beginning  at  the  point  ( —  1,  0). 
We  easily  find 

r  cos  6  _    r  sin  6    _  r^  1 

1-/^2  -  2R  cos  e  ^  l-2ii:sine  +  ^2  =  n-2ii;sine  +  i22 ' 

where  X=  R  cos  0,  F=  ^  sin  0.  Moreover,  for  variations  along  the  circumference,  we 
have  \lr  =  2(f)  ;  whence,  substituting  and  denoting  by  F{x),  =/(2  tan~i.r),  the  value  of 
the  potential  at  a  point  on  the  axis  of  real  quantities  whose  abscissa  is  x,  we  ultimately 
find 


,  „  ,  1       r  R  Sm  0  r,  ,     s     ^ 

u  (R,  0)  =  -  I        -7^ — - — n 5  -T  (x)  ax. 


as  the  value  of  the  potential-function  u  sX  -a.  point  [R,  0)  in  the  upper  half  of  the  plane, 
when  it  has  assigned  values  F{x)  at  points  along  the  axis  of  real  variables. 

222.  The  function  u  has  now  been  determined,  by  means  of  the  general 
conditions  within  an  area  and  the  assigned  boundary  values,  for  each  space 
obtained  by  the  method  indicated  in  |  221.  But  the  determination  is 
unique  and  distinct  for  each  space  thus  derived;  and,  if  two  such  spaces 
have  a  common  part,  there  are  distinct  functions  xi.  We  now  proceed  to 
shew  that  when  two  spaces,  for  each  of  which  alone  a  function  u  can  be 
determined,  have  a  common  part  which  is  not  merely  a  point  or  a  line, 
then  the  function  u  is  uniquely  determined  for  the  combined  area  by  the 
assignment  of  finite,  uniform  and  continuous  values  {or  partially  discontinuous 
values,  as  in  §  219)  along  the  boundary  of  the  combined  area. 

Let  the  spaces  be  Tj  and  T2  having  a  common  part  T,  so  that  the  whole 
space  can  be  taken  in  the  form  T^  +  T2  —  T; 
and  suppose  that  the  boundaries  of  T^  and 
Tg  cut  at  finite  angles  at  A  and  B,  that  is, 
that  they  do  not  touch  one  another  there. 
Let  the  part  of  the  boundary  of  Ti  without 
T2  be  Zo,  and  the  part  within  T2  be  Lo :  and 
similarly,  for  the  boundary  of  T^,  let  Zj  de- 
note the  part  within  T^  and  Xg  the  part  ~^  ^jg.  go. 
without  it.     Then  the  boundary  of 

T.  +  T^-T 

is  made  up  of  Zq  and  X3 :  the  boundary  of  T  is  made  up  of  L^  and  Lo. 

Let  any  series  of  values  be  assigned  along  Lq  and  L3  subject  to  the 
conditions  of  being  uniform,  finite  everywhere,  and  discontinuous,  if  at  all, 
only  at  a  limited  number  of  isolated  points.  The  method  of  §  218  can  be 
used  to  remove  these  discontinuities,  whether  they  occur  at  points  of 
continuous  curvature,  or  at  points  (such  as  A  and  B,  fig.  80)  of  discontinuous 
curvature ;  for  this  purpose  we  take  a  new  function  of  the  form 

U  —  ItUr,    =  U, 


222.]  OF    AREAS  481 

where  each  of  the  functions  m,.  has  one  and  only  one  discontinuity,  taken  to 
be  the  same  as  u.  The  values  of  U  are  uniform,  finite  everywhere,  and 
continuous  everywhere ;  they  give  the  boundary  values  of  the  function  U  to 
be  determined  for  the  whole  area,  and  will  be  called  the  assigned  values.  In 
particular,  let  mj  be  the  assigned  value  at  A,  Wg  the  assigned  value  at  B: 
these  being  continuous  in  passing  through  the  respective  points  of  discon- 
tinuous curvature  regarded  as  belonging  to  the  contour  made  up  of  io  and  L^ 
only. 

The  process  consists  in  the  determination  of  functions  for  the  regions  T^ 
and  T2  alternately,  using  in  each  case,  for  boundary  values  along  L^  and  L^ 
respectively,  the  values  of  the  preceding  function  as  determined. 

Assume  for  a  boundary  value  along  L^  from  AtoB,&,  succession  of  values 
passing  continuously  from  m-y  to  nio, — say  in-^-[-  (1  —  X)  ??io,  with  \  decreasing 
from  1  to  0  :  the  actual  form  is  not  material,  and  we  shall  merely  suppose 
(though  even  this  is  not  necessary)  that  no  value  falls  below  the  minimum  or 
rises  above  the  maximum  of  the  assigned  values  along  L^  and  Z3.  Let  U^ 
denote  the  function,  which  is  uniquely  determined  for  the  region  T^  by  the 
general  conditions  for  the  area  and  by  values  along  the  boundary,  constituted 
by  the  assigned  values  along  Xq  and  the  assumed  values  along  L^,.  The 
values  acquired  by  U-^  along  the  line  L^  in  this  region  are  uniform,  finite,  and 
continuous ;  the  value  at  A  is  ii^  and  the  value  at  B  is  ma ;  at  any  inter- 
mediate point,  the  value  is  less  than  the  maximum  and  greater  than  the 
minimum  of  the  boundary  values. 

Let  U2  denote  the  function,  which  is  uniquely  determined  for  the  region 
T.2  by  the  general  conditions  for  the  area  and  by  values  along  the  boundary, 
constituted  by  the  assigned  values  along  L^  and  by  the  values  of  U-^  acquired 
along  Zi.  The  values  acquired  by  U^  along  the  line  X,  in  this  region  are 
uniform,  finite,  and  continuous ;  the  value  at  A  is  m-^  and  the  value  at  B  is 
ma;  at  any  intermediate  point,  the  value  is  less  than  the  maximum  and 
greater  than  the  minimum  of  the  boundary  values. 

Generally,  let  U^n-x  denote  the  function*,  which  is  uniquely  determined  for 
the  region  T^  by  means  of  boundary  values,  consisting  of  the  assigned  values 
along  Lq  and  of  the  values  acquired  by  U2n-2  along  L^,  beginning  at  A  with 
m^  and  endmg  at  B  with  m^^.  Similarly,  let  Uon  denote  the  function,  which 
is  uniquely  determined  for  the  region  T^  by  means  of  boundary  values, 
consisting  of  the  assigned  values  along  Lo,  and  of  the  values  acquired  by  C/gn-i 
along  Xi,  beginning  at  A  with  mj  and  ending  at  B  with  m^.  By  taking 
n=2,  3,  4, ...,  we  have  a  succession  of  functions,  t/3,  TJ^,  Us,  U^,  ... 

Now  consider  the  function  U^—U-y,  for  T^.  It  satisfies  the  general 
conditions.     It  is  zero  along  L^;  it  is  zero  at  A,  and  at  B;  along  L^  it  is 

*  All  the  functions  are  to  be  determined  subject  to  the  general  conditions  for  the  respective 
areas ;  the  specific  mention  of  the  general  conditions  will  be  omitted. 

F.  F.  31 


482  COMBINATION  [222. 

uniform,  finite,  and  continuous,  so  that  it  takes  a  continuous  series  of  values 
from  0  at  ^  to  0  at  B.  Along  L^,  there  may  be  negative  values  and  positive 
values ;  let  —  C  denote  the  minimum,  and  +  D  the  maximum,  among  these. 
(Here  G  may  be  0,  if  D  is  not  zero  :  or  D  may  be  0,  if  C  is  not  zero :  the  case 
when  D  =  0,  C  =  0  together  has  already  been  discussed,  §  220,  II.)  Then 
along  Li,  the  value  of  11^  —  U^  is  (§  220,  V.) 

-C  +  q{D  +  G), 

where  0<i2i^^<Qi<l;  here  R^  is  zero  if  G  is  zero,  and  otherwise  i^j  >  0  : 
it  is  always  less  than  Q^,  and  Qi  is  less  than  1.  Let  —  G^  denote  the  smallest 
value  of  U^  —  Ui  along  Li ,  and  +  -Di  its  greatest  value  ;  Dj  cannot  be  less  than 
zero,  nor  d  greater  than  zero,  the  terminal  values*  at  A  and  £.     Then 

-C,  =  -G  +  R,{D  +  G),    n,  =  -G  +  Q,(D  +  G); 

where,  in  general,  Cj  and  D^  are  both  greater  than  0 :  if  C  be  zero,  then  Ri  is 
zero.     Also,  let 

pi  =  Qi  —  Ri> 
so  that  0  <  pi  <  1 :  we  have 

A  +  0,  =  p,(D  +  (7). 

Similarly  as  regards  the  function  U^—  U^,  for  T^.  It  satisfies  the  general 
conditions.  It  is  zero  along  L^;  it  is  zero  at  A,  and  at  B;  along  L^,  it  takes 
the  values  of  f/g  —  f/^,  (for  11^  takes  the  values  of  U^,  and  Uo  the  values  of  U-^, 
so  that  along  L^  it  is  uniform,  finite,  and  changes  continuously  from  0  at  JL, 
through  a  minimum  —  G^  and  a  maximum  D^,  to  0  at  B.  Then  along  L^,  a 
line  in  the  area  of  T^,  the  value  of  11^  —  U^  is  (§  220,  V.) 

-C',  +  9(A  +  C'0, 

where  0  <  jRs  <  ^  <  Qa  <  1 ;  here  R2  is  zero  if  Gi  is  zero,  and  otherwise  i^g  >  0  ; 
it  is  always  less  than  Qo,  and  Q2  is  less  than  1.  Let  —  G2  denote  the  smallest 
value  of  f/4  —  Uo  along  Zg,  and  +  D2  its  greatest  value  along  the  line.     Then 

-G2  =  -G,  +  R2(D,  +  G,),    D2  =  -G,  +  Q2{D,  +  G,); 

where,  in  general,  Cg  and  Dg  are  both  greater  than  0 :  if  C'l  be  zero,  then  Ro 
is  zero.     Also,  let 

,  so  that  0  <  p.,  <  1 :  we  have 

D2  +  G2  =  p2{D,  +  G,). 

*  It  is  at  this  step  in  each  of  the  stages  that  advantage  accrues  from  (i)  having  modified 
initially  the  assigned  values,  so  that  no  discontinuity  occurs  at  A  or  at  B,  and  (ii)  having 
secured  continuity  in  value  through  the  points  A  and  B,  both  along  Li  and  L2,  for  the 
successive  functions.  By  these  conditions,  we  secure  that  A  and  B  do  not  need  to  be  excluded 
by  small  arcs,  as  in  the  earlier  part  of  §  219  (the  points  A  and  B  would  otherwise  remain 
excluded  throughout,  and  would  not  be  part  of  the  boundary  at  the  end) ;  and  we  secure  that  Qi 
is  certainly  less  than  unity  at  each  step. 


222.]  OF   AREAS  483 

And  so  on,  alternately,  for  the  functions  connected  with  the  two  regions. 
The  functions  U<,_n,  U^n+i  (for  successive  values  of  n)  satisfy  the  general 
conditions.  In  T^,  the  function  U^n+i  —  U'zn-i  is  greater  than  —  C^-^,  and  less 
than  -Dsw-sj  while  along  ij  it  ranges  continuously  between  —  C2,i_i  and  An-i 
(with  0  at  ^  and  0  at  B),  where 

—  (^211-1  =^        ^291—2  +  -ti'-zn—i  \-L'2n—2  "f"  ^271—2)  j 

-^2?l— 1  ^^  ~  ^271—2  "1"  ^5211—1    \-L'2n-2  "T  ^2n—2)) 

where  Q271-1,  R^n-i  are  the  major  and  the  minor  factorial  fractions  for  the 
distribution  ranging  between  —C271-2  and  1)2^-2  (with  0  at  A  and  0  at  B) 
along  Z2.  In  T2,  the  function  U2n+2—U'2n  is  greater  than  —Gzn-i,  and  less 
than  Dzn+i,  while  along  L^  it  ranges  continuously  between  —  Cg^  and  D2n 
(with  0  at  J.  and  0  at  B),  where 

—  ^211  =  ■"  ^2,1-1  +  R-in  (j-^m-i  +  C^2n-i)] 

J^-2n  —  ~  ^271—1  +  ^211  {J-'2n-i  +  ^2?i— i)j 

where  Q.2»n  -^271  are  the  major  and  the  minor  factorial  fractions  for  the 
distribution  ranging  between  —  Co^-i  and  A»i-i  (with  0  at  ^  and  0  at  B) 
along  Xj. 

Now    let  pni  =  Qm  -  Rm, 

for  all  values  of  m,  odd  and  even  :  we  have 

0  <  p,n  <  I' 
Then  Doii-i  +  C^2«-i  =  p^n-i  (^211-2  +  ^271-2), 

■i-'2>i      "I"  ^271       ^  Pin      V-^271— 1  +  ^2n—i)  3 

hence,  taking  account  of  the  value  of  Di  +  C\,  we  have 

Dm  +  C,n  =  pip-i."  pm  (D  +  C). 

Since  each  of  the  quantities  p  is  a  positive  quantity,  known  to  be  less  than  1, 
Tve  have 

Lim{pip2...  pm)  =  0, 

and  therefore  Lim  (i),„  +  C„i)  =  0. 

In  Tj,  the  range  of  value  of  the  function  U 2,1+1  —  ^271-1  is  equal  to  -D271-2  +  C'2n-2 
along  L.2,  and  is  equal  to  D.^n-i  +  Cm-i  along  A;  and  in  T^,  the  range  of 
value  of  the  function  U 2,1+2  — ^^2,1  is  equal  to  Dan-i  +  <^2w-i  along  L^,  and  is 
•equal  to  D^n  +  C'o^  along  Zg.  Hence,  as  the  number  of  operative  constructions 
is  made  to  increase  indefinitely,  there  are  limits  to  which  the  functions  with 
an  odd  suffix  and  functions  with  an  even  suffix  approach  along  ii  and  Zg. 
Let  TJ'  denote  the  limit  of  functions  with  an  odd  suffix  along  Zj,  and  U"  that 
of  functions  with  an  even  suffix  along  L^. 

31—2 


484  COMBINATION   OF   AREAS  [222. 

Both  of  these  limits  are  finite.  To  prove  this,  let  M  denote  the  maximum 
and  m  the  minimum  of  the  assigned  values ;  so  that  the  range  in  value  of  Ui 
is  not  greater  than  M  —  m.     We  have,  along  Zj , 

U'  =  U,+{U,-U,)  +  {U,-U,)  +  ...  adinf., 

U'<M  +  D,  +  I},  +  ... 

^M+{D  +  G)(Pj+  pip2p3  +  plp2PspiP5  +...). 

Among  the  quantities  p^,  p^,  ps, ...,  all  of  which  are  less  than  1,  let  cr  be  the 
greatest;  then 

Pi  +  P1P2P3  +  ...  ^cr  +  a^  +  a^  +  ... 

a 
so  that  U'<M  +  -^ (D  +  G). 

1  —  O"" 

Thus  the  upper  limit  of  U'  is  finite.     Also,  denoting  by   V  the  range  in 
value  of  U',  we  have 


^M-m  +  (D,  +  C,)  +  (D3+G,)  +  ... 
^M-m  +  --^{D  +  G); 

1  —  O" 

which  is  finite.     Hence  the  upper  limit  and  the  lower  limit  for  V  are  both 
finite  ;  and  therefore  U'  is  finite. 

Similarly,  we  have,  along  L2, 

U"=U,  +  {U,-U,)  +  {Ue-U,)+...  adinf., 

^M  +  (D,+  G,)  +  (D,  +  G,)+... 
^M+{I)+C){p,p2  +  pip,psP,  +  ...) 

<M  +  --^^(D  +  G); 
1  —  (J- 

and,  as  before       U"  ^M-m  +  ^r-^^{I)  +  G). 

1  —  0" 

Both  of  these  are  finite ;  hence  U"  is  finite. 

Now  in  determining  U'  for  Tj  and  regarding  it  as  the  limit  of  U^n+i,  we 
have  its  values  along  Z,  as  the  values  of  Uon,  that  is,  of  U"  in  the  limit ;  and 
in  determining  U"  for  T^  and  regarding  it  as  the  limit  of  U.n+o,  we  have  its 
values  along  L^  as  the  values  of  U^n+i,  that  is,  of  U'  in  the  limit.  Hence  over 
the  whole  boundary  of  T,  the  region  common  to  2^1  and  To,  we  have  U'=  U"; 


222.]  BY   ALTERNATING  PROCESS  485 

and  therefore  (by  §220,  VI.)  we  have  U'=TJ"  over  the  whole  area  of  the 
common  region  T. 

Lastly,  let  a  function  U  be  determined  for  the  region  T^ ,  having  the 
assigned  values  along  L^  and  the  values  of  V  along  Lo.  Then  the  function 
TJ—U'  satisfies  the  general  conditions;  it  has  zero  values  round  the  whole 
boundary  of  T^,  and  therefore  (by  §220,  VI.)  it  is  zero  over  the  whole  region 
Tj.     Hence  U'  is  the  function  for  I\. 

Similarly,  determining  a  function  CT"  for  T2,  which  has  the  assigned  values 
along  X3  and  the  values  of  U"  along  L^,  we  have  U  =U"  everywhere  in  T^, 
so  that  U"  is  the  function  for  T«. 

The  functions  U'  and  U"  satisfy  the  general  conditions  for  T^  and  T2 
respectively;  and  these  two  regions  have  a  common  portion  T  over  which 
JJ'  and  U"  have  been  proved  to  be  the  same.  Hence,  by  Lemma  I.  of  §216, 
they  determine  one  and  the  same  function  for  the  whole  region  made  up  of 
Ti  and  T.^.  This  function  U  satisfies  the  general  conditions  and,  along  the 
boundary  of  the  whole  region,  assumes  values  that  are  assigned  arbitrarily, 
subject  only  to  the  general  limitations  of  being  everywhere  finite  and, 
except  for  finite  discontinuities  at  isolated  points,  uniform  and  continuous. 
The  proposition  is  therefore  established. 

•  .  This  method  of  combination,  dependent  upon  the  alternating  process 
whereby  a  function  determined  separately  for  two  given  regions  having  a 
common  part  is  determined  for  the  combination  of  the  regions,  is  capable,  of 
repeated  application.  Hence  it  follows  that  a  function  exists,  subject  to  the 
general  conditions  within  a  given  region  and  acquiring  assigned  finite  values 
along  the  boundary  of  the  region,  when  the  region  can  be  obtained  by 
combinations  of  areas  that  can  be  conformally  represented  upon  the  area 
of  a  circle. 

Note.  Let  A,  B,  G  he  three  non-intersecting  simple  closed  curves,  such 
that  C  lies  within  B  and  B  within  A.  The  area  bounded  by  the  curves  A  and 
G  can,  by  a  similar  method,  be  combined  with  the  whole  area  enclosed  by  B ; 
and  we  can  make  the  same  inference  as  above,  as  to  the  existence  of  a  potential 
function  for  the  whole  area  enclosed  by  A,  when  it  exists  for  the  areas  that 
are  combined. 

223.  At  the  beginning  of  the  discussion  it  was  assumed  that  the  areas, 
in  which  the  existence  of  the  function  is  to  be  proved,  lie  in  a  single  sheet 
(§216)  or,  in  other  words,  that  no  branch -point  occurs  within  the  area. 

It  is  now  necessary  to  take  the  alternative  possibility  into  consideration : 
a  simple  example  will  shew  that  the  theorem  just  proved  is  valid  for  an  area 
containing  a  branch-point,  except  in  one  unessential  particular. 

Let  the  area  be  a  winding-surface  consisting  of  m  sheets :  the  region  in 
each  sheet  will  be  taken  circular  in  form,  and  the  centre  c  of  the  circles  will 


486  BRANCH-POINT   IN   THE   AREA  [228. 

be  the  winding-point,  of  order  m  —  1.  Such  a  surface  is  simply  donnected 
(§  178);  and  its  boundary  consists  of  the  m  successive  circumferences  which, 
owing  to  the  connection,  form  a  single  simple  closed  curve.  Using  the 
substitution 

z-c  =  RZ^, 

we  have  a  new  ^-surface  which  consists  of  a  circle,  centre  the  ^-origin  and 
radius  unity :  it  lies  in  one  sheet  in  the  ^-region  and  has  no  branch-points ; 
its  circumference  is  described  once  for  a  single  description  of  the  complete 
boundary  of  the  winding-surface.  The  correspondence  between  the  two 
regions  is  point-to-point :  and  therefore  the  assigned  values  along  the  bound- 
ary of  the  winding-surface  lead  to  assigned  values  along  ^he  Z-circumference. 
Any  function  w  oi  z  changes  into  a  function  1^  of  ^:  hence  u  changes 
into  a  real  function  U  satisfying  the  general  conditions  in  the  -^-region ; 
and  conversely. 

But  a  function  U,  satisfying  the  general  conditions  over  the  area  of  a 
plane  circle  and  acquiring  assigned  finite  values  along  the  circumference,  is 
uniquely  determinate ;  hence  the  function  u  is  uniquely  determined  on  the 
circular  winding-surface  by  satisfying  the  general  conditions  over  the  area 
and  by  assuming  assigned  values  along  its  boundary. 

It  is  thus  obvious  that  the  multiplicity  of  sheets,  connected  through 
branch-lines  terminated  at  branch-points  and  (where  necessary)  at  the  single 
boundary  of  the  surface  consisting  of  the  sheets,  does  not  affect  the  validity 
of  the  result  obtained  earlier  for  the  simpler  one-sheeted  area ;  and  therefore 
the  function  u,  acquiring  assigned  values  along  the  boundary  of  the  simply 
connected  surface,  and  satisfying  the  general  conditions  throughout  the  area 
of  the  surface  which  may  consist  of  more  than  a  single  sheet,  is  uniquely  deter- 
minate. 

There  is,  as  already  remarked,  one  unessential  particular  in  which 
deviation  from  the  theorem  occurs  when  the  region  contains  a  branch-point. 
At  a  branch-point  a  function  may  be  finite*,  but  all  its  derivatives  are  not 
necessarily  finite ;  and  therefore  at  such  a  point  a  possible  exception  to  the 
general  conditions  arises  as  to  the  finiteness  of  value  of  the  derivatives 
and  the  consequent  satisfying  of  the  equation  SJ^u  =  0 :  no  exception,  of 
course,  arises  as  regards  the  uniformity  of  the  derivatives  on  the  Riemann's 
surface.  The  exception  does  not  necessarily  occur ;  but,  when  it  does  occur, 
it  is  only  at  isolated  points,  and  its  nature  does  not  interfere  with  the  validity 
of  the  proposition.  We  shall  therefore  assume  that,  in  speaking  of  the 
general  conditions  through  the  area,  the  exception  (if  necessary)  from  the 
general  conditions,  of  finiteness  of  value  of  the  derivatives  at  a  branch-point, 
is  tacitly  implied. 

*  Infinities  of  the  function  itself  at  a  branch-point  will  fall   under  the  general  head  of 
iafinities  of  the  function,  discussed  afterwards  (in  §  229). 


223.] 


MULTIPLY   CONNECTED   SURFACES 


487 


Hence  we  infer,  by  taking  combinations  of  circles  in  a  manner  some- 
what similar  to  the  process  adopted  for  successive  circles  of  convergence 
in  the  continuation  of  a  function  in  §34,  that  a  function  u  exists,  subject  to  the 
general  conditions  luithin  any  simply  connected  surface  and  acquiring  assigned 
finite  values  along  the  boundary  of  the  surface. 

224.  The  functions,  which  have  been  discussed  so  far  in  the  present 
connection,  are  functions  having  no  infinities  and,  except  possibly  at  points 
on  the  boundaries  of  the  regions  considered,  no  discontinuities :  they  are 
uniform  functions.  And  the  regions  have,  hitherto,  been  supposed  simply 
connected  parts  of  a  Riemann's  surface,  or  simply  connected  surfaces.  When 
the  surface  is  multiply  connected,  we  resolve  it  by  a  canonical  system 
(§  181)  of  cross-cuts  as  follows. 

We  also  proceed  to  introduce  the  cross-cut  constants,  and  so  to  consider 
the  existence  of  functions  which  have  the  multiform  character  of  the  integrals 
of  uniform  functions  of  position  on  the  Riemann's  surface.  The  functions 
will  still  be  considered  to  be  uniform,  finite  and  continuous  except  at  the 
cross-cuts :  their  derivatives  will  be  supposed  uniform,  finite  and  continuous 
everywhere  in  the  region,  and  subject  to  the  equation  ^^u  =  0:  and  boundary 
values  will  be  assigned  of  the  same  character  as  in  the  previous  cases.  As 
moduli  of  periodicity  are  to  be  introduced,  the  unresolved  surface  is  no  longer 
one  of  simple  connection :  we  shall  begin  with  a  doubly  connected  surface. 

Let  such  a  surface  T  be  resolved,  in  two  different  ways,  into  a  simply 
connected  surface:  say  into  T^  by  a  cross-cut  Qi,  and  into  Tg  by  a  cross- 
cut Q.,.     Mark  on  I^j  and  on  To  the  directions  of  Qo,  and  of  Q-^  respectively:  the 


Fig.  81. 

notations   of  the   boundaries   are  indicated  in  the  figures,  and   T'  is  the 
region  between  the  lines  of  Q^  and  Q.2. 

It  will  be  shewn  that  a  function  u  exists,  determined  uniquely  by  the 
following  conditions  : 

(i)  The  first  and  the  second  derivatives  are  throughout  T  to  be 
uniform,  finite  and  continuous,  and  to  satisfy  ^hi  =  0 :  but  no  conditions 
for  them  are  assigned  at  points  on  the  boundary : 

(ii)  The  (single)  modulus  of  periodicity  is  to  be  K,  which  will  be 
taken  as  an  arbitrary,  real,  positive  constant :  the  value  of  any  branch  of  u  at 


488  POTENTIAL   FUNCTIONS   WITH  [224. 

a  point  on  the  positive  edge  is  therefore  to  be  greater  by  K  than  its  vahie  at 
the  opposite  point  on  the  negative  edge : 

(iii)  Some  selected  branch  of  u  is  to  assume  assigned  vahies  along 
a  and  h',  typically  represented  by  H,  and  assigned  values  along  a  and  h, 
typically  represented  by  G.  These  boundary  values  are  to  be  finite  every- 
where, though  they  may  be  discontinuous  at  a  finite  number  of  isolated  points 
on  the  boundary ;  such  discontinuity  will  arise  through  the  modulus. 

In  Ti,  for  zero  values  along  a,  b,  a',  h'  and  for  unit  values  along  Q~ 
and  Qi+,  let  the  fractional  factor  for  the  line  Q^  be  q^ :  and  similarly  in  To, 
for  zero  values  along  a,  b,  a,  b'  and  for  unit  values  along  Qo"  ^'^d  Q2+, 
let  the  fractional  factor  for  the  line  Qi  be  q2,  where  q^  and  5.,  are  positive 
proper  fractions. 

For  the  simply  connected  region*  T^  determine  a  function  Ui,  satisfying 
the  general  conditions  and  having  as  its  boundary  values,  H  along  a'  and  b', 
G  along  a  and  b,  arbitrarily  assumed  values  represented  by  0  (the  maximum 
value  being  M^  and  the  -minimum  value  being  m^)  along  Qp  and  values 
6  +  K  along  Q{^:  the  function  so  obtained  is  unique.  Let  the  values 
along  the  line    Q.2  in  T^  be  denoted  by  m/. 

For  the  region  T.,  determine  a  function  u.2,  satisfying  the  general  con- 
ditions and  having  as  its  boundary  values,  H  along  a'  and  b',  G  —  K  along 
a  and  b,  u-^  —  K  along  ^2"  and  m/  along  Qg"*" :  the  function  so  obtained  is 
unique.  Let  its  values  along  .the  line  Q^  in  To  be  denoted  by  w/,  the 
maximum  value  being  M.2  and  the  minimum  value  being  7?i2- 

For  the  region  2\  determine  a  function  u^,  satisfying  the  general  conditions 
and  having  as  its  boundary  values,  H  along  a'  and  b',  G  along  a  and  b,  u^ 
along  Qi~  and  ^2'+-^^^"  along  Q{^:  the  function  so  obtained  is  unique.  Let  its 
values  along  the  line  Q2  in  T^  be  denoted  by  u^.  Then  the  function  u.^  —  u^ 
satisfies  the  general  conditions  in  T^ ;  it  is  zero  along  a'  and  b',  a  and  b  :  it  is 
ti2  —  6  along  Qi~  and  also  along  Q{^,  and  Uo  —  6  ^  JS'L  —  ??ii  and  ^  m^  —  M^. 

A  difference  of  limits  for  u-/—  u^'  arises  according  to  the  relative  values  of 
ifa  and  mi,  of  mg  and  M^ ;  evidently  M2  —  m-^  >  vu  —  Mi. 

(i)  If  7712  —  -^i  be  positive,  then  ilig  —  ^'^1  is  positive  and  equal,  say,  to 
A, ;  the  boundary  values  for  W3  —  Uj_  may  range  from  0  to  X,  and  we  have 
Us  —  111  >  0  <  q^X  along  Q2. 

(ii)  If  ??i2  —  Ml  be  negative  and  equal  to  —  e,  then  M2  —  m^  is  either 
positive  or  negative. 

(a)     If  ifg  —  nil  be  negative,  then  the  boundary  values  for  U3  —  Ui 
may  range  from  0  to  —  e,  that  is,  boundary  values  for  Uj  —  u^  may  range  from 

*  In  the  special  case,  when  Tj  is  bounded  by  concentric  circles  and  the  cross-cut  is  made 
along  a  diameter,  the  region  can  be  represented  eonformally  on  the  area  of  a  circle  :  see  a  paper 
by  the  author,  Quart.  Journ.  Math.,  vol.  xxvi,  (1892),  pp.  145—148. 


224.]  MODULI   OF    PERIODICITY  489 

0  to  e,  and  we  have  u-[  —  u^>0  <  qi€  along  Q2,  which  may  be  expressed  in  the 

form 

\  U3' -  Ut' \  <  q-^e, 

where  e  is  the  greatest  modulus  of  values  along  the  boundary. 

(b)  If  M2  —  nil  be  positive,  let  its  value  be  denoted  by  rj :  then  the 
boundary  values  for  Ug  —  Ui  may  range  from  77  to  —  e.  The  boundary  values 
for  U3  —  Ui  +  e  may  range  from  0  to  ?;  +  e,  and  it  is  a  function  satisfying  all  the 
internal  conditions  :  hence  M3  —  Ui  +  e^qi{r)  +  e),  and  therefore 

W3  -  "1  <  qiV  -  (1  -  2i)  e  ^  ^iV- 
Again,  the  boundary  values  of  u^  —  11$  +  rj  may  range  from  77  +  e  to  0,  and  it  is 
a  function  satisfying  all  the  internal  conditions :  hence  -w^  —  Wg  +  V'^qiiv  +  ^)y 
and  therefore 

«i  - '<3  <  gie  -  (1  -  5i)  7;  ^  5ie. 

Hence  at  points  where  u^  >  th ,  so  that  u.^  —  u-^  is  positive,  we  have  u^  —  Ui^qii]; 
and  at  points  where  Wg <  Ui,  so  that  U]_  —  u^  is  positive,  we  have  u^  =  u^^qie. 

Every  case  can  be  included  in  the  following  result  * :  If  //.  be  the  greatest 
modulus  of  the  values  of  u.2  —  d  along  the  two  edges  of  Qi  in  Tj ,  then 

along  Q2,  so  that  q^/u,  is  certainly  the  greatest  modulus  of  u-/  —  Ui  along  Qa- 

225.  For  the  region  T2  determine  a  function  u^,  satisfying  the  general 
conditions  and  having  as  its  boundary  values,  H  along  a'  and  b',  Q  —  K  along 
a  and  b,  u^  —  K  along  Q.^  and  w/  along  Qa"^ :  the  function  so  obtained  is 
unique.  Let  its  values  along  the  line  Q^  be  denoted  by  w/.  Then  the 
function  u^  —  11.2  satisfies  the  general  conditions  in  T2 :  it  is  zero  along  a  and 
b',  a  and  b:  it  is  Ug'  —  u/  along  Qo"  and  also  along  Qs"*",  and  along  Q2  we  have 

I  Us  -  Ui  \<qifi. 
Hence,  after  the  preceding  explanations,  we  have  along  Qi  in  T2 

i  u,'  -  U2  I  <  q^qifJ^- 
Proceeding  in  this  way  for  the  regions  alternately,  we  have  for  T^  a  function 
^271+1.  the  boundary  values  of  which  are,  H  along  a  and  b',  G  along  a  and  b, 
u'zn  along  Qi~  and  u'^n  +  K  along  Q{^ :  and  along  Qg 

I  ^^  2W+1    u  211—1 1  ^  q\'  q2    f^ } 

and  for  Tg  a  function  112,1+2  >  the  boundary  values  of  which  are,  H  along  a  and 
b',  G  —  K  along  a  and  6,  u'on+i  —  K  along  Q.,"  and  u'^n+i  along  Qg"*" :  and 
along  Qi 

I  ^^'2,1+2  -  u'2n  I  <  qi^q-Z'H'- 

*  Another  method  of  proceeding,  different  from  the  method  in  the  text,  depends  upon  the 
introduction  of  the  minor  fractional  factor  (§  322)  for  the  cross-cut,  having  the  same  relation 
to  minimum  values  as  qi  to  maximum  values ;  but  it  is  more  cumbersome,  as  it  requires  the 
continuous  consideration  of  successive  cases,  and  the  method  is  adequately  indicated  by  the 
process  of  §  222. 


490  POTENTIAL   FUNCTION   FOR  [225. 

Thus  both  the  function  Uon+i  along  Q2  and  the  function  U2n  along  Qy 
approach  limiting  values;   let  them  be  u'  and  u"  respectively. 

These  limiting  values  are  finite.     For 

Uon+i  =  Wi  +  (Us  -  i/i)  +  ('Us  -  U3)  +  . . .  +  (ihn+i  "  ^2/1-1)  ; 

in  the  limit,  when  n  is  infinitely  large,  the  sum  of  the  moduli  of  the  terms  of 
the  series  at  points  along  Q^ 

<  (M,  +K)  +  q.iJi  +  q^%fji  +  q,%J^i  +  . . . 


<  M,  +K  + 


i-q^q^' 


so  that  the  series  converges  and  the  limit  of  u.2n+i,  viz.  u',  is  finite.     Similarly 
for  u". 

Now  consider  the  functions  in  the  portions  T  —  T'  and  T'  of  the 
region  T. 

For  T  —  T'  we  have  iion ,  (that  is,  u"  in  the  limit),  with  values  H 
along  a'  and  h',  u  along  Qo+:  and  also  u.yn+i,  (that  is,  u'  in  the  limit), 
with  values  H  along  a  and  b'  and  u"  along  Qi~:  thus  u'  and  ii"  have 
the  same  values  over  the  whole  boundary  oi  T  —  T'  and,  therefore,  throughout 
that  portion  we  have  u  =  u". 

For  T'  we  have  1(2,1,  (that  is,  u"  in  the  limit),  with  values  G^  —  ^  along 
a  and  b,  and  u'  —  K  along  Q.2~ :  and  also  Won+i ,  (that  is,  u'  in  the  limit),  with 
values  G  along  a  and  b  and  ii"  +  K  along  Q{^.  Thus  over  the  whole  boundary 
of  T'  we  have  u'  —  u"  =  K:  and  therefore  within  the  portion  T'  we  have 

Lastly,  for  the  whole  region  T  we  take  u  =  u.  In  the  portion  T  —  T'  we 
have  ii  =  u'=u",  and  in  the  portion  T'  we  have  it=  ii  =  u"  +  K ;  that  is,  the 
function  is  suck  that  in  the  region  T^  the  value  changes  from  u"  at  Q~  to  u"  +  K 
at  Q{^,  or  the  modulus  of  periodicity  is  K. 

Hence  the  function  is  uniquely  determined  for  a  doubly  connected  surface 
by  the  general  conditions,  by  the  assigned  boundary  values,  and  by  the 
arbitrarily  assumed  real  modulus  of  periodicity. 

226.  We  now  consider  the  determination  of  the  function,  when  the 
surface  S  is  triply  connected  and  has  a  single  boundary. 

Let  *S^  be  resolved,  in  two  different  ways,  into  a  doubly  connected  surface. 
Let  ^1  be  a  cross-cut,  which  changes  the  surface  into  one  of  double 
connectivity  and  gives  two  pieces  of  boundary:  and  let  Q.y  be  another 
cross-cut,  not  meeting  the  direction  of  Qi  anywhere  but  continuously 
deformable  into  Q^,  so  that  it  also  changes  the  surface  into  one  of  double 
connectivity  with  two  pieces  of  boundary.     Then,  in  each  of  these  doubly 


226.]  MULTIPLY   CONNECTED   SURFACES  491 

connected  surfaces,  any  number  of  functions  can  be  uniquely  determined 
which  satisfy  the '  general  conditions,  each  of  which  assumes  assigned 
boundary  values,  that  is,  along  the  boundary  of  >S  and  the  new  boundary, 
and  possesses  an  arbitrarily  assigned  modulus  of  periodicity. 

The  combination  of  these  functions,  by  an  alternate  process  similar  to 
that  for  the  preceding  case,  leads  to  a  unique  function  which  has  an 
assigned  modulus  of  periodicity  for  the  cross-cut  Q^.  The  conditions 
which  determine  it  are :  (i),  the  general  conditions :  (ii),  the  values  along 
the  boundary  of  the  given  surface,  (iii)  the  value  of  the  modulus  of 
periodicity  for  the  cross-cut,  which  resolves  the  surface  into  one  of  double 
connectivity,  and  the  modulus  of  periodicity  for  the  cross-cut,  which 
resolves  the  latter  into  a  simply  connected  surface,  that  is,  by  assigned 
moduli  of  periodicity  for  the  two  cross-cuts  necessary  to  resolve  the 
original  surface  S  into  one  that  is  simply  connected. 

Proceeding  in  this  synthetic  fashion,  we  ultimately  obtain  the  result 
that  a  real  function  u  exists  for  a  surface  of  connectivity  2jj  -H  1  with  a  single 
boundary,  uniquely  determined  by  the  following  conditions : 

(i)     its  derivatives  within  the  surface  are  everywhere  uniform,  finite 

and  continuous,  and  they  satisfy  the  equation  Vhc  =  0 ; 
(ii)  it  assumes,  along  the  boundary  of  the  surface,  assigned  values 
which  are  always  finite  but  may  be  discontinuous  at  a  limited 
number  of  isolated  points  on  the  boundary ; 
(iii)  the  function  within  the  surface  is  everywhere  finite  and,  except  at 
the  positions  of  cross-cuts,  is  everywhere  uniform  and  continuous : 
the  discontinuities  in  value  in  passing  from  one  edge  to  another 
of  the  cross-cuts  are  arbitrarily  assigned  real  quantities. 

.  227.  The  question  next  arises  as  to  the  existence  of  a  function  ic  upon  a 
Riemann's  surface  of  connectivity  2p  +  1  that  has  no  boundary,  the  function 
satisfying  (i)  and  (iii)  of  the  foregoing  conditions.  The  existence  can  be 
established  as  follows*. 

On  some  sheet  of  the  surface,  take  two  concentric  circles  of  radii  r  and  r 
(where  r  >  r),  choosing  them  so  that  the  outer  circle  (and  therefore  also  the 
inner  circle)  encloses  no  singularity  and  meets  no  cross-cut;  clearly  the 
magnitude  of  r'  will  be  at  our  disposal,  and  it  will  be  supposed  finite  (not 
zero).  Let  the  circumferences  of  the  circles  be  denoted  by  G  and  C ;  denote 
the  part  of  the  Riemann's  surface  outside  C  by  /S",  and  the  circular  area 
within  C  on  the  Riemann's  surface  by  S.  Thus  8'  and  8  are  Riemann's 
surfaces,  each  with  a  single  boundary ;  and  they  have  a  common  annulus. 

Assume  any  set  of  finite  and  continuous  values  along  C.  Determine  for 
the  bounded  Riemann's  surface  8'  a  function  m/,  which  acquires  these  values 

*  Schwarz,  Ges.  Werke,  t.  ii,  p.  306;  Picard,  Cours  d' Analyse,  t.  ii,  p.  470. 


492  POTENTIAL   FUNCTION   ON   A  [227. 

along  C,  satisfies  the  general  conditions  everjrwhere  in  8',  and  at  the  various 
cross-cuts  possesses  arbitrarily  assigned  real  moduli  of  periodicity.  This 
function  u-^  is  unique ;  and  it  acquires  finite  and  continuous  values  along  G, 
which  lies  within  the  region  of  its  existence. 

Determine  for  the  bounded  Riemann's  surface  8  a  function  iii,  which 
acquires  along  C  the  values  that  are  acquired  along  that  circle  by  u^  ,  and 
which  satisfies  the  general  conditions  everywhere  in  S.  (As  no  cross-cut 
occurs  within  8,  all  the  moduli  of  periodicity  may  be  regarded  as  zero.)  This 
function  u^  is  unique :  and  it  acquires  finite  and  continuous  values  along  C, 
which  lies  within  the  region  of  its  existence. 

Determine  for  8'  a  function  u^,  which  acquires  along  C"  the  values 
acquired  by  u-^,  satisfies  the  general  conditions  everywhere  in  8',  and  at  the 
various  cross-cuts  possesses  the  same  arbitrarily  assigned  moduli  of  periodicity 
as  u-^.  This  function  u.2  is  unique ;  and  it  acquires  finite  and  continuous 
values  along  C,  which  lies  within  the  region  of  its  existence. 

Determine  for  8  a  function  U2,  which  acquires  along  C  the  values  that  are 
acquii'ed  along  that  circle  by  u^,  and  which  satisfies  the  general  conditions 
everywhere  in  8;  as  there  are  no  cross-cuts  within  8,  there  are  no  moduli  of 
periodicity.  This  function  Uo.  is  unique  :  and  it  acquires  finite  and  continuous 
values  along  C,  which  lies  within  the  region  of  its  existence. 

And  so  on,  in  alternate  succession  for  the  spaces  8'  and  8.  We  thus 
obtain  a  sequence  of  functions  u^,  u^,  ...,  u^,  •■■,  which  satisfy  the  general 
conditions  within  8'  and  possess  the  arbitrarily  assigned  moduli  of  periodicity 
at  the  various  cross-cuts  :  and  a  sequence  of  functions  Ui,U2,  ...,Un,  •■•,  which 
satisfy  the  general  conditions  within  8,  an  area  that  contains  no  cross-cuts. 
Moreover,  we  have 

Un=Un-x  along  G'\ 
iLn  =  tin      along  C  J  ' 
as  values  along  the  boundaries   of  aS^'  and  8  respectively,  assigned  to  the 
functions  in  the  respective  sequences. 

Now  (by  Ex.  1,  Lemma  II.  §  216)  we  have 

[•■ItT  r  277 

Un,  (r,  (j))d(j)=        u,n  {r,  (f>)  dcj). 

Jo  .        •'  0 

But  Urn  {r ,  (f>)  =  u\n+i  (r,  4>\  Oil  accouut  of  boundary  values  :  thus 


TStt 

U,n  {r,  4))d(j>=  U'm+i  (r,  (j))  d(f) 

J   0 


■277 


u'm+1  (r,  (j>)  d(f), 


by  Ex.  2,  Lemma  II.  §  216.     Also  u'm+i  (r,  (f>)  =  Um-^i  (?•,  0),  on  account  of 
boundary  values :  and  therefore 


u,n  (r,  (f>)  dcf)  =       Um+i  {r,  <^)  d<p. 
Jo  -0 


227.]  SURFACE   WITHOUT   A   BOUNDARY  493 

The  functions  Wj,  ii2,  ■■■  exist  over  the  whole  of  the  interior  of  the  circle  of 
radius  r ;  hence  (by  I.  §  220)  we  have 

Urn  (0)  =  Um+i  (0), 
or  all  the  functions  ii-^,  itz,  ...  have  the  same  value  at  the  centre  of  this  circle. 

We  proceed  to  shew  that  these  functions  u  and  u'  converge  to  the  same 
limit  in  the  infinite  sequence.     Let 

SO  that  U,r^{r,  ^)  =  U,n    (r,  (f>)] 

along  the  circles  C  and  C  respectively.     At  the  common  centre,  we  have 

and  therefore,  if  Mm  he  the  maximum  value  of  |  Um  (r,  (f))\  along  the  circle  C, 
we  have  (by  Ex.  1,  §  218) 

4  r' 

I  U^m  (r\  (b)  \  ^  -Mm  -  sin-1  - 

TT  r 

along  the  circle  C".  Now  in  the  initial  assumption  of  the  circles  C  and  G', 
we  have  merely  made  r  <  r,  provided  r'  does  not  become  an  indefinitely  small 
quantity.     Suppose  now  that  the  inner  circle  is  chosen  so  that 

4    .       r' 

—  sin~^  —  <  (T, 

TT  r 

where  cr  is  a  finite  positive  quantity  less  than  unity.     Then 

I  Vm  (r,  (f>)  I  <  a-M„,. 
Consequently,  we  have 

I  U',n+i  (.r,  (/>)  I  =  i  U,n  (r',  0)  I 

<  a  Mm. 

Now  the  function  U\n+i,  which  exists  in  the  Riemann  surface  S'  outside  C, 
is  such  that  it  satisfies  the  general  conditions  within  >S";  moreover,  it  has  no 
moduli  of  periodicity,  for  the  functions  w'r„+2  and  m',^+i  have  the  same 
arbitrarily  assigned  moduli  of  periodicity ;  hence  (§  220)  the  maximum  values 
of  i!7'w,+i  and  the  minimum  values  of  U'^+j  lie  on  its  boundary  which  is  the 
circle  C,  and  therefore  |  U\n+i  \  within  S'  is  less  than  the  maximum  value  of 
j  U'm+i  1  along  C.     Accordingly 

I  U\n+i  {r,  (f))\<\  U'rn+i  (r,  (f))  I 

<  0-M,n, 

that  is,  f^m+i  {f^,  <i>)  I  <  o-Mm 

<a-\Um{r,  (f>)\. 


494  FUNCTIONS   ON   A   RIEMANN's   SURFACE  [227. 

Hence,  if  the  maximum  value  of  j  u.2  —  u-^  \  along  the  circle  C  be  N,  we  have 
1  ^1  (^'>  <^)  I  ^  -^ ;  ^-nd  therefore 

\U^{r,4>)\<a-^-^N, 
I  U^  (r,  j>)\<<T^    N,' 
\UJ(r',cf>)\<a-^-'N, 

Moreover,  we  have 

Un  =  Ui  +  (^2  -  'Wi)  +  (Ms  —  Mq)  +  • . .  +  (W„  —  Un-i) 

and  therefore  |  w„  |  <  j  Mj  |  +  |  C/i  j  +  . . .  +  |  Un-i  \ ; 

1  —  a^ 

SO  that  I  M„  I  <    Ml  +  iV  ^j along  C, 

'       '  i  —  cr 

1  _  o-m+i  ^ 

<  I  Ml   +  iV  — ^i along  G  ; 

1  —  cr  "^ 

and  similarly  for  Un'.     Hence,  in  the  infinite  sequence,  the  functions  u  and  u' 
converge  to  a  finite  limit ;  and  since 

Un  =  Un-i  along  C,     Un  =  Un  along  C , 
this  finite  limit  is  the  same  along  both  circles  C  and  C. 

Because  m  —  w'  =  0  along  C  and  along  C,  it  follows  (§  220)  that  m  —  w'  =  0 
throughout  the  annulus.  But  u  exists  in  S'  without  C ,  and  u  exists  over 
the  whole  of  8  within  G ;  hence  (Lemma  I.  §  216)  w  and  u'  define  a  single 
function  for  S  and  8'  combined,  that  is,  for  the  Riemann's  surface  without 
any  boundary. 

It  thus  is  proved  that  a  function  exists  satisfying  all  the  assigned 
conditions ;  but  as  the  values  initially  assumed  along  G'  for  the  function  u^ 
were  arbitrary  to  some  extent,  it  is  possible  that  the  function  which  satisfies 
all  the  assigned  conditions  may  be  far  from  unique.  To  determine  this 
issue,  let  u  and  v  denote  two  functions  on  the  Riemann's  surface,  which 
everywhere  satisfy  the  general  conditions  and  which  possess  the  arbitrarily 
assigned  moduli  of  periodicity  at  the  cross-cuts.  Consider  the  function  u  —  v. 
Its  moduli  of  periodicity  are  zero :  that  is,  owing  to  the  other  characteristics 
of  u  and  v,  the  function  u  —v  is  uniform,  finite,  and  continuous  over  the 
whole  of  the  unbounded  Riemann's  surface,  and  it  therefore  (§§  220,  231) 
is  a  constant.     We  thus  infer  the  theorem : 

Real  functions  exist  on  a  Riemann's  surface,  finite  everytvhere  on  the  surface, 
and  {except  as  to  an  additive  constant)  uniquely  determined  by  their  modidi  of 
periodicity  at  the  cross-cuts,  which  modidi  are  arbitrarily  assigned  real 
constants. 

It  will  be  proved  that  the  moduli  cannot  all  be  zero  (|  231) — a  result  so 
far  anticipated  in  the  discussion  of  the  uniqueness  save  as  to  the  additive 
constant. 


228.]  WHICH   ARE   LINEARLY   INDEPENDENT  495 

228.  The  following  important  proposition  may  now  be  deduced  : — 

Of  the  real  functions,  which  satisfy  the  general  conditions  and  are  finite 
everywhere  on  the  Riemann's  surface,  and  are  determined  by  arbitrarily 
assigned  moduli  of  periodicity,  there  are  2p  and  no  nfiore  that  are  linearly 
independent  of  one  another ;  and  every  other  such  function  can  be  expressed, 
except  as  to  an  additive  constant,  as  a  linear  combination  of  multiples  of  these 
functions  luith  constant  coefiicients. 

Taking  into  account  only  real  functions,  which  satisfy  the  general 
conditions  and  are  everywhere  finite,  we  can  obtain  an  infinite  number  of 
functions  by  assigning  arbitrary  moduli  of  periodicity. 

"When  one  function  u^  has  been  obtained,  with  tuj^i,  Wj^o,  ...  Wi^op  as  its 
arbitrarily  assigned  moduli,  another  function  Uo  can  be  obtained  with 

as  its  arbitrarily  assigned  moduli  of  periodicity,  which  are  not  the  moduli  of 
kiUi,  where  k\  is  a  constant.  A  third  function  u.  can  then  be  obtained,  with 
<03,i5  «3,2j  •••)  (^s,2p  as  its  arbitrarily  assigned  moduli  of  periodicity,  which  are 
not  the  moduli  of  k\Ui  +  k^Uo,  where  ^'i  and  k.  are  constants ;  and  so  on, 
provided  that  the  number  of  functions  obtained,  say  q,  is  less  than  2p. 
When  q  <  2p,  another  function  can  be  obtained  whose  moduli  of  periodicity 

are  different  from  those  of  1  Avw,..     But  when  q  =  2j),  so  that  2p  definite 

r=l 

functions,  linearly  independent  of  one  another,  have  been  obtained,  it  is 
possible  to  determine  constants  ^'i,  ko,  ...,  k.^p,  so  that 

2/9 

,.=1 

(for  m  =  1,  2,  ...,  2p),  where  Oi,  Ho,  ...,  Ogp  are  arbitrary  constants. 

Let  U  be  the  potential  function,  which  satisfies  the  general  conditions 
and  is  finite  everywhere  on  the  surface  and  is  determined  by  the  arbitrarily 
assigned  constants  O^,  fig,  ...,  Hg^;  then  the  function 

% 

r=l 

has  all  its  moduli  of  periodicity  zero,  it  is  everywhere  finite  and,  because  its 
moduli  are  zero,  it  is  uniform  and  continuous  everywhere  on  the  surface.  It 
is  therefore,  by  §  220,  a  constant;  and  therefore 

U=    %  krUr  +  A, 
r=l 

proving  the  proposition. 

229.  The  only  remaining  condition  of  §  214  to  be  considered  is  the 
possible  possession,  by  the  function  u,  of  infinities  of  assigned  forms,  at 
assigned  positions  on  the  surface. 


496  FUNCTIONS    ON   A   RIEMANN's   SURFACE  [229. 

Let  the  infinity  at  a  place  on  the  surface,  where  z  is  equal  to  Cr,  be 
represented  by  the  real  part  of  ^{z,  c^),  where 

and  let  this  real  part  be  denoted*  by  9t^(^,  c^);  then  %l—^^{z,  Cr)  has  no 
infinity  at  z  =  Cr.  Proceeding  in  the  same  manner  with  the  other  assigned 
infinities  at  all  the  assigned  points,  we  have  a  function 

U=U-    t    jR(/)(^,  Cr), 
r=l 

which  has  no  infinities  on  the  surface.  Its  derivatives  everywhere  (save  at 
branch-points)  are  finite,  uniform  and  continuous,  and  satisfy  the  equation 
V^w  =  0.  If  T  be  a  typical  representation  of  the  assigned  boundary  values 
of  u,  and  if  <1>  be  the  corresponding  typical  representation  of  the  assigned 
boundary  values  of  S  ?R<p  (z,  Cr),  then  T  —  <E>  is  a  typical  representation  of 

r=l 

the  boundary  values  of  IT. 

The  moduli  of  periodicity  of  U  may  arise  through  two  sources :  (1) 
arbitrarily  assigned  real  moduli  of  periodicity  at  the  2p  cross-cuts  of  the 
canonical  system  (§  181),  that  are  necessary  to  resolve  the  original  surface  into 
one  that  is  simply  connected:  (2)  the  various  moduli  9i  {2'TriBr),  arising  from 
the  infinities  Cr  in  the  surface,  the  occurrence  of  which  infinities  renders  these 
additional  moduli  necessary  for  the  various  additional  cross-cuts  that  must  be 
made  in  resolving  the  surface.  Then  U  has  all  these  moduli  as  its  moduli  of 
periodicity,  it  is  finite  everywhere  on  the  surface  and,  except  for  its  moduli  of 
periodicity,  it  is  uniform  and  continuous  on  the  surface;  hence  it  is  a  function 
uniquely  determinate,  which  is  a  constant  if  all  the  moduli  be  zero. 

It  therefore  follows  that  the  determination  of  u  is  unique,  that  is,  that  a 
real  function  u  on  the  Riemanns  surface  is  determined  by  the  general  conditions 
at  all  points  on  the  surface  except  infinities,  hy  the  assignment  of  specified  forms 
of  infinities  at  isolated  points,  and  hy  the  possession  of  arbitrarily  assigned 
moduli  of  periodicity  at  the  cross-cuts  which  would  have  to  be  made  in  order 
to  resolve  the  surface  into  one  that  is  simply  connected.  And,  when  all  the 
moduli  are  zero,  the  real  function  u  is  uniform. 

Now  w,  =  u-\-  iv,  is  determined  by  u  save  as  to  an  arbitrary  additive 
constant.  Hence,  summarising  the  preceding-  results,  we  infer  the  existence 
of  the  following  classes  of  functions  on  the  surface  : — 

(A)     Functions  which  are  finite  everywhere  on  the  surface  and,  except 
at  the  lines  of  the  cross-cuts  which  suffice  to  resolve  the  surface 

*  The  form  of  <p  {z,  c,.)  implies  that  the  series  giving  the  infinite  terms  has  negative  integral 
exponents ;  the  case,  in  which  the  exponents  are  proper  fractions  so  that  the  point  is  a  branch- 
point, is  covered  by  the  transformation  of  §  223  when  the  modified  form  of  <p  explicitly  satisfies 
the  tacit  implication  as  to  form. 


229.]  WITH   ASSIGNED   INFINITIES  497 

into  one  that  is  simply  connected,  uniform  and  continuous ; 
the  functions  have,  at  these  cross-cuts,  moduli  of  periodicity, 
the  real  parts  of  which  are  arbitrarily  assigned  constants : 

(B)  Functions  which  have  a  limited  number  of  assigned  singularities 
(either  algebraical,  or  logarithmic,  or  both)  at  assigned,  isolated 
points,  and  which  otherwise  have  the  characteristics  of  the 
functions  defined  in  (A). 

The  existence  of  the  various  kinds  of  functions,  considered  in  this  chapter  in 
connection  with  a  special  form  of  Riemann's  surface,  will  be  established  for 
any  given  Riemann's  surface  in  the  next  chapter. 

^x.     Shew  that  a  function  of  the   complex   argument  s  is  determined,  save  for  a 
constant,  within  a  rectangle  drawn  upon  a  plane  representing  the  values  of  z,  by  the 

conditions  (i)  of  being  infinite  like ,  (ii)  of  having  values  at  opposite  points  of  a  pair 

of  opposite  sides  whose  difference  is  a  real  quantity  constant  for  that  pair  of  sides. 

If  ^{z),  K  (z)  be  two  such  functions,  the  former  being  infinite  like  and  having 

z  —  a 

6i,  62  for  its  real  differences  at  the  two  pairs  of  sides,  the  latter  being  infinite  like 

and  having  ^1,  ^2  foi"  its  real  differences,  prove  that 

^(.)-7r(.)  +  ^ifc^. 

is  a  function  whose  values  at  opposite  boundary  points  are  the  same. 

(Math.  Trip.,  Part  II.,  1894.) 


32 


CHAPTEE  XVIII. 

Applications  of  the  Existence-Theorem. 

230.  We  proceed  to  make  some  applications  of  the  existence-theorem 
as  established  in  the  preceding  chapter  in  connection  with  any  Riemann's 
surface,  which  is  supposed  given  geometrically  in  an  arbitrary  way.  We 
shall  first  consider  it  in  relation  with  the  functions  usually  known  as  Abelian 
transcendents. 

The  existence  of  various  classes  of  functions  of  position  has  been  established. 
Let  functions  which,  satisfying  the  general  conditions,  are  finite  everywhere 
on  the  Riemann's  surface  and  have  assigned  moduli  of  periodicity  at  the  2p 
cross-cuts,  be  called  functions  of  the  first  kind,  in  analogy  with  the  nomen- 
clature of  §§205 — 211 ;  let  functions  which,  satisfying  the  general  conditions, 
have  assigned  algebraic  infinities  on  the  Riemann's  surface  and  have 
assigned  moduli  of  periodicity  at  the  2^  cross-cuts,  be  called  functions  of 
the  second  hind;  and  let  functions  which,  satisfying  the  general  conditions, 
have  assigned  logarithmic  and  algebraic  infinities  *  and  have  assigned  moduli 
of  periodicity  at  the  '2p  cross-cuts  as  well  as  the  proper  moduli  in  connection 
with  the  logarithmic  infinities,  be  called  functions  of  the  third  kind.  These 
classes  of  functions  evidently  contain  the  integrals  of  the  three  respective 
kinds  which  arise  through  algebraic  functions.  The  three  classes  of  functions 
U,  thus  proved  to  exist  on  a  Riemann's  surface  and  characterised  by  the 
property  that,  at  the  same  position  on  opposite  edges  of  a  cross-cut,  the 
values  differ  by  a  quantity  Avhich  is  constant  along  the  cross-cut,  are  such 
that  dU/dz  is  a  uniform  function f  of  position  on  the   surface.      Thus,  by 

1 193,  we  have 

dtl      „  .        . 
-r-  =  E{w,  2), 

where  E  is  a  rational  function  of  its  arguments ;  and  therefore 

U=\R(w,z)  dz, 

*  The  logarithmic  infinities  must  be  at  least  two  in  number,  by  §  210. 

t  It  should  be  noted  that  this  property  does  not  belong  to  all  functions  on  the  Eiemann's 
surface ;  for  instance,  it  does  not  belong  to  a  function,  which  is  the  product  of  functions  of  the 
second  kind  and  third  kind,  or  to  a  function  which  is  not  linear  in  functions  of  the  three  kinds 
specified. 


280.] 


APPLICATIONS   OF   THE   EXISTENCE-THEOREM 


499 


a  result  that  indicates  the   importance   of  integrals  of  rational   functions 
associated  with  a  Riemann's  surface. 

First,  let  P  and  Q  be  two  functions  of  x  and  y,  the  derivatives  of  which 
are  finite,  uniform  and  continuous  at  all  points  (except  possibly  branch-points) 
on  the  given  Riemann's  surface  and  satisfy  the  equation  V-u  =  0.  Let  the 
functions  themselves  be  finite  and,  except  at  cross-cuts,  uniform  and 
continuous  on  the  surface:  and  let  their  moduli  of  periodicity  be  A-^,  ..., 
Ap,  Bi,  ...,  Bp]  A-[,  ...,  Ap',  Bi,  . . . ,  Bp,  for  the  cross-cuts  a^,  ...,  ap,  h^,  ...,bp 
respectively,  the  moduli  for  the  cross-cuts  c  being  zero.  (If  P  and  Q  should 
have  infinities  on  the  surface,  as  will  be  the  case  in  later  applications,  so  that 
in  their  vicinity  portions  of  the  surface  are  excluded,  thereby  requiring  other 
cross-cuts  for  the  resolution  of  the  surface  into  one  that  is  simply  connected, 
other  moduli  will  be  required ;  but,  in  the  first  instance,  P  and  Q  have 
merely  the  2p  assigned  moduli.) 

When  the  surface  is  resolved  by  the  2p  cross-cuts  into  one  that  is  simply 
connected,  the  functions  P  and  Q  are  uniform,  finite  and  continuous  over 
the  resolved  surface.     Proceeding  as  in  §  16  and  §  216,  we  have 


dP^Q_dQdPx 

doc  dy      doc  dy  J 


-g^- 


P  TT-TT-  docdv 
docdy        ^ 

F  :^dx—  j  IP ^-^  docdy 


-ri'^-H- 


where  the  double   integrals    extend  over  the  whole  area  of  the   resolved 

surface,  and  the  single  integrals  extend  positively  round 

the  whole  boundar}^     This  boundar}^  is  composed  of  a 

single  curve,  composed  of  both  edges  of  each  of  the 

cross-cuts;  and  the  positive  directions  of  the  description 

are  indicated  in  the  figure,  at  a  point  of  intersection  of 

two  cross-cuts.    - 

As  explained  in  §  196,  the  negative  edge  of  the  cross-  -p.     g^ 

cut  Qr  is  GE  and  the  positive  edge  is  DF;  the  negative 
edge  of  the  cross-cut  br  is  EF,  and  the  positive  edge  is  CD.     Then  we  have 

Pjy-Pj.^Pc-PE  =  B,,       PF-PE^PB-Pc=Ar; 

and  similarly  for  the  function  Q. 

Consider  the  integral  JPdQ,  taken  along  the  two  edges  of  the  cross-cut 
Ur :  let  P_  and  P+  denote  the  functions  along  the  negative  and  the  positive 

32—2 


500  PROPERTIES   OF   A  [230. 

edges  respectively,  so  that  P+  —  P_  =  J.^.     The  value  of  the  integral  for  the 
two  edges  is 

P+dQ,  taken  in  the  direction  F...D 

J  F 
rJB 
+       P-dQ,  taken  in  the  direction  G...E 

J  c 

=       (P+  -  P_) dQ,  taken  in  the  direction  F...D 

J  F 

CD 

=  AJ      dQ  =  Ar{QD-QF)  =  ArBr'. 

J  F 

Similarly,  when  the  value  of  the  integral  for  the  two  edges  of  the  cross-cut  6,- 
is  taken,  we  have 

P+dQ,  taken  in  the  direction  D...G 

.'  D 

cF  *_  _ 

+       P-dQ,  taken  in  the  direction  E...F 

J  E 

[^  .  .  .' 

=       (P+  —  P_) dQ,  taken  in  the  direction  D ...G 

}  D 

=  Br\''  dQ^Br  {Qc  -Ql>)  =  -  BrAr'. 
J  D 

And  the  value  of  the  integral  for  the  combination  of  the  two  edges  of  any 
cross-cut  c  is  zero. 

Hence  summing  for  the  whole  boundary  of  the  resolved  surface,  we  have 

[pdQ  =  I  {ArB;  -  BrAr'), 

J  r=l 

and  therefore  '  ■ 

subject  to  the  assigned  conditions. 

This  theorem  is  of  considerable  importance :  and  the  conditions,  subject 
to  which  it  is  valid,  permit  P  and  Q  (or  either  of  them)  to  be  real  or  complex 
potential  functions  of  oc  and  y  or  to  be  a  function  of  z. 

231.  As  a  first  application,  let  P  and  Q  be  real  potential  functions  such 
that  P  -I-  iQ  is  a  function  of  z,  say  w,  evidently  a  function  of  the  first  kind. 
Let  its  moduli  for  the  cross-cuts  be 

cog  +  ivg  at  tts,  for  s  =  1,  2,  ... ,  _p ; 

and  Wg+ivs    at  &«,  for  s  =  l,  2,  ...,p. 

Since  P  -F  iQ  is  a  function  of  ^  -1-  iy,  we  have,  by  §§  7,  8, 

dP^dQ       _dP^dQ 

dx      dy  '         dy      dx' 


231.]  SYSTEM   OF   MODULI  501 

The  double  integral  then  becomes 

which  cannot  be  negative,  because  P  is  real ;  it  is  a  quantity  that  is  positive 
except  only  when  P  (and  therefore  w)  is  a  constant  everywhere.  In  the 
present  case 

Ar  =  <Wri    Bg  =  Wg   ;       Ar   =  Vr,    Bg   =  Vg  , 
V 

so  that  %  ((OrVr  —  ct)/u,.)  is  always  positive.     Hence  : 

r=l 

If  a  function  w,  everywhere  finite  on  a  Riemann's  surface,  have  cog  +  ivg  at 
ag  {for  s  =  1,  2,  ...,p)  and  <o,'  +  ivg  at  bg  {for  s  =  l,2,  ...,p)  as  its  moduli, 
the  cross-cuts  a  and  b  being  the  2p  cross-cuts  necessary  to  resolve  the  surface 
into  one  that  is  simply  connected,  the  quantity 

S    {o)r^r'  ~  (i),'V).)  ' 
r=l 

is  always  positive,  unless  %u  is  a  constant :  and  then  it  is  zero. 
This  proposition  has  the  following  corollaries. 

CoROLLAEY  I.  A  function  of  z  of  the  first  kind  cannot  have  its  moduli  of 
periodicity  for  ai,  ...,  Up  all  zero. 

For  if  all  these  moduli  were  to  vanish,  then  each  of  the  quantities  cOr  and 

each  of  the  quantities  Vr  would  be  zero :    the  sum  S  (wrV  —  w/i;^)  would 

then  vanish,  which  cannot  occur  unless  w  be  a  constant. 

Corollary  II.  A  function  of  z  of  the  first  kind  cannot  have  its  moduli  of 
periodicity  for  b^,  ...,  bp  all  zero ;  it  cannot  have  its  moduli  of  periodicity  all 
purely  real,  or  all  purely  imaginary,  or  some  zero  and  all  the  rest  either 
purely  real  or  purely  imaginary. 

The  different  cases  can  be  proved  as  in  the  preceding  Corollary, 

Note.  One  important  inference  can  at  once  be  derived,  relative  to 
functions  of  the  first  kind  that  have  only  two  moduli  of  periodicity, 
Hi  and  Xlg. 

Neither  of  the  moduli  may  vanish;  for  if  one,  say  flj,  were  to  vanish 
then  wjQ^o  would  be  a  function  having  one  modulus  zero  and  the  other  unity. 

The  ratio  of  the  moduli  may  not  be  real.  If  it  were  real,  then  wjQ.-,  would 
be  a  function  having  one  modulus  unity  and  the  other  real.  Both  of  these 
inferences  are  contrary  to  Corollary  II. ;  and  therefore  the  ratio  of  the  two 
moduli  .is  a  complex  constant,  the  real  part  of  which  may  vanish  but  not  the 
imaginary  part. 


502  PROPERTIES   OF  [231v 

The  association  of  this  result  with  the  doubly-periodic  functions  is 
immediate. 

Ex.  Shew  that,  if  two  functions  of  the  first  kind  have  the  same  moduli  of  periodicity, 
their  difference  is  a  constant :  and  that,  if  TF  be  a  value,  at  any  point  of  the  surface,  of  a 
function  of  the  first  kind  with  moduh  ©i,  a^,  ...,  o),,,,  all  the  functions  of  the  first  kind, 
which  have  those  moduli,  are  included  in  the  form 

r=l 

where  the  coefiicients  m.  are  integers  and  J.  is  a  constant. 

232.  As  a  second  application,  let  P  be  a  function  of  z  and  Q  also  a 
function  of  z ;  evidently,  with  the  restriction  of  the  proposition,  P  and  Q 
must  be  functions  of  the  first  kind,  when  no  part  of  the  surface  is  excluded 
from  the  rano'e  of  variation  of  z.     Then 

.dP_dP        .dQ^dQ 
dx      dy  '        dx      dy  ' 

so  that  at  every  point  on  the  surface  we  have 

dPoQ_dQdP^^ 

dx  dy      dx  dy 

Consequently  the  double  integral 

and  therefore,  if  a  function  of  the  first  kind  have  moduli  A^, ...,  Ap,  B^, ...,  Bp, 
and  if  any  other  function  of  the  first  kind  have  moduli  A-^,  ...,  Ap,  P/,  .,.,  Bp 
at  the  C7'oss-cuts  a  and  b  ^respectively,  then 

i  (^,.P,'-P,^/)  =  0. 

?•  =  ! 

233.  Next,  let  Q  be  a  function  of  z  of  the  first  kind,  as  in  the  preceding 
case ;  but  now  let  P  be  a  fimction  of  z  of  the  second  kind,  so  that  all  its 
infinities  are  algebraical.  The  points  where  the  function  is  infinite  must  be 
excluded  from  the  surface :  a  corresponding  number  of  cross-cuts  will  be 
necessary  for  the  resolution  of  the  surface  into  one  that  is  simply  connected. 
The  modulus  of  periodicity  of  P  for  each  of  these  cross-cuts  is  zero,  (as  in 
Ex.  8  of  §  199,  which  is  an  instance  of  a  function  of  this  kind),  no  additional 
modulus  being  necessary  with  an  algebraical  infinity. 

Then  over  the  resolved  surface,  thus  modified,  the  functions  P  (z)  and 
Q  (z)  are  everywhere  uniform,  finite  and  continuous :  and  therefore,  as 
before 


233.]  MODULI  503 

the  double  integral  extending  over  the  whole  of  the  resolved  surface  and  the 
single  integral  extending  round  its  whole  boundary.  But,  at  all  points  in 
the  resolved  surface,  we  have 

dPdQ_dQdP^^ 

dx  dy       dec  dy        '      ■■ 

and  therefore,  as  before,  the  double  integral  vanishes.  Hence  jPdQ,  taken 
round  the  whole  boundary,  vanishes. 

The  boundary  is  made  up  of  the  double  edges  of  all  the  cross-cuts  a,  h, 
and  those,  say  I,  which  are  introduced  through  the  infinities,  and  of  the  small 
curves  round  the  infinities. 

As  in  §230,  the  value  of  the  integral  for  the  two  edges  of  a.r  is  J.  ,.5/; 
and  its  value  for  the  two  edges  of  hr  is  —BrAr'.  The  value  of  the  integral 
for  the  two  edges  of  any  cross-cut  I  is  zero,  because  the  subject  of  integration 
is  the  same  along  the  edges  which  are  described  in  opposite  directions. 

To  find  the  value  round  one  of  the  small  curves,  say  that  which  encloses 
an  infinity  represented  analytically  by  a  value  Cg  of  z,  we  take,  in  the  imme- 
diate vicinity  of  Cg, 

Z-Cs 

where  p{z  —  Cg)  is  a  converging  series  of  positive  integral  powers  of  ^  —  Cg.  In 
that  vicinity,  let 

Q  =  Qg  ■{-  (z  —  Cg)  Qs  +  higher  powers  of  z  —  Cg, 

so  that  Qs  is  dQ/dz  for  z  =  Cs;  thus 

dQ  =  (Qg'  +  positive  powers  of  ^;  —  Cg)  dz. 

Hence  along  the  small  curve 

dz 
PdQ  =  EgQg' +  q(z-  Cg) dz, 

Z-  Cg 

where  q{z  -  Cg)  is  a  converging  series  of  positive  integral  powers  of  z  —  Cg. 
The  value  of  the  integral  round  the  curve  is  27riHgQ/. 

Summing  these  various  parts  of  the  integral  and  remembering  that  the 
whole  integral  is  zero,  we  have 

I  (ArBr'-BrA;)  +  27rilHgQg'  =  0, 

)•  =  ! 

there  being  as  many  terms  in  the  last  summation  as  there  are  simple 
infinities  of  P. 


The  equation 

p 


S  {ArB;  -  Br  a;)  -f-  2'7Ti2Hg  (^)        =  0 

r=l  s  V"'^  '  z=Cf 


504  EELATIONS   BETWEEN   MODULI  [233. 

is  the  relation  which  subsists  between  the  moduli  A',  B'  of  a  function  Q  (z)  of 
the  first  kind  and  the  modidi  A,  B  of  a  function  P (z)  of  the  second  kind, 
all  the  infinities  of  which  are  simple. 

The  simplest  illustration  is  furnished  by  the  integrals  that  were  considered  in  Ex.  6 
and  Ex.  8  of  §  199.  ^ 

Let  P  be  the  function  of  Ex.  8,  usually  denoted  by  E  {z),  being  the  elliptic  integral 
of  the  second  kind  ;  it  is  infinite  for  2  =  oo  in  each  sheet.  In  the  upper  sheet  we  have, 
for  large  values  of  1 2 1 , 

P=E  {z)=kz  (l+positive  integral  powers  of  -  j  ; 
and  for  the  same  in  the  lower,  we  have 
P 


E  {z)=  —  kz  { 1  +  positive  integral  powers  of  -  j 


Let  Q  be  the  function  of  Ex.  6,  usually  denoted  by  F  {z),  being  the  eUiptic  integral 
of  the  first  kind,  finite  everywhere.  We  easily  find,  for  large  values  of  \z\  in  the  upper 
sheet,  that 

dQ=dE{z)  —  j-x  (l+positive  integral  powers  of  -j  dz, 


kz^ 
and,  for  large  values  of  |  s  |  in  the  lower,  that 

dQ  =  dF  (2)  =  —  -r-j  (  1  +  positive  integral  powers  of  -  1  dz. 

Then  for  large  values  of  |  s  |  in  the  upper  sheet,  we  have 

PcZ$=  —  f  l  +  positive  integral  powers  of  -j 

dz' 
=  — Y  (l  +  positive  integral  powers  of  s'), 

where  zz'  =  \;   and  we  may  consider  the  Riemann's  surface  spherical.     Hence  the  value 
round  the  excluding  curve  in  the  upper  sheet  is  —  27ri. 

Similarly  the  value  round  the  excluding  curve  in  the  lower  sheet  is  —  27^^'. 

Now  A-^  and  B^,  the  moduli  of  P,  are  4:E  and  2i  {K'  -  E')  respectively  ;  A{  and  B.^,  the 
moduli  of  Q,  are  4X  and  '2.iK'  respectively.     Hence 

4.E .  2iK'  -  AK .  2^■  {K'  -  E')  -  47^^■ = 0, 

leading  to  the  Legendrian  equation 

EK'  +  E'E-KK'  =  \tz. 

234.  Before  proceeding  to  the  relations  affecting  the  moduli  of  periodicity 
of  functions  of  the  third  kind,  we  shall  make  some  inferences  from  the 
preceding  propositions. 

It  has  been  proved  that  functions  of  the  first  kind,  special  examples  of 
which  arose  as  integrals  of  algebraic  functions,  exist  on  a  Riemann's  surface. 
They  are  everywhere  finite  and,  except  for  additive  multiples  of  the  moduli, 
they  are  uniform  and  continuous ;  and  when,  in  addition  to  these  properties, 
the  real  parts  of  their  moduli  of  periodicity  are  arbitrarily  assigned,  the 


234.]  LINEARLY   INDEPENDENT   FUNCTIONS  505 

fanctions  are  uniquely  determinate.     Hence  the  number  of  such  functions  is 
unlimited :  they  are,  however,  subject  to  the  following  proposition : — 

The  number  of  linearly  independent  functions  of  the  first  kind,  that  exist  on 
a  given  Riemanns  surface,  is  equal  to  p ;  where  2^  +  1  is  the  connectivity  of 
the  surface.     And  every  function  of  the  first  kind  on  that  surface  is  of  the 

p 
form  C+  2  CqtUg,  where  C  is  a  constant,  the  coefiicients  Cj,  ...,Cp  are  constants, 
3=1 

a7id  Wi,  ...,Wp  are  p  linearly  independent  functions. 

Let  q  sets  of  linearly  independent  real  quantities,  each  set  containing 
2p  non-vanishing  constants,  be  arbitrarily  assigned  as  the  real  parts  of  the 
moduli  of  periodicity  of  functions  of  the  fii'st  kind,  which  are  thence  uniquely 
determined.  Let  the  functions  be  w^,  w.^,  ...,  Wq]  and  let  the  real  parts  of 
their  moduli  be  (&)i_i,  (Wj^,)  ••• .  <Wi,2^)>  (&>2,i,  6^2,2,  •••>  «2,2f)))  •••>  ('"g,!.  ft>q,-2,  •■■,c^q,2p)- 
The  modulus  of  wv  at  the  cross-cut  G^n  has  its  real  part  denoted  by  &)^_,„  : 
when   the   modulus   is   divided   into   real    and   imaginary  parts,  let   it  be 

^r,  m    1    ^^  r,  m  • 

If  any  set  of  g  arbitrary  complex  constants  be  denoted  by  Ci,  ...,  Cq,  where 
Cg  is  of  the  form  Og  +  i^g,  then,  at  the  cross-cut  Cm,  the  real   part  of  the 

modulus  of  2  Cr^i'r  is  the  real  part  of  1  Cr((Or,m  +  *'&>';•, wX  that  is,  it  is  equal  to 

r=l  r=l 

Ol<«l,»i+  •••  +  ^q(^q,m-  0lf^\,m—   •••  —  0q(^'q,m, 

holding  for  ??i  =  l,  2,  ...,  2p,  and  therefore  giving  2^)  expressions  in  all. 

Now  let  a  set  of  real  arbitrary  quantities  A^,  A.^,  ...,  A^p  be  assigned  as 
the  real  parts  of  the  moduli  of  periodicity  of  a  function  of  the  first  kind, 
which  is  uniquely  determined  by  them ;  and  consider  the  equations 

-^1=  «!&>!,  1-1-  a2&)2,l+  ...  -I-  OfgW^^l  —  ZSj  ft)'i_i  —  yS2  <»'2,1—  •••  —y^^tu'j^i  1 


A.p=  ai<Oi,22J  +  a2&>2,22)+  •••  +  '^qfOq,2p  -  ^lO)\,op  —  ^2(^'2,2p  -  •••  —  ^q<»'q, 


2p) 


First,  let  q<p:  the  2q  constants  a  and  Q  can  be  determined  so  as  to  make 
the  right-hand  sides  respectively  equal  to  2q  arbitrarily  assigned  constants  A. 
The  right-hand  sides  of  the  remaining  equations  are  then  determinate  con- 
stants; and  therefore  the  remaining  equations  will  not  be  satisfied  when  the 
remaining  constants  A  are  arbitrarily  assigned. 

The  function  determined  by  the  moduli  A  has  some  of  its  moduli  different 
from  those  of  the  function  ^cw,  when  q  <  p;  hence,  when  q  functions 
Wi,  ...,  Wq,  where  q<p,  have  been  obtained,  we  can  obtain  another  function, 
and  so  on ;  until  q  =  p. 

But,  when  q  =  p,  then  the  foregoing  2p  equations  determine  the  quantities 
a  and  /3,  whatever  be  the  quantities  A.     Let  W  be  the  function  of  the  first 


506  INDEPENDENT    FUNCTIONS  [234. 

kind,  determined  by  the  constants  A  as  the  arbitrarily  assigned  real  parts  of 

its  moduli  of  periodicity  :  then 

p 

F-    S    CsWs, 
s  =  l 

where  the  coefficients  c  are  constants,  has  the  real  parts  of  all  its  moduli  of 
periodicity  zero :  it  is  therefore,  by  Cor.  II.  §  231,  a  constant,  so  that 

Tf  =  Ci^i  +  ...  +  Cp'Wp  +  G, 

where  C  is  a  constant.  Therefore  the  number  of  linearly  independent 
functions  of  the  first  kind  is  p ;  and  every  function  of  the  first  kind  is  of 

the  form 

p 

C  +    1    CsWs. 

It  has  been  assumed,  in  what  precedes,  that  the  determinant  of  the  quanti- 
ties CO  and  ft)'  does  not  vanish.  This  possibility  is  not  excluded  merely  by  the 
arbitrary  choice  of  the  quantities  &) ;  because  the  quantities  co'  are  determined 
for  w,  and  w  is  dependent  on  v.  If,  however,  the  determinant  should  vanish, 
then,  by  taking  the  quantities  a  and  (3  proportional  to  the  minors  of  co  and  co' 
respectively  in  the  determinant,  all  the  quantities 

E 

Z    {C(gO)g,n  —  I3s(0  sm) 
s=l 

could  be  made  to  vanish.     These  quantities  are  the  real  parts  of  the  moduli 

^  . 

of  periodicity  of  2  CgiVg  which,  because  they  all  vanish,  is  a  constant,  that  is, 

the  quantities  Wg  are  not  linearly  independent  of  one  another- — an  inference 
contrary  to  their  construction.  Hence  the  determinant  of  the  quantities  co  does 
not  vanish. 

Note.  It  may  be  remarked,  in  passing,  that  each  function  w,  being  of  the 
first  kind,  gives  rise  to  two  real  potential  functions,  which  are  everywhere 
finite  and  have  moduli  of  periodicity  at  the  cross-cuts :  one  of  the  functions 
being  the  real  part  of  iv,  the  other  arising  from  its  imaginary  part. 
Hence  from  the  p  linearly  independent  functions  of  the  first  kind,  there  are 
altogether  2p  linearly  independent  real  potential  functions.  This  number  is 
the  same  as  the  total  number  of  real  potential  functions  considered  in  §  228  : 
hence  each  of  them  can  be  expressed  as  a  linear  function  of  the  members  of 
that  former  system,  save  possibly  as  to  an  additive  constant.  Conversely,  it 
follows  that  linear  combinations  of  the  members  of  that  former  system  can  be 
taken  in  pairs,  so  as  to  furnish  p  (and  not  more  than  p)  linearly  independent 
functions  of  z  of  the  first  kind. 

235.  The  functions  so  far  obtained  are  very  general :  it  is  convenient  to 
have  a  set  of  functions  of  the  first  kind  in  normal  forms.  The  foregoing 
analysis  indicates  that  linear  combinations   of  constant    multiples   of  the 


235.]  NORMAL   FUNCTIONS   OF   THE    FIRST   KIND  507 

functions,  being  themselves  functions  of  the  first  kind,  are  conveniently 
considered  from  the  point  of  view  of  their  moduli  of  periodicity :  and  the 
simpler  the  aggregate  of  these  moduli  is,  the  simpler  will  be  the  expressions 
for  the  functions  determined  by  them.  Some  conditions  have  been  shewn 
(§  231)  to  attach  to  the  aggregate  of  the  moduli  for  any  one  function  of  the 
first  kind,  and  a  condition  (§  232)  for  the  moduli  of  different  functions ;  these 
are  the  conditions  that  limit  the  choice  of  linear  combinations. 

Let  CiWi+  ...  +CpiVp  be  a  linear  combination  of  the  functions  Wi,  ...,  Wp 
which  have  (On,  ...,  «,.^  (r  =  1,  ...,p)  as  the  moduli  of  periodicity  for  the 
cross-cuts  ai,  ...,  a.p.     Then  A,  where  A  is  the  determinant 


A  = 


&>n,   C0i2,    ,   0)ip 


pi  >    f"p2  >    )    f"pp 

cannot  vanish :  for  otherwise  by  taking  constants  Cj,  ...,Cp  proportional  to  the 

p 
first  minors,  we  should  obtain  a  function  S  CgiVg,  having  all  its  moduli  for 

the  cross-cuts  a-^,  ...,  ap  zero  and  therefore,  by  §  231,  merely  a  constant,  so 
that  Wi,  ...,  iVp  would  not  be  linearly  independent.     Hence  A  does  not  vanish. 

Next,  we  can  choose  constants  c  so  that  the  moduli  of  periodicity  vanish 

p 
for  the  function  S  CgWs  at  all  the  cross-cuts  a,  except  at  one,  say  a,.,  and 

that  there  it  has  any  assigned  value,  say  iri.     For,  solving  the  equations, 

0  =  Cia>s,i  + c.20)s^o  + ...  +  CpCOg^p,     (for  s  =  1,  2,  .,.,  p,  except  r); 

Tri  =  Ci  (Or,  1  -f  Co  &)r,  2  +  •  •  •  +  Cp(Or,p, 

the  determinant  of  the  right-hand  side  does  not  vanish,  and  the  constants  c, 
say  Cr^i,  c,.,2, ...,  Cr,p,  are  determinate.  The  function  Cr^iW-^  +  0^,2^2  +  •■•  +  Gr,pWp, 
say  Wy,  then  has  its  moduli  zero  for  ttj,  ...,  ttr-i,  ^r+u  ...,  a^:  it  has  the 
modulus  iri  for  a^;  it  has  moduli,  say  5^,1,  ...,  B^^p  at  b^,  ...,bp  respectively. 
And  the  function  is  determinate  save  as  to  an  additive  constant. 

This  combination  can  be  effected  for  each  of  the  values  1,  ...,p  of  r: 
and  thus  p  new  functions  will  be  obtained.  These  p  functions  are  linearly 
independent :   for,  if  there  were  a  relation  of  the  form 

GjW^  +  C2W2  +  . . .  +  GpWp  =  constant, 

p 
the  modulus  of  the  function  2  GrWr  at  the   cross-cut  a^  would   be    zero 

r=l 

because  the  function  is  a  constant ;  and  it  is  Cs'jri,  so  that  all  the  coefficients 
0  would  be  zero. 


508  NOEMAL  FUNCTIONS   OF  THE   FIRST  KIND 

The  functions  W,  thus  obtained,  have  the  moduli : — 


[235. 


«l 

«2 

dp 

&1 

&2 

&P 

w. 

7^^■ 

0 

0 

A.1 

A,  2 

a:p 

^2 

0 

7^^ 

I     0 

A,l 

^2,2 

-^2,p 

w. 

0 

0 

TTl 

-^p.l 

^P,2   ! 

^P,P 

These  functions  are  called  normal  functions  of  the  first  kind :  they  are  a 
complete  system  linearly  independent  of  one  another,  and  are  such  that 
every  function  of  the  first  kind  is,  except  as  to  an  additive  constant,  a  linear 
combination  of  constant  multiples  of  them. 

The  quantities  B  are  not  completely  independent  of  one  another.     Since 
Wj,  Wf  are  functions  of  the  first  kind  we  have,  by  §  232, 

>•  =  ! 

which,  for  the  normal  functions,  takes  the  form 

•jriBjf  —  iriBj'j  =  0, 

that  is,  Bjf  =  Bj'j.  Hence  the  moduli  B  with  the  same  pair  of  integers  for 
sufiix  are  equal  to  one  another. 

This  is  a  first  relation  among  the  moduli.     Another  is  given  by  the 
following  theorem : — 

Let    B„i^n  =  Pm,n  + 'i(^m,n,    (sO    that    Pm,n  =  Pn,my    Cl^d    CT^^n  =  (^7i,m)  '•     then,    if 

Ci,  ...,  Cp  he  any  real  quantities,  the  expression 

pnCi^  +  2/!3i2CiC2  +  p^Co'+  ...  +  PppCp% 

is  negative,  unless  the  quantities  c  vanish  together. 

The  function  c-iW-^  +  c..W<i-\- ...  +CpWp  +  C  is  a  function  of  the  first  kind 
with  moduli  (say)  co,.  +  ivr  at  a,.,  where  r  =  l,  ...,  p,  and  moduli  Wg  +  ivg  at  h^, 

V 

where  s~\,  ...,p.     Then,  by  §231,  the  sum   S  (w^.u/ —  w/u,.)  is  positive, 

except  when  the  function  is  a  constant,  that  is,  except  when  Ci,  ...,  Cp  all 
vanish.     But 

«,.  +  iVr  ■=■  CrTri, 

SO  that  (Or  =  0,  Vr  =  TTCy ',    and 

<w/  +  iv.s  =  CiB^s  +  CzB-i^s  +  . . .  +  CpBp^s, 
so  that  ft)/  =  Ci/3i,s  +  C2po,s  +  . . .  +  Cppp^s- 


235.]  FUNCTIONS   OF   THE   SECOND   KIND  509 


•P 


Hence  the  sum  S  —  Cy7r(ci/3],r  +  C2/32,r  +  •••  +  c^p^,,.) 

is  positive  and  therefore  the  sum  S   S  p^sCrC,,  is  negative.     This  (with  the 

r=\s=\ 

property  p^„  =  pnm)  is  the  required  result. 

These  properties  of  the  periods,  all  due  to  Riemann,  are  useful  in  the 
construction  of  the  Theta-Functions. 

-  For  the  ordinary  Jacobian  elliptic  functions  in  which  'p  =  \,  there  is  only 
one  integral  which  is  everywhere  finite  :  its  periods  are  4j5r,  liK'.  To  express 
it  in  the  normal  form,  we  take  cF  {z),  choosing  c  so  that  the  period  at  0^  is 

purely  imaginary  and  =  iri;  hence  c  =  t^>  and  the  normal  integral  is 

TriFjz) 

ttK' 
The  other  period  of  this  function  is  —  ^rfr,  which,  when  k  is  real  and  less  than 

unity,  is  a  negative  quantity ;  it  is  the  value  of  p^  and  satisfies  the  condition 
that  puCi  is  negative  for  all  real  quantities  c. 

236.  It  has  been  proved  that  functions  exist  on  a  Riemann's  surface,, 
having  assigned  algebraic  infinities  and  assigned  real  parts  of  its  moduli 
of  periodicity,  but  otherwise  uniform,  finite  and  continuous.  The  simplest 
instance  of  these  functions  of  the  second  kind  occurs  when  the  infinity  is  an 
accidental  singularity  of  the  first  order. 

Let  the  single  infinity  on  the  surface  be  represented  by  ^^  =  c  :  let  Ec  {z} 
be  the  function  having  5  =  c  as  its  algebraical  infinity,  and  having  the  real 
parts  of  its  moduli  of  periodicity  assigned.  If  Eg  {z)  be  any  other  function 
with  that  single  infinity  and  the  real  parts  of  its  moduli  the  same,  then 
Eci^)  —  Ec  (z)  is  a  function  all  the  real  parts  of  whose  moduli  are  zero;  it 
does  not  have  c  for  an  infinity  and  therefore  it  is  everywhere  finite :  by  §  231,  it 
is  a  constant.  Hence  an  elemeyitary  function  of  the  second  kind  is  determined, 
save  as  to  an  additive  constant,  by  its  infinity  and  the  real  parts  of  its  moduli. 

Again,  it  can  be  proved,  as  for  the  special  case  in  §  208,  that  an  elementary 
function  of  the  second  kind  is  determined,  save  as  to  an  additive  function  of 
the  first  kind,  by  its  infinity  alone  :  hence,  if  E  (z)  be  any  elementary  function,, 
having  its  infinity  at  ^^  =  c,  we  have 

E  (z)  =  Ec(z)  +  \W,+  ...  +  \pWp  +  A, 

where  \,  ...,\p,A  are  constants,  the  values  of  which  depend  on  the  special 
function  chosen.  Let  E^z)  have  TriC^,  ...,  TriCp  for  its  moduli  at  the  cross- 
cuts «!,...,  a^  respectively :   and  let  the  function  E  (z)  be  chosen  so  as  to- 


510  NOEMAL   FUNCTION    OF   THE   SECOND    KIND  [236. 

have  all  its  moduli  at  a-^,  ...,ap  equal  to  zero :  then  \,.  =  —  6',.  and  £" (z)  is 
given  by 

The  special  function  of  the  second  kind,  which  has  all  its  moduli  at  the  cross- 
cuts tti,  ...,  Op  equal  to  zero,  is  called  the  normal  function  of  the  second  kind. 
It  is  customary  to  take  unity  as  the  coefficient  of  the  infinite  term,  that  is, 
the  residue  of  the  normal  function. 

This  normal  function  is  determined,  save  as  to  an  additive  constant,  by  its 
infinity  alone.   For  HE  {z)  and  E'{z)  be  two  such  normal  functions,  the  function 

E{z)  -  E'{z) 
is  finite  everywhere;  its  moduli  are  zero  at  ttj,  ...,  o^;  hence  (§231)  it  is  a 
constant. 

Normal  functions  of  the  second  kind  will  be  used  later  (§  240)  in  the 
construction  of  functions  with  any  number  of  simple  infinities  on  the  surface. 

Let  the  moduli  of  this  normal  function  E  (z)  of  the  second  kind  be 
Bj,  ...,  Bp  for  the  cross-cuts  h^,  ...,  bp.  Then  applying  the  proposition  of 
§233  and  considering  the  integral  [EdW,.,  we  have  A^=  ...  =Ap  =  0 ;  also 

Ai  =  ...  =  A  r—i  —  A  r^i  =  ...  =  Ap  =  0, 

and  Ar  =  7ri.     The  relation  therefore  is 

„     .     ^    .fdWr\ 
-  Br-m  -h  27^^  I  -~         =  0, 
V  djZ  J  z=c 

where,  in  the  immediate  vicinity  of  ^^  =  c, 

E{z)  =  j--^+p{z-c), 
p  being  a  converging  series  of  positive  powers.     Thus 

dW 
or,  as  —j-^  is  an  algebraic  function  (§  241)  on  the  surface,  the  periods  of  a 

normal  function  of  the  second  kind  at  the  cross-cuts  h  ai^e  algebraic  functions 
of  its  single  infinity. 

In  th«  case  of  the  Jacobian  ellii^tic  integrals,  the  integral  of  the  second  kind  has  at 
z=  00  an  infinity  of  the  first  order  in  each  sheet  (Ex.  8,  §  199).  The  moduli  of  this  integral, 
denoted  by  E  (z),  are  4E  and  2i  (K'  -  E')  for  a^  and  b^  respectively ;  hence  the  normal 
integral  of  the  second  kind  is 

F{z)  being  the  (one)  integral  of  the  first  kind.     This  is  the  function  Z{z).     Its  modulus  is 
zero  for  aj ;    for  hi ,  it  is 

^i{K'-E')-^^2iK', 

which  is  -r^{KK'  - E' K - EK'),  that  is,  the  modulus  is  —^- 


237.]  NORMAL   FUNCTION    OF   THE   THIRD    KIND  511 

237.  The  other  simple  class  of  function,  which  exists  on  a  Riemann's 
surface  with  assigned  infinities  and  assigned  real  parts  of  its  moduli,  is  that 
which  is  represented  by  the  elementary  integral  of  the  third  kind.  It  has 
two  points  of  logarithmic  infinity  on  the  surface*,  say  Pi  and  P^;  let  these 
be  represented  by  the  values  Ci  and  Ca  of  z.  On  division  by  a  proper  constant, 
the  function,  which  may  be  denoted  by  Ilia,  takes  the  forms 

-  log  {z  -  Ci)  +  pi{z-  Ci),     +  log  {z  -  Ca)  +  i?2  (^  -  C2), 

in  the  immediate  vicinities  of  Pi  and  of  Pg  respectively,  where  p^  and  p2  are 
converging  series  of  positive  integral  powers. 

The  points  Pi  and  P2  can  be  taken  as  boundaries  of  the  surface,  as  in 
Ex.  7  in  §  199.  A  cross-cut  fi-om  P2  to  Pi  is  then  necessary  for  the  resolution 
of  the  surface  :  and  the  period  for  the  cross-cut  is  ^iri,  being  the  increase  of  the 
function  in  passing  from  the  negative  to  the  positive  edge  of  the  cross-cut. 

Then  with  this  assignment  of  infinities  and  with  the  real  parts  of  the 
moduli  at  the  cross-cuts  tti,  ...,  a^,  61, ... ,  6^  arbitrarily  assigned,  functions  Ilia 
exist  on  the  Riemann's  surface. 

As  in  the  case  of  the  function  of  the  second  kind,  it  is  easy  to  prove  that  a 
function  Oio  of  the  third  kind  is  determined,  save  as  to  an  additive  constant,  by 
its  two  infinities  and  the  assignment  of  its  moduli :  and  that  it  is  determined, 
save  as  to  an  additive  function  of  the  first  kind,  by  its  infinities  alone. 

Among  the  infinitude  of  elementary  functions  of  the  third  kind,  having 
the  same  logarithmic  infinities,  a  normal  form  can  be  chosen  in  the  same 
manner  as  for  the  functions  of  the  second  kind.  Let  IIio  be  an  elementary 
function  of  the  third  kind,  having  Pi  and  Po  for  its  logarithmic  infinities  : 
let  its  moduli  of  periodicity  be  ^iri  for  the  cross-cut  P1P2;  Trt'Ci,  ...,  iriCp  for 
ai,  ...,  ap  respectively;  and  other  quantities  for  h^,  ...,  hp  respectively.     Then 

^,,=  Ii,,-G,W,-...-GpWp 

is  an  elementary  function  of  the  third  kind,  having  zero  as  its  modulus  of 
periodicity  at  each  of  the  cross-cuts  a^,  ...,  ap.  This  function  is  the  normal 
form  of  the  elementary  function  of  the  third  kind. 

If  -5712'  and  ffiTia  be  two  normal  elementary  functions  of  the  third  kind  with 

the   same  logarithmic  infinities   and  the  same  period  liri  at  the   cross-cut 

PiPo,  then 

■SJia  —  "STia 

is  a  function  without  infinities  on  the  surface ;  its  modulus  for  Pi  Pg  is  zero, 
and  its  modulus  for  each  of  the  cross-cuts  ai,  ...,ap  is  zero;  and  therefore 

*  The  representation  of  a  single  point  on  the  Riemann's  surface  by  means  solely  of  the  value 
of  z  at  the  point  will  henceforward  be  adopted,  without  further  explanation,  in  instances  when  it 
cannot  give  rise  to  ambiguity.  Otherwise,  the  representation  with  full  detail  of  statement  will 
be  adopted. 


512  MODULI    OF   NOKMAL   ELEMENTAEY   FUNCTION  [237. 

it  is  a  constant.     Hence  a  normal  elementary  function  of  the  third  kind  is 
determined,  save  as  to  an  additive  constant,  by  its  infinities  alone. 

Ex.  The  sum  of  three  normal  elementary  functions  of  the  third  kind,  having  as 
their  logarithmic  infinities  the  respective  pairs  that  can  be  selected  from  three  points, 
is  a  constant. 

238.  A  relation  among  the  moduli  of  an  elementary  function  of  the  third 
kind  can  be  constructed  in  the  same  way  as,  in  §  233,  for  the  function  of  the 
second  kind. 

Let  the  surface  be  resolved  by  the  2p  cross-cuts  a^,  ...,  a^,  \,  . . . ,  hp  and  by 
the  cross-cut  P^P^,  joiiiing  the  excluded  infinities  of  an  elementary  function 
Ilia  of  the  third  kind.  Let  iv  be  any  function  of  the  first  kind ;  then  over  the 
resolved  surface,  we  have 

dx    dy       dy    dx 

everywhere  zero;    and  therefore  jlii^dw  round  the  whole  boundary  of  the 
resolved  surface  is  zero,  as  in  §  233. 

Let  the  moduli  of  Ilja  be  A^,  ...,  Ap,  B^^,  ...,  Bp,  and  those  of  w  be 
J./,  ...,  Ap,  B^,  ...,  Bp   for  the  2^  cross-cuts  a  and  h  respectively. 

The  whole  boundary  is  made  up  of  the  two  edges  of  the  cross-cuts  a,  the 
two  edges  of  the  cross-cuts  h,  the  two  edges  of  the  cross-cut  P^P^,  and  the 
small  curves  round  Pj  and  Pg. 

The  sum  of  the  parts  contributed  to  jTl^zdw  by  the  edges  of  all  the  cross- 
cuts a  and  h  is,  as  in  preceding  instances, 

i{A,B:-A:B,). 

The  direction  of  integration  along  PjPa  that  is  positive  relative  to  the  area 
in  the  resolved  surface  is  indicated  by  the  arrows ;  the 
portion  of  ^W-^^dw  along  the  two  edges  of  the  cut  is       J,     ^  ^ 

^2  r  ^1  . 

ITia+cZw;  4-      Yl^^rdw 

Ci  .'  C2  Fig.  83. 

r<?2  /'^2 

=       (ni2+  -  Hir)  dw  =  27rt      dw  =  27ri  {w  (c.)  -  vi  (cj)}. 

Lastly,  the  portion  of  the  integral  for  the  infinitesimal  curve  round  Pi  is  zero, 

by  I.  of  §  24,  because  the  limit  of  (2  -  Cj)  His  -j-  for  z  =  Cj  vanishes,  Pi  being 

assumed  not   to  be  a  branch-point ;    and  similarly  for  the  portion  of  the 
integral  contributed  by  the  infinitesimal  curve  round  P^. 


238.]  OF   THE   THIRD   KIND  513 

As  the  integral  JTli2dw  vanishes,  we  therefore  have 

I  {AsB;  -  As'Bs)  +  27^^■  {w  (c,)  -  w  (c,)]  =  0, 

s=l 

which  is  the  relation  required. 

The  most  important  instance  occurs  when  Ilia  is  the  normal  elementary 
function  of  the  third  kind  (and  then  A^,  A2,  ...,  Ap  all  vanish),  and  w  is  a 
normal  function  of  the  first  kind,  say  Wr ;  then 

Hence,  if  B^  be  the  modulus  at  br  of  the  normal  elementary  integral  -stu,  we 
have 

Br=2{Wr{C2)-Wr{c,)], 

so  that  the  moduli  of  the  normal  elementary  function  of  the  third  kind  can  be 
expressed  in  terms  of  normal  functions,  of  the  first  kind,  of  its  logarithmic 
discontinuities. 

The  important  property  of  functions  of  the  third  kind,  known  as  the 
interchange  of  argument  and  parameter,  can  be  deduced  by  a  similar  process. 

Let  III.,  be  an  elementary  function  with  logarithmic  discontinuities  at 

Ci  and  Cg,  with  Inri  as  its  modulus  for  the  cross-cut  CiCj,  and  with 

Ai,  ...,  Ap,  i>i,  ...,  Bp 

as  its  moduli  for  the  cross-cuts  a-^,  ...,  Up,  b^,  ...,  6^;  and  let  1134  be  another 
elementary  function  with  logarithmic  discontinuities  at  Cg  and  C4,  with  ^iri  as 
its  modulus  for  the  cross-cut  C3C4,  and  with  A-^,  ,..,  Ap,  5/,  ...,Bp'  as  its 
moduli  for  the  cross-cuts  a-^,  ...,  ap,  b^,  ...,bp. 

Then  when  the  infinities  are  excluded  and  the      /^nK^ i-W> 

surface  is  resolved  so  that  both  IT  12  and  1134  ^         G  O      +  m 

are  uniform  finite  and  continuous  throughout  f  Ar  r 

the  whole  surface,  we  have  '^^     +        Hi      d"^ 

9 II 12  51134  _  9II34  dU-^2  ^  Q  Fig.  84. 

dx     dy         dx     dy  ' 

everywhere  in  the  resolved  surface ;  and  therefore,  as  in  the  preceding 
instances,  /  012^^1134  round  the  whole  boundary  vanishes. 

The  whole  boundary  is  made  up  of  the  double  edges  of  the  cross-cuts  a 
and  the  cross-cuts  6,  and  of  the  configuration  of  cross-CTits  and  small  curves 
round  the  points.  The  modulus  of  both  Ilia  and  1134  for  the  cut  AG  is, 
zero ;  the  modulus  of  II12  for  the  cut  C3C4  is  zero,  and  that  of  1134  for  the  cut 
C1C2  is  zero. 

The  part  contributed  to  /IIi2C^n34  by  the  aggregate  of  the  edges  of  the 
p 
cross-cuts  a  and  6  is  S  {AgBg  —  AgB^,  as  in  preceding  cases. 

s=\ 
F.  F.  33 


514  INTERCHANGE   OF   ARGUMENT  [238. 

The  part  contributed  by  the  small  curve  round  Ci  is  zero,  because  the 
limit,  for  z  =  d,  of  (z  —  d)  ITis  -j^  is  zero.  Similarly  the  part  contributed  by 
the  small  curve  round  Ca  is  zero. 

The  part  contributed  by  the  two  edges  of  the  cross-cut  C1C2  is 

=  27^^•  r  dU^  =  27ri  [U^  (c,)  -  U,,  (c,)}. 

J  c, 

The  part  contributed  by  the  two  edges  of  the  cross-cut  AG  is 

the  subject  of  integration  does  not  change  in  crossing  from  one  edge  to  the 
other,  and  therefore  this  part  is  zero. 

For  points  on  the  small  curve  round  C3,  we  have 
dJls4  =  —  — \-  p(z  —  C3)  dz, 

Z        C3 

where p  is  a  converging  series  of  integral  powers  of  z—  Cs:  and  therefore  for 
points  on  that  curve 

U^^dUsi  = '-^-^  dz  +  q(z-  Cg)  dz, 

where  q  (z  —  Cs)  is  a  converging  series  of  positive  integral  powers  of  z  —  Cg. 
Hence  the  part  contributed  to  JYIi^dns^  by  the  small  curve  round  C3  in  the 
direction  of  the  arrow,  which  is  the  negative  direction  for  integration  relative 
to  C3,  is  27rin  12(03). 

Again,  for  points  on  the  small  curve  round  C4,  we  have 

dz 
dUsi  = h  Pi{z  —  Ci)  dz. 

Proceeding  as  for  C3,  we  find  the  part  contributed  to  {^y-^d'U.-.i^  by  the  small 
curve  round  C4,  which  is  negatively  described,  to  be  —  27rini.,  (C4). 

Lastly,  the  sum  of  the  parts  contributed  by  the  two  edges  of  the  cross-cut 

C3C4  IS 

'*ni,(^n34++  [''ni2c^n34- 


238.]  AND   PARAMETER  515 

But  though  1134  has  a  modulus  for  the  cross-cut  C3C4,  its  derivative  has  not  a 
modulus  for  that  cross-cut :  we  have  dU..^-^/dz  =  dlis^jdz,  and  therefore  the 
last  part  contributed  to  jYl^^d^M  vanishes. 

The  integral  along  the  whole  boundary  vanishes ;  and  therefore 
I  (^,B;  -  A^B,)  +  27ri  {U^ (cs)  -  n,, (c,)]  +  27rm,, (c,)  -  27riU,, (C4)  =  0, 

a  relation  between  the  moduli  of  two  elementary  functions  of  the  third  kind. 

The  most  important  case  occurs  when  both  of  the  functions  are  normal 
elementary  functions.  We  have  A^,  ...,  Ap  zero  for  tn-ja,  and  Ai,  ...,  Ap  zero 
for  13-34 ;  and  the  relation  then  is 

tD-34  (Ca)  -  -3734  (Ci)  =  tSTi.^  (C4)  -  tn-12  (Cs), 

which  is  often  expressed  in  the  form 

re-2  rc-i 

-    Ci  •'  C3 

the  paths  of  integration  in  the  unresolved  surface  being  the  directions  of 
cross-cuts  necessary  to  complete  the  resolution  for  the  respective  cases. 
Hence  the  normal  elementary  integral  of  the  third  kind  is  unaltered  in  value 
hy  the  interchange  of  its  limits  and  its  logarithmic  infinities. 

Ex.  1.  Denoting  by  Ei  and  E^  the  normal  elementary  integrals  of  the  second  kind 
which  have  their  single  simple  infinities  at  Cj  and  c^  respectively,  shew  that  the  value  , 

of  —j^  at  Co  is  equal  to  the  value  of  —r^  at  c, . 
dz  '         ^  dz  ^        - 

Ex.  2.     Denoting  by  E^  the  normal  elementary  integral  which  has  its  single  infinity 

,at  C3,  prove  that  the  value  of        '^  at  c^  is  E-i{c2)- E^{ci). 

239.  From  the  simple  examples,  discussed  in  §  199  and  elsewhere,  it  has 
appeared  that  when  a  function  w  is  defined  as  the  integral  of  some  function 
of  z,  the  integral  being  uniform  except  in  regard  to  moduli  of  periodicity,  a 
process  of  inversion  is  sometimes  possible  whereby  z  becomes  a  function  of  w, 
■  either  uniform  or  multiform.  But  in  all  the  cases,  in  which  z  thus  proves  to 
be  a  uniform  function,  the  number  of  periods  possessed  by  w  is  not  gTeater 
than  two;  and  it  follows,  from  §  110,  that,  when  w  possesses  more  than  two 
periods,  z  can  no  longer  be  regarded  as  a  uniform  function  of  w. 

A  question  therefore  arises  as  to  the  form,  if  any,  of  functional  inversion, 
when  w  has  more  than  two  independent  periods  and  when  there  are  more 
functions  w  than  one. 

Taking  the  most  general  case  of  a  Riemann's  surface  of  class  p,  let 
w-^,  w^,  ...,  Wp  denote  the  jp  functions  of  the  first  kind.  Let  there  be  q  inde- 
pendent variables  z-^^,  ...,  Zq,  where  q  is  not,  of  initial  necessity,  equal  to  p; 

33—2 


516  PROBLEM   OF  [239. 

and,  by  means  of  any  q  of  the  functions  of  the  first  kind,  say  Wi,  ...,Wq,  form 
q  new  functions,  also  of  the  first  kind,  and  defined  by  the  equations 

-y,.  =  lU^  {z-^  +  Wr  (^2)  -\-  ...  -\-Wr  (Zq), 

where  r  =  1,  2,  ...,  q.  We  make  the  evident  limitation  that  q  is  not  greater 
than  p,  which  is  justifiable  from  the  point  of  view  of  functional  inversion. 
Then  the  functions  v,.  are  multiform  on  the  surface  with  constant  moduli  of 
periodicity ;  they  have  the  same  periods  as  Wr,  say  t»,-,i,  cor,2,  •••>  c^r,2p- 

The  various  values  of  Wr  (^m)  differ  by  multiples  of  the  periods  :  so  that,  if 
Wr(zm)  be  the  value  for  an  exactly  specified  ^^-path  (as  in  §  110),  the  value 
for  any  other  ^j^-path  is 

This  being  true  for  each  of  the  integers  w  =  1,  2,  ... ,  q,  it  follows  that,  if 

Q 

ms=  %  nm,s,     {s  =  l,2,...,2p), 

m  =  l 

q 
and  if  v.y  be  the  value  of  Z  w,. (z^n)  for  the  exactly  specified  paths  for  z^,  •••,  2q^ 

then  the  general  value  of  v,.  for  any  other  set  of  paths  for  the  variables  is 

Vr  +  mift),.^i  4-  1112,(0 r, 2.  +  ...  +  ^2pf^r,2p> 

holding  for  r  =  1,  2,  ...,  q.  The  integers  7im,s,  and  therefore  the  integers  Wg,. 
are  evidently  the  same  for  all  the  functions  v. 

The  reason  which,  in  the  earlier  case  (§  110),  prevented  the  function  w  from 
being  determinate  as  a  function  of  z  alone  was,  that  integers  could  be  determ- 
ined so  as  to  make  the  additive  part  of  w,  dependent  upon  the  periods,  less 
than  any  assigned  quantity  however  small :  say  less  than  an  infinitesimal 
quantity.     It  is  necessary  to  secure  that  this  possibility  be  excluded. 

Let  (0\^fi  =  a^^fj,  +  i^\^fj,,  where  the  quantities  a  and  /3  are  real:  then  we 
have  to  prevent  the  possibility  of  the  additive  portions  for  all  the  functions  v 
being  infinitesimal.  In  order  to  reduce  the  additive  part  to  an  infinitesimal 
value  for  each  of  the  functions  v,  it  would  be  necessary  to  determine  integers 
oiii,  niz,  ...,  m^p  so  that  the  2q  quantities 

rriyfir,!  +  vn^^r,2  +  •  •  ■  +  'ni2p^r,2p 
for  r  =  1,  .,.,  q,  all  become  infinitesimal. 

If  q  be  less  than  p,  the  2p  integers  can  be  so  determined.  In  that  case^ 
the  general  possibility  of  functional  inversion  between  the  q  functions  v  and 
the  q  variables  z  would  require  that  the  quantities  z  are  so  dependent  upon 
the  quantities  v  that  infinitesimal  changes  in  the  latter,  carried  out  in  an 


239.]  INVERSION  517 

infinite  variety  of  ways  and  capable  of  indefinite  repetition,  would  leave  the 
quantities  z  unchanged.  The  position,  save  that  we  have  q  variables  instead 
of  only  one,  is  similar  to  that  in  §  110 :  we  do  not  regard  the  functions  v  as 
having  determinate  values  for  assigned  values  of  z-^,  ...,  Zg,  but  the  values  of 
Vi,  ...,Vq  are  determinate,  only  when  the  paths  by  which  the  independent 
variables  acquire  their  values  are  specified.  And,  as  before,  the  inversion  is 
not  possible. 

If  q  be  not  less  than  p,  so  that  it  must  in  the  present  circumstances  be 
equal  to  p,  then  the  2p  integers  cannot  be  determined  so  that  the  2p  quanti- 
ties all  become  infinitesimal.  They  can  be  determined  so  as  to  make  any 
2p—l  of  the  quantities  become  infinitesimal ;  but  the  remaining  quantity 
is  finite  as,  indeed,  should  be  expected,  because  the  determinant  of  the 
constants  a.  and  /3  is  different  from  zero*. 

If  then  there  be  p  variables  Zi,  ...,  Zp,  and  p  functions  Vi,  ...,  Vp,  defined 
by  the  equations 

Vr  =  lUr  (^i)  +  W,.  (Z.2)  +  ...+  W,.  (Zp), 

for  ?^  =  1,  2,  ...,  j^,  then  the  values  of  the  functions  v  for  assigned  values  of 
the  variables  z,  whatever  be  the  paths  by  which  the  variables  attain  these 
values,  are  of  the  form 

for  r  =  1,  2,  ...,  p  ;  and  it  has  been  proved  that  the  2p  integers  m  cannot  be 
determined  so  that  all  the  additive  parts,  dependent  upon  the  periods,  become 
infinitesimal.  Hence  the  functions  v-^,  ...,  Vp  are,  except  as  to  additive 
multiples  of  the  periods  (the  numerical  coefiicients  in  these  multiples  being 
the  same  for  all  the  functions),  uniform  functions  of  the  variables  z^,  ...,  Zp-, 
and  they  are  finite  for  all  values  of  the  variables.  Conversely,  we  may  regard 
the  quantities  z  as  functions  of  the  quantities  W],  ...,  Vp,  which  have  2p  sets 
of  simultaneous  periods  Wi^g,  &>2,6.,  ...,  &)^,s  for  s  =  l,  2,  ...,  2p:  that  is,  the 
variables  z  are  2j9-ply  periodic  functions  oi p  variables  v^,  ...,  Vp.  This  result 
is  commonly  called  the  inversion-pi^ohlem  for  the  Abelian  transcendents. 

In  effecting  the  inversion  of  the  equations 

dvi  =  u\  {z-^) dzi  +  ■z^i'  (^s) dz.2+  ...  +  Wi  (zp) dzp\ 


dvp  =  lUp  {z-)  dzi  +  tUp  {z.^  dz^-Jr  ...  +  w/ (Zp) dzp] 

the  actual  form,  which  is  adopted,  expresses  all  symmetric  functions  of 
the  quantities  ^■j,  ...,%  as  uniform  functions  of  the  variables,  so  that,  if 
z-^,  Z.2,  . . . ,  Zp  he  the  roots  of  the  equation 

(/)  {Z)  =  ZP  +  P,Zp-'  +  P,ZP-'  +  ...+Pp  =  0, 

*  The  2p  potential-functions,  arising  from  the  p  functions  w,  would  otherwise  not  be  linearly 
independent. 


518  ABELIAN    FUNCTIONS  [239. 

then*  Pi,  ...,  Pp  are  uniform  multiply-periodic  functions  of  the  variables 
Vi,  ...,Vp.  Consequently,  all  rational  symmetric  functions  of  z-^,  ...,Zp  are 
uniform  multiply-periodic  functions  oi  v-^,  . . . ,  Vp. 

Frequent  reference  has  been  made  to  the  functions  determined  by  the  equation 

w^-  R(z)  =  w^  —  {z-ao)  (z-  cci) ...{z- a2p)  =  0. 

f  U  (z) 
It  has  been  proved  that  an  integral  of  the  form   I  — ^  dz  is  an  integral  of  the  first  kind, 

provided  U  {z)  be  any  polynomial  function  of  degree  not  higher  than  ^-1,  and  that 
the  otherwise  arbitrary  character  of  U  {z)  makes  it  possible  to  secure  the  necessary 
p  integrals  by  allowing  the  suitable  choice  of  the  coefficients.  "Weierstrass  takes  the 
equations,  which  lead  to  the  inversion,  in  the  following  formt  : — 

The  constants  a  are  different  from  one  another  and  can  have  any  values  :  and  it  is 

convenient  to  take 

P {x)  =  {x- ai)  {x -a3)...{x-  a^p-i), 

Q{x)  =  {x-aQ){x-a2)...{x-a.2p-2){x-ao^p\ 

so  that  P{x)  Q{x)  —  R{x).     If  the  coefficients  a  be  real,  it  is  assumed  that 

ao  >  «i  >  0^2  >  •••  >  «^2p- 

The  equations  which  give  the  new  variables  are 

^                 P  (0i)  dzr                       P{z.,)dz2           ,  ^         P(zp)dzp 

au-,=    ,  -I-     ,  -r -t-     - 


(^1  -ai)'jR  {zx)  (S2  -ai)\/R  (z^)  (zp  -  ay)  s/R  (Zp) 

du  =         ^  ^^'^  ^^1  +  Pi^2)dzz  _^         _^         P  (zp)  dzp 


(^1  -  as)  Vii  (si)  (02  -  «3)  "^R  (^2)  {z,p  -  as) "-' R  (^p) 


P{z^)dz^            ,           Piz2)dz2                                  P{zp)dzp 
a\ip  = ■  H ; — ,„  ,    .  + H 


(^1  -  «2p-i)  "^R  (%)      (22  -  ^2^-1)  "^R  (22)  {h  -  «2p-i)  ■^R  (s), 

and  when  integration  takes  place,  the  arbitrary  constants  are  defined  by  the  equations 

Wi,  ?<2;  •••5  Wp  =  0  (with  periods  for  moduli), 
when  Sj,   235  •••)  %=«!)  «3)  •••>  «2p-i  respectively. 

The  p  variables  z  are  the  roots  of  an  algebraical  equation  of  degree  jo,  the  coefficients  in 
which  are  (multiply-periodic)  uniform  functions  of  the  variables  u.  The  functions,  arising 
out  of  the  equations  in  this  form,  are  discussed  J  in  Weierstrass's  two  memoirs,  just 
quoted. 

Note  1.  The  results  thus  far  established  in  this  chapter  lie  at  the  basis  of  the  theory 
of  Abelian  functions.  The  fuller  establishment  of  that  theory  and  its  development 
are  beyond  the  range  of  the  present  treatise. 

So  far  as  concerns  the  general  theory,  recourse  must  be  had  to  the  fundamental 
memoirs  of  Abel,  Jacobi,  Hermite,  Riemann  and  Klein,  and  to  treatises,  in  addition  to 

*  For  further  considerations  see  Clebsch  und  Gordan,  Theorie  der  AheVschen  Functionen, 
Section  vi. 

t  Equivalent  to  that  given  in  Crelle,  t.  lii,  (1856),  pp.  285  et  seq. ;  it  is  slightly  diiierent  from 
the  form  adopted  by  him  in  Crelle,  t.  xlvii,  (1854),  p.  289. 

X  Some  of  the  results  are  obtained,  somewhat  differently,  in  a  memoir  by  the  author,  Phil. 
Trans.,  (1883),  pp.  323—368. 


239.]  UNIFORM    FUNCTIONS   ON   RIEMANN'S   SURFACE  5l9 

those  by  Neumann  and  by  Clebsch  and  Gordan  already  cited,  by  Prym,  Krazer,  Konigs- 
berger,  Briot,  and  Stahl.  The  most  comprehensive  of  all  is  Baker's  treatise  Abel's  theorem 
and  the  allied  theory^  including  the  theory  of  the  theta  functions,  (Cambridge,  1897). 

Moreover,  as  our  propositions  have  for  the  most  part  dealt  with  functions  of  only  a 
single  variable,  it  is  important  in  connection  with  the  Abelian  functions  to  take  account 
of  Weierstrass's  memoir*  on  functions  of  several  vaciables. 

Note  2.  We  have  discussed  only  very  limited  forms  of  integrals  on  the  Riemann's 
surface :  and  any  professedly  complete  discussion  would  include  the  theorem  that  \iv'dz, 
where  ^o'  is  a  general  function  of  position  on  the  surface,  can  be  expressed  as  the  sum  of 
some  or  all  of  the  following  parts  : — 

(i)      algebraical  and  logarithmic  functions  ; 

(ii)     Abelian  transcendents  of  the  three  kinds  ; 

(iii)    derivatives  of  these  transcendents  with  regard  to  parameters  ; 

but  such  a  discussion  is  omitted  as  appertaining  to  the  investigations  relative  to  Abelian 
transcendents.  Supplementary  Notes  will  be  found  at  the  end  of  this  Chapter  XVIII., 
giving  an  account  of  Abel's  theorem,  and  indicating  a  mere  beginning  of  the  theory  of 
Abelian  transcendents. 

For  the  particular  case  in  which  the  integral  iio'dz  is  an  algebraical  function  of  2,  see 
Briot  et  Bouquet,  TheoHe  des  fonctions  elliptiques,  (2™«  e'd.),  pp.  218- — 221  ;  Stickelberger, 
Crelle,  t.  Ixxxii,  (1877),  pp.  45,  4fi  ;  and  Humbert,  Acta  Math.,  t.  x,  (1887),  pp.  281—298, 
by  whom  further  references  are  given. 

240.  There  are  functions  belonging  to  class  {B)  in  §  229,  other  than 
those  already  considered.  In  particular,  there  are  functions  with  assigned 
infinities  on  the  surface  and  with  the  real  parts  of  all  their  moduli  of 
periodicity  for  the  canonical  system  of  cross-cuts  equal  to  zero.  But  it 
does  not  therefore  follow  that  all  the  moduli  of  periodicity  vanish  ;  in  order 
that  their  imaginary  parts  may  vanish,  so  as  to  make  the  moduli  of 
periodicity  zero,  certain  conditions  would  require  to  be  satisfied. 

We  shall  limit  the  ensuing  discussion  to  some  sets  of  these  functions 
with  zero  moduli,  and  shall  assign  the  conditions  necessary  to  secure  that 
the  moduli  shall  be  zero.  We  shall  assume  that  all  their  infinities  are 
algebraic  ;  the  functions  are  then  uniform  everywhere  on  the  surface,  and, 
except  at  a  limited  number  of  isolated  points,  where  they  have  only 
algebraic  infinities,  are  finite  and  continuous.  They  are,  in  fact,  algebraic 
functions  of  z. 

Two  classes  of  these  functions  are  evidently  simpler  than  any  others. 
The  first  class  consists  of  those  which  have  a  limited  number,  say  m,  of 
isolated  accidental  singularities  and  which  are  not  infinite  at  any  of  the 
branch-points ;  the  other  class  consists  of  those  which  have  no  infinities 
except  at  the  branch-points.  These  tAvo  classes  will  be  briefly  discussed 
in  succession. 

*  First  published  in  1886  ;  Ahliandhingen  aus  der  Functionenlehre,  pp.  105 — 164 ;  Ges.  Werke, 
t.  ii,  pp.  135 — 188.  See  also  the  author's  Lectures  introductory  to  the  theory  of  functions  of  tivo 
complex  variables  (Camb.  Univ.  Press,  1914). 


520  UNIFORM    FUNCTIONS    IN   TERMS   OF  [240. 

Let  w  be  a  uniform  function  having  accidental  singularities  at  the 
points  Ci,  ...,Cm  and  no  other  infinities;  and  for  simplicity,  assume*  that 
each  of  them  is  of  the  first  order.  Also  let  the  normal  function  of  the 
second  kind,  having  c,.  for  its  sole  infinity,  be  Z^.     Then 

where  /Si,  ...,  ^m  are  constants  at  our  disposal,  is  a  function,  having  infinities 
of  the  same  class  and  at  the  same  points  as  tv  has;  the  function  is  otherwise 
finite  everywhere  on  the  surface  and  therefore,  by  properly  choosing  the 
constants  /S,  we  have  the  function 

VJ-{^rZ,+  ...+^,r,Z„,) 

finite  everywhere  on  the  surface,  so  that  it  is  a  function  of  the  first  kind. 

Now  because  its  modulus  vanishes  at  each  of  the  cross-cuts  a  in  the 
resolved  surface,  it  is  a  constant,  so  that 

tU=^,Z,+  ...+|3n^Z,,,+0o■ 
The  modulus  of  w  is  to  vanish  at  each  of  the  cross-cuts  6^.     Let  (f)r(z)=     ,  ^, 

so  that  <f)r  (z)  is  an  algebraic  function  on  the  surface :  assigning  the  condition 
that  the  modulus  of  tu  at  the  cross-cut  br  shall  vanish,  we  have 

/3i(/),  (Ci)  +  /3,<j>r  {c.^  +  ...+  ^mA  (Cm)  =  0, 
an  equation  which  must  hold  for  all  the  values  r  =  1,  ...,p. 

When  the  quantities  c  represent  quite  arbitrary  points,  thei^e  must  be 
at  least  p+  1  of  them ;  otherwise,  as  the  equations  are  independent  of  one 
another,  they  can  be  satisfied  only  by  zero  values  of  the  constants  /5,  a  result 
which  renders  the  uniform  function  evanescent.  If  m  >  p,  the  equations 
determine  p  of  the  coefficients  /S  linearly  in  terms  of  the  remaining  on  -  p  : 
when  these  values  are  substituted,  the  resulting  expression  for  w  contains 
^Yi  —  p  -^  I  constants,  viz.,  the  remaining  m  —  p  constants  yS,  and  the  constant 
/3o.  The  coefficient  of  each  of  the  m  —  p  constants  /3  is  a  function  of  z,  which 
has  p+l  accidental  singularities  of  the  first  order,  p  of  which  are  common 
to  all  the  functions,  so  that  w  then  is  an  arbitrary  linear  combination 
of  constant  multiples  of  m  —  p  functions,  each  of  which  possesses  ^3  +  1 
accidental  singularities  and  can  be  expressed  in  the  form 

A,. (2;)=  Zi,  Zo,    ,  Zp,  Zp^r 

<^l  (Ci),    </>!  (C2),    ,    </>!  (Cp),    4>i  (Cp+r) 

4>2{Ci),    </)2(Co),     ,    4>2{Cp)-    (p-2(Cp+r) 

^p{Ci),    <\>p{CoX (i>p{Cp),    (f)piCp+r) 

If  a  pole,  say  at  cj,  is  of  order  s,  then  /Si^i  would  be  replaced  by 


OCi 

in  the  expression  for  tv ;  and  so  for  other  cases. 


71+72  g^+...+-,.g,^. 


240.]  NORMAL   FUNCTIONS   OF   THE    SECOND   KIND  521 

When  the  quantities  c  are  not  completely  arbitrary,  but  are  such  that 
relations  among  them  can  be  satisfied  so  as  no  longer  to  permit  the  preceding 
forms  to  be  definite,  we  proceed  as  follows. 

The  most  general  way  in  which  the  preceding  forms  cease  to  be  definite 
is  by  the  dependence  of  some  of  the  equations 

^,(f>r  (Ci)  +  /3,(br  (Ca)  +  . . .  +  /3,n<f>r  (Cm)  =  0 

on  the  remainder.  Let  q  of  them,  say  those  given  by  r  =  l,  ...,  q,  be 
dependent  on  the  remaining  p  —  q,  so  that  0<q<p:  then  the  conditions 
of  dependence  can  be  expressed  by  equations  of  the  form 

(f>,.  (Cn)  =  ^1,  r4>q+l  (^n)  +  ^2,r<f)q+2  M  +  •  •  •  +  Ap_q^,.(f>p  (C„), 

for  7'  =1,  2,  ...,  q  and  n  =  1,  2,  ...,  m. 

The  functions  of  the  first  kind  W,  through  which  the  functions  <^  are 
derived,  are  a  complete  set  of  normal  functions :  when  any  number  of  them 
is  replaced  by  the  same  number  of  independent  linear  combinations  of  some 
or  all,  the  first  derivatives  are  still  algebraic  functions.  We  therefore  replace 
the  functions  W-^,  TTo,  ...,  Wq  by  Wi,  Wg,  ...,  Wq,  where 

for  r  =  1,  2,  ...,  g,  so  that,  for  all  values  of  z, 

^r{^)  =  ^r{2)-A,^r(i>q+ii2)-A,^r<f>q+2(2)-  . ..  -  Ap^q^.^pi^)- 

Hence  the  functions  <I>i,  <i>o,  ...,  ^q  vanish  at  each  of  the  points  Cj,  Ca,  ...,  c,„. 

The  original  system  of  j)  equations  in  ^j,  ...,  <pq,  4>q+i,  ...,  <^p,  when  made 
a  system  of  equations  in  <l>3,  ...,  <l>g,  ^q+i,  ...,  (f)p  is  equivalent  to 

A  %  (Ci)  +  l3,^r  (C,)  +  ..•+  An  <^,-  (C,n)  =  0) 
A  0s  (Ci)  +  A  0,  (C.,)  +  . . .  +  /3„,  (f)s  (Crn)  =  Oj  ' 

for  r  =  1,  ...,  q  and  s  =  q  +  1,  ...,  p.  The  first  q  of  these  are  evanescent ;  and 
therefore  their  form  is  the  same  as  if  we  had  initially  assumed  that  each  of 
the  functions  (f}^,  ...,  (f)q  vanished  for  each  of  the  points  z  =  Ci,  ...,  c^,  the  two 
assumptions  being  in  essence  equivalent  to  one  another  on  account  of  the 
property  of  linear  combination  characteristic  of  functions  of  the  first  kind. 

Suppose,  then,  that  q  of  the  functions  (f),  derived  through  functions  of  the 
first  kind,  vanish  at  each  of  the  points  Cj,  ...,  c^;  the  .number  of  surviving 
equations  of  the  form 

/Si  (fir  (Ci)  +  /3,<^,  (C.)  +  . . .  +  /3,«0r  (C,n)  =  0 

is  p-  q,  and  they  involve  m  arbitrary  constants  /S.  Hence  they  determine 
p  —  q  of  these  constants,  linearly  and  homogeneously,  in  terms  of  the  other 
7n—p  +  q.     When  account  is  taken  of  the  additive  constant  ^o,  then*  the 

*  This  is  usually  known  as  Eiemann-Roch's  Theorem.     It  is  due  partly  to  Eiemann  and 
partly  to  Eoch ;    see  references  in  §  242. 


522 


RIEMANN-ROCH  S 


[240. 


function  tu  contains  m.—p-\-q^-l  arbitrary  constants ;  and  it  is  a  linear  com- 
bination of  arbitrary  multiples  of  m—p  +  q  functions,  each  having  p  —  q--\-\ 
accidental  singidarities  of  the  first  order,  p  —  qof  which  are  common  to  all 
the  functions  in  the  combination. 

The  functions  under  consideration,  being  linear  combinations  of  normal 
functions  Z  of  the  second  kind,  have  no  infinities  except  at  the  accidental 
singularities ;  the  branch-points  of  the  surface  are  not  infinities.  And  it 
appears,  from  the  theorem  just  proved,  that  there  are  functions  having  only 
p  —  q+1  accidental  singularities,  each  of  the  first  order,  so  that  the  total 
number  is  less  than  p  +  1.  A  question  therefore  arises  as  to  what  is  the 
inferior  limit  to  the  number  of  accidental  singularities  that  can  be  possessed 
by  a  function  which  is  uniform  on  the  Riemann's  surface  and  which,  except 
at  these  accidental  singularities,  is  everywhere  finite  and  continuous  on  the 
surface. 

Let  this  limit  be  denoted  by  /a  ;  then  the  p  equations 

A<^r(ci)+...+/3^(^,.(c^)  =  0, 

for  r  =  l,  2,  ...,p,  must  determine  yu,  — 1  of  the  constants  13  in  terms  of  the 
remaining  constant  /3,  say,  B.  The  function  thence  determined  contains  two 
constants,  viz.,  the  surviving  constant  j3  and  the  additive  constant,  its  form 
being 


A+B 


</>i  (Ci), 


■Z„ 


z. 

</>2  (Cm) 


<^^_i(Ci),*0^_i(C2),    ,   <^^_i(c^) 

Among  the  points  Ci,  Cj, ,  c^,  the  relations 

(j>i(c,),         <^i(Co),    ,         <^a(c^) 


for  r  =  0,  1,  ...,p  —  fX;  must  be  satisfied,  that  \s,p  —  /u.  +  l  relations  must  be 
satisfied  *. 

Since  there  are  fi  points  c  among  which  p  —  /x+l  relations  are  satisfied, 
it  follows  that  the  number  of  surviving  arbitrary  constants  c  is,  in  general, 
equal  to  fi  -(p  —  /jl+I),  that  is,  to  2/j,  —  p  —  1.  These  occur  as  arbitrary  con- 
stants in  the  inferred  function,  independently  of  the  two  constants  A  and  B : 
so  that  the  number  of  arbitrary  constants,  in  the  function  with  //,  accidental 
singularities,  is  2yu,  —  p  —  1  +  2,  that  is,  2yu,  —  p  +  1. 

*  This  result  implies  that  the  relations  are  independent  of  one  another,  which  is  the  case  in 
general :  but  it  is  conceivable  that  special  relations  might  exist  among  the  branch-points,  which 
would  affect  all  these  numbers. 


240.]  THEOREM  523 

Again,  the  number  of  infinities  of  a  uniform  function  of  position  on  a 
Riemann's  surface  is  equal  to  the  number  of  its  zeros  (|  194),  and  also  to 
the  number  of  points  where  it  assumes  an  assigned  value ;  and  all  these 
properties  are  possessed  by  any  function,  with  which  lu  is  connected  by 
any  lineo-linear  relation.     If  u  be  one  such  function,  then  another  is 

au  +  b 
lu  — 7  , 

■u  —  a 

where  a,  h,  d  are  arbitrary  constants ;  and  therefore  lu  contains  at  least 
three  arbitrary  constants,  when  it  is  taken  in  the  most  general  form  that 
possesses  the  assigned  properties. 

But  it  has  been  shewn  that  the  number  of  independent  arbitrary  con- 
stants in  the  general  form  of  w  is  2/j,  —  p+l.  This  number  has  just  been 
proved  to  be  at  least  three,  and  therefore 

2fi-p  +  l^S, 
or  yu,  ^  1  +  ^p. 

Thus  the  integer  equal  to,  or  next  greater  than,  \  -{-  hp  is  the  smallest 
number  of  isolated  accidental  singularities  that  an  algebraical  function  can 
have  on  a  Riemanns  surface,  on  the  supposition  that  it  has  no  infinities  at 
the  branch-points*. 

Note.  A  method  of  decomposing  rational  functions  on  a  Riemann's 
surface,  so  that  the  elements  are  normal  functions  of  the  second  kind,  is 
given  above ;  another  method  of  constructing  a  rational  function  is  as 
follows. 

It  was  seen  that  the  number  of  simple  poles  of  a  rational  function,  when 
all  of  them  are  arbitrarily  assigned,  cannot  be  less  than  p  +  1;  to  consider 
the  simplest  case,  we  accordingly  assign  ^  +  1  arbitrary  points,  which  shall 
be  infinities  (and  the  only  infinities)  of  a  rational  function.  Take  the  most 
general  polynomial  P (w,  z)  of  order  n—  2  in  w  and  z  combined,  and  make  it 
vanish  at  the  ^  +  1  assigned  points.  Also  make  P  vanish  at  each  of  the 
multiple  points  of  the  curve  /=  0  in  such  a  way  that,  when  the  point  is  of 
multiplicity  A,  for  /=  0,  it  is  of  multiplicity  X  — 1  for  P  =  0;  consequently 
such  a  point  counts  for  A,(X—  1)  intersections  among  the  points  common  to 
/=0,  P  =  0.  Hence  the  number  of  mtersections  common  to/=0,  P  =  0, 
other  than  the  multiple  points  of  /  =  0,  and  the  p  +  l  arbitrary  points,  is 

=  n{n-2)-(p  +  l)-^\(X-l). 

*  This  result  applies  only  to  a  completely  general  surface  of  class  p.  And,  for  special  forma 
of  siu'face  of  class  p,  a  lojver  limit  for  /j.  can  be  obtained;  thus,  in  the  case  of  a  two-sheeted 
surface,  the  limit  is  2.     (See  Klein-Fricke,  i,  p.  556.) 


524  CONSTEUCTION   OF   FUNCTIONS  [240. 

But,  as  in  §  182,  we  have* 

|9  =  i(w-l)(?i-2)-2U(X-l); 
and  therefore  the  number  of  remaining  intersections  is 

=  n  (n  -  2)  -  ( j9  +  1)  -  {n  -  1)  {n  -2)  +  2p 

=  n  +  p  —  S. 
Now  a  polynomial  in  w  and  z  of  order  n-2  contains  in(n-l)  terms. 
The  number  of  relations  among  the  constants,  necessary  to  secure  that 
a  point  on  a  curve  is  of  order  k,  is  ^k(k  +  l);  so  that  the  number  of 
relations  among  the  constants,  necessary  to  make  each. of  the  multiple  points 
of /=0  a   multiple  point  of  the   proper   order  for  P  =  0,   is   l^\(\-l). 

Since 

^n{n  -l)-l^\(X-l)-(n  +  p-  3) 

=  i?r  (a  -  1) - ^{n  -  l){n  -  2)  +  p  -  {7i+p  -  3) 

=  2, 

it  follows  that  there  are  two  independent  polynomials,  which  can  be  drawn 
through  the  multiple  points  of /=0,  vanishing  to  the  proper  order  at  each 
of  them,  and  through  the  n+p  -3  points  common  to  /=  0,  P  =  0,  which 
are  other  than  the  p  +  1  arbitrary  zeros  of  P.  Clearly  P  itself  can  be 
taken  as  one  of  these  polynomials;  let  P^  denote  another  independent 
of  P.  In  general.  Pi  does  not  pass  through  any  other  zero  of  P ;  in 
order  to  make  it  do  so,  one  other  relation  among  the  constants  would 
be  necessary,  and  then  there  would  be  only  a  single  polynomial  passing 
through  the  multiple  points  to  the  proper  order,  through  the  n+p-S 
points,  and  through  the  other  zero  of  P:  the  single  polynomial  being 
P  itself. 

Thus 

_  Pi  (w,  z) 
^'      P{iv,z)' 

is  a  function,  which  has  the  p  +  1  assigned  points  for  poles,  because  they 
are  zeros  of  P  and  not  of  Pj ;  all  the  other  zeros  of  P  on  the  surface 
are  zeros  of  Pi  to  the  same  order,,  and  they  therefore  are  not  jDoles  of  g. 
Thus  a  rational  function  has  been  constructed,  which  has  ^  +  1  assigned 
points  as  poles,  and  it  has  no  other  poles. 

241.  The  other  simple  class  of  uniform  functions  on  a  Riemann's 
surface  consists  of  those  which  have  no  infinities  except  at  the  branch- 
points of  the  surface. 

They  will  not  be  considered  in  any  detail :  we  shall  only  briefly  advert 

*  A  multiple  point  of  order  X  on  a  curve  is  equivalent  to  i\(\  — 1)  double  points  (Salmon's 
Higher  Plane  Curves,  §  40) :  hence  the  aggregate  of  equivalent  double  points  is  SiX  (\  -  1). 


241.]  ALGEBRAIC  FUNCTIONS  525 

to  those  which  consist  of  the  first  derivatives  of  functions  of  the  first  kind. 
This  set  is  characterised  by  the  theorem : — 

These  functions  (f)  (z)  are  infinite  only  at  branch-points  of  the  surface,  and 
the  total  number  of  infinities  is  2p  —  2+  2n.  For,  let  w  (z)  be  the  most 
general  integral  of  the  first  kind,  and  let 

Near  an  ordinary  point  a  on  the  surface  we  have 
w  (z)  =  w  (a)  +  (z  —  a)  P  (z  —  a), 

where  P  is  a  converging  series  that  may,  in  general,  be  assumed  not  to 
vanish  for  z  =  a]  hence 

<l>  (z)  =  P(z  -a)  +  (z  -  a) P' {z-  a); 

that  is,  cf)  (z)  is  finite  at  an  ordinary  point. 

Near  z=  ao  (supposed  not  to  be  a  branch-point)  we  have,  if  k  be  the 
value  of  lu  there, 

where  P  [-)  may,  in  general,  be  assumed  not  to  vanish  for  z  =  cc  ;  so  that 


1  -  -      1  - 

-  P  {{z  -  y)  '"I '"  +  -  (z  -  jY"  P'  {{z  -  ryy 


z-      \z  I      z"        \z , 

and  therefore  (^  (2^)  has  a  zero  of  the  second  order  at  z  =  cc , 

Near  a  branch-point  7,  where  m  sheets  of  the  surface  are  connected,  we 
have 

IV  {z)  -  w  (7)  =  {z-  7)'"  P  {{z  -  7)'»|, 

where  P  may,  in  general,  be  assumed  not  to  vanish  for  ^  =  7 :  hence 

4>  (^)  =  (^  -  7) 

so  that  <^  (ir)  is  infinite  at  2;  =  7,  and  the  infinity  is  of  order  ni  —  1. 

Hence  the  total  number  of  infinities  is  2  (m  —  1),  where  m  is  the  number 
of  sheets  connected  at  a  branch-point,  and  the  summation  extends  over  all 
the  r  branch-points.  But  2p -f- 1  =  S  (m  —  1)  —  2?i -|- 3,  and  therefore  the 
number  of  infinities  is  2p  —  2  -r  2n. 

•  We  can  now  prove  that  the  number  of  zeros  of  (f)  (z)  in  the  finite  part 
of  the  surface  is  2p  -  2,  of  tuhich  p  —  I  can  be  arbitrarilij  assigned. 

The  total  number  of  zeros  is  2p  —  2-\-2n,  being  equal  to  the  number 
of  infinities  because    (^{z)  is  an  algebraic   function.      But   (^{z)  has  been 


526  ALGEBRAIC   FUNCTIONS  [241. 

proved  to  have  a  zero  of  the  second  order  when  ^  =  qo  ;  and  this  occurs  in 
each  of  the  n  sheets,  so  that  2?i  (and  no  more)  of  the  infinities  of  (p  (z) 
are  given  by  z=oo .  There  thus  remain  2p  —  2  zeros,  distributed  in  the 
finite  part  of  the  surface. 

Moreover,  the  most  general  function  (f)  (z)  of  the  present  kind  is  of  the 

form 

<l>  (z)  =  C,  (/)i  (z)  +C,(f>,{z)+...  +  Cp(f>p  (z), 

where  <^i  (z),  ...,  (f)p  (z)  are  derived  through  the  normal  functions  of  the  first 
kind.  The  p  —  1  ratios  of  the  constants  C  can  be  chosen  so  as  to  make 
(b  (z)  vanish  for  p  —  1  arbitrarily  assigned  points.  Hence  an  algebraic 
function  arising  as  the  derivative  of  an  integral  of  the  first  kind  is  determined, 
save  as  to  a  constant  factor,  by  the  assignment  of  p)—l  of  its  zeros  in  the 
finite  part  of  the  plane. 

Note*.  It  may  happen  that  the  assumptions  as  to  the  forms  of  the  series 
in  the  vicinity  of  a  particular  point  a,  of  oo  ,  and  of  7,  are  not  justified. 

If  (/)  (a)  vanish,  we  may  regard  a  as  one  of  the  2p)  —  2  zeros. 

If  5  =  00  on  one  sheet  be  a  zero  of  «^  {z)  of  order  higher  than  two,  say 
2  +  s,  we  may  consider  that  s  of  the   2p  —  2   zeros  are  removed  from  the 
finite  part  of  the  surface  to  coincide  with  ^  =  00  . 
j^ 

If  P  [{z  —  7)"*}  vanish  for  2-  =  7,  the  order  of  the  infinity  for  ^  {z)  is 
reduced  from  m  —  1  to,  say,  m  —  s—1;  we  may  then  consider  that  s  of  the 
2p  —  2  zeros  coincide  with  the  branch-point. 

242.  When  the  integer  ^  of  §  240  is  greater  than  zero,  so  that  a 
rational  function  having  m  assigned  simple  poles  can  be  expressed  as  a 
linear  combination  of  m  —  p  +  q  functions  each  possessing  only  p  -\-l—q 
poles,  then  the  rational  function  is  called f  a  special  function,  to  distinguish 
it  from  the  most  general  case,  when  g  =  0  and  the  points  c  are  quite 
arbitrary.  In  the  case  of  a  special  function,  having  Ci,  ...,  c,,,,  for  its  (simple) 
poles,  these  points  are  such  that  q  of  the  functions  ^,  say 

^1,    02,    ...,    (^q, 

vanish  at  each  of  them.  Now  in  §  241  it  was  proved  that  the  number  of 
zeros  of  any  function  0  for  finite  values  of  ^  is  2p—2;  consequently,  in 
the  case  of  a  special  function, 

m^2p  —  2, 

or  the  degree  of  a  special  function  is  not  greater  than  2p  —  2.  Moreover, 
q  denotes  the  number  of  distinct  adjoint  curves  of  order  n  —  3,  which  pass 
through  the  poles  of  the  special  function. 

*  See  Klein-Fiicke,  vol.  i,  p.  545. 

t  Klein-Fricke,  Vorl.  ii.  d.  Th.  d.  ell.  Modulfunctionen,  t.  i,  p.  552. 


242.]  SPECIAL   ALGEBRAIC    FUNCTIONS  527 

Denoting  the  special  function  by  g,  and  selecting  any  one  of  the  q 
adjoint  polynomials  of  order  n  —  3,  which  occur  in  the  q  functions  <^,  say 
U-i^{w,  z),  consider  the  product  gU^^.  This  product  is  finite  at  each  of  the 
points  Ci,  ...,  Cm,  because  those  points  are  zeros  of  U^ ;  and  g  is  not  elsewhere 
infinite  on  the  surface.     Consequently  the  integral 

dw 

is  finite  everywhere  on  the  surface,  and  it  is  therefore  an  integral  of  the 
first  kind,  say 

[V,dz 

9/"  '      - 
dw 

where  Fj  is  the  appropriate  adjoint  polynomial  of  order  w  —  3.     Thus 

or  a  special  function  is  expressible  as  the  quotient  of  one  adjoint  polynomial 
of  order  n  —  S  by  another*. 

Consider,  in  particular,  the  function 

_  a^cf),  +  Of202  +  .  .  ■  +  aq({)q 

where  a-i^,\..,o.q  are  arbitrary  constants.  Each  of  the  quantities  (pi,...,(f>q 
vanishes  at  the  points  Ci,...,Cni-  The  only  infinities  of  gi  are  the  zeros 
of  (f)i,  which  are  only  2^  —  2  for  finite  values  of  z;  and  m  of  these  2^—2 
zeros  are  not  infinities  of  g^ ,  so  that  gi  has  only  2p  —  2  —  m  infinities,  say 

where  m  =2p  —  2  —  m.  Now,  by  the  Riemann-Roch  theorem  of  |  240,  the 
most  general  rational  function,  which  has  m  simple  poles  (and  no  athers) 
on  the  surface,  contains  m'  —p  +  q'  +  l  arbitrary  constants,  where  q'  is  the 
number  of  the  functions 

(or  the  number  of  linear  combinations  of  them),  which  vanish  at  all  the 
points  kx,  k2,  ...,k\n'.  The  arbitrary  constants  in  gi  are  a^,  ...,  aq,  being 
q  in  number ;  hence  •  » 

q^m'  —p  +  q  +  1. 

Now  treat  the  m'  points  k-^^,  ...,km'  in  the  same  way  as  the  m  points 
Ci,  ...,c-m  have  been  treated;  and  let  the  analogous  function  be  constructed. 
Let  </)/ {z),  ... ,  (j)'q' (z)  be  the  q'  quantities  (being  linear  combinations  of  the 

*  Practically  given  by  Eiemann,  Ges.  Werke,  p.  111. 


528  THE   BEILL-NOTHER   LAW   OF   RECIPROCITY  [242. 

functions   {^i, </)p),  which   vanish   at   each   of  the  points  k■^,  ...,h^n'\    the 

analogous  function  is 

which  has  only  Ci,...,c„,  (and  not  k\,.,.,k,n')  for  poles.     It  has  m  ^oles ; 

at  each  of  these,  q  functions  </>  (or  linear  combinations  of  them)  vanish ;  and 

it  contains  q  constants,  so  that,  by  another  application  of  the  Riemann-Roch 

theorem,  we  have 

q  ^m  —  p  +  q  +  1. 

But  m  +  m  =  2p  —  2,  so  that  m  —  p  +  1  +  m'  -  p  +  1  =  0;  that  is,  the  preceding 
relations  betAveen  q  and  q'  are  equalities,  so  that 

q  =  m'  —p  +  q'  +  1, 

q'  =  m—p  +  q  +  l, 
and  therefore* 

2(q  —  q')  =  m'  —  m, 

which  is  called  the  Brill-Nother  latu  of  reciprocity.     It  is  a  complement  of 
the  Riemann-Roch  theorem. 

Since  q  is  actually  equal  to  m'  —p  +  q  +1,  and  does  not  merely  possess 
it  for  an  upper  limit,  it  follows  that 

a](^i  +  ...  +  aq^q 

is   a   special   function,    which   contains    the   largest   admissible   number    of 
arbitrary  constants. 

Note.  The  preceding  investigations  deal  solely  with  those  rational  functions  on  a 
Riemann's  surface  which  have  their  poles  of  the  first  order.  When  we  have  to  deal  with 
functions  which  have  poles  of  order  higher  than  unity,  the  investigations  are  much  more 
complicated  and  really  belong  to  the  general  theory  of  Abelian  functions.  They  will  be 
found,  together  with  references,  in  Baker's  Abelian  Functiotis,  ch.  in. 

The  simplest  result  is  contained  in  the  following  example. 

£x.  If  a  rational  function  is  infinite  at  only  a  single  point  c  on  a  Eiemann  surface 
the  order  of  its  infinity  being  m,  and  if  the  point  c  is  perfectly  arbitrary,  then  m  must  be 
greater  than  p.  (Weierstrass.) 

243.  The  existence  of  functions  that  are  uniform  on  the  surface  and, 
except  at  points  where  they  have  assigned  algebraical  infinities,  are  finite 
and  continuous,  has  now  been  proved ;  we  proceed,  as  in  §  99,  to  shew  how 
algebraical  functions  imply  the  existence  of  a  fundamental  equation,  now  to 
be  associated  with  the  given  surface. 

The  assigned  algebraical  infinities  may  be  either  at  the  branch -points, 
or  at  ordinary  points  which  are  singularities  only  of  the  branch  associated 
with  the  sheet  in  which  the  ordinary  points  lie,  or  both  at  branch-points  and 
at  ordinary  points. 

*  Brill  u.  Nother,  Math.  Ann.,  t.  vii,  (1874),  p.  283. 


243.]  FUNDAMENTAL   EQUATION   FOR   THE   SURFACE  529 

Let  the  surface  have  n  sheets;  on  the  surface  let  the  points  Ci,  Cg,  •.■■,Cm, 
be  ordinary  infinities  of  orders  q^,  q.y,  ...,  q^  respectively — we  shall  restrict 
ourselves  to  the  more  special  case  in  which  g^,  q^,  ...,  qm  are  finite  integers, 
thus  excluding  (merely  for  the  present  purpose)  the  case  of  isolated  essential 
singularities;  and  let  the  branch-points  a^,  a^,,  ...  be  of  orders  pi,  p^,  ...  as 
infinities*  and  of  orders  n  —  1,  r^  —  l,  ...  as  winding-points. 

Let  w-i,  W2,  ...,  Wn  be  the  n  values  of  the  function  for  one  and  the  same 
arithmetical  value  of  z  ;  and  consider  the  function  {w  —  w^)  {w  —  w^  ...{tu  —  Wn). 
The  coefficients  of  w  are  symmetric  functions  of  the  values  Wi,  ...,  w^  of  the 
assigned  function. 

An  ordinary  point  for  all  the  branches  w  is  an  ordinary  point  for  each  of 
the  coefficients. 

An  ordinary  singularity  of  order  q  for  any  branch,  which  can  occur  only 
for  one  branch,  is  an  ordinary  singularity  of  the  same  order  for  each  of  the 
symmetric  functions ;  and  therefore,  merely  on  the  score  of  all  the  ordinary 
singularities,  each  of  these  symmetric  functions  can  be  expressed  as  a  mero- 
morphic  function  the  denominator  of  which  is  the  same  polynomial  function 

m 

of  degree  2  q^  in  z. 

In  the  vicinity  of  the  branch-point  cv^,  there  are  r-^  branches  obtained  from 

{z-a,)  ^^P{{z-a,)% 

1 
where  P  is  finite  when  z  =  a^  by  assigning  to  (z  —  a^)^'  its  r^  various  values. 
Then,  as  in  §  99,  the  point  a^  is  no  longer  a  branch-point  of  any  of  the 
symmetric  functions ;  for  some  of  the  symmetric  functions  the  point  tti 
is  an  accidental  singularity  of  order  p^,  but  for  no  one  of  tjiem  is  it  a 
singularity  of  higher  order.  Hence,  merely  on  the  score  of  the  infinities  at 
branch -points,  each  of  the  symmetric  functions  can  be  expressed  as  a  mero- 
morphic  function  the  denominator  of  which  is  the  same  polynomial  function 
of  degree  2_pi  in  ^. 

No  other  points  on  the  surface  need  be  taken  into  account.  If,  then,  P  (z) 
be  the  denominator  of  the  coefficients  arising  through  the  isolated  algebraical 

TO 

singularities,  so  that  P  (z)  is  of  degree    S  qs  in  z,  and  if  Q  (z)  be  the  de- 

.9=1 

nominator  of  the  coefficients  arising  through  the  infinities  at  the  branch- 
points, then 

P  (z)  Q  (z)  (tU  -  Wi)  (W  -  tU^)  ...{lU  -  Wn) 

*  A  branch-point  a  is  said  to  be  an  infinity  of  order  p  and  a  winding-point  of  order  r-1, 

_2  1 

when  the  affected  branches  in  its  Yicinity  can  be  expressed  in  the  form  {z-a)    ''P  {(^-a)'"}, 

where  P  is  finite  when  z  =  a. 

F.   F.  .  -  34 


530  FUNDAMENTAL   EQUATION  [243. 

is  a  rational  integral  function  of  w  and  z',  say  f{w,  z),  which  is  evidently 

m 

of  degree  n  in  w  and  of  degree  "^  qs-\-  Xp  in  z. 

s=l 

Its  only  roots  are  iu  =  Wi, ...,  Wn',  that  is,  the  function  w  on  the  Riemann's 
surfece  is  determined  as  the  root  of  the  equation  f(w,  z)  =  0;  and  therefore 
the  equation  f(w,  z)  =  0  is  a  fundamental  equation,  to  be  associated  with 
the  surface. 

Ex.  1.  Shew  that  a  fundamental  equation  for  a  three-sheeted  surface,  having  e"^"*'^^  (for 
<m  =  0,  1,  ...,  5)  for  branch-points  each  of  the  first  order,  is 

and  that  a  fundamental  equation  for  a  four-sheeted  surface  haying  the  same  branch-points 
each  of  the  same  order  is 
^   ,  w*--.(6-l-3v'2s2)w2-4vl2\/2m=3  +  v'2  22_|9v'424.  (Thomee.) 

Every  algebraic  function  on  the  surface  requires  its  own  fundamental 
equation;  but,  as  the  branch-points  are  the  same  for  any  surface,  no 
fundamental  equation  can  be  regarded  as  unique.  Having  now  obtained 
one  fundamental  equation  for  algebraic  functions  on  the  surface,  all  the 
investigations  in  Chap.  XVI.  may  be  applied. 

The  preceding  sketch,  in  §§  240 — 243,  of  algebraic  functions  is  intended  only  as  an 
introduction ;  the  developments  are  closely  connected  with  the  theory  of  Abelian  functions 
and  of  curves.     The  propositions  actually  given  are  based  upon 

Eiemann,   Theorie  der  AheVschen  Fmictionen,  Oes.    Werke,  pp.  100 — 102 ; 
Roch,  Crelle,  t.  Ixiv,  (1865),  pp.  372—376; 

Klein's   Vorlesungen  iiher  die  Theorie  der  elliptischen  Modulfunctionen,  (Fricke),  vol.  i, 
pp.  540—549; 
for  further  information  reference  should  be  made  to  the  following  sources: — 
Brill  und  Noether,  Math.  Ann.,  t.  vii,  (1874),  pp.  269—310; 
Lindemann,   Untersiichungen   Uber   den   Riemann-Roch'schen    Satz,  (Leipzig,  Teubner, 

1879),  40  pp. ; 
Brill,  Math.  Ann.,  t.  xxxi,  (1888),  pp.  374—409;  ib.,  t.  xxxvi,  (1890),  pp.  321—360; 
Baker's  treatise,  quoted  §  239  and  at  the  end  of  §  242 ; 

Ap]Dell  and  Goursat,   Theorie  des  fonctions  algehriques  et  de  leurs  integrates,  (Paris, 
Gauthier-Villars,  1895). 

Ex.  2.  Prove  that  the  algebraic  equation  which  subsists  (§  118)  between  two 
functions  u  and  i;  of  a  variable  z,  doubly-periodic  in  the  same  periods,  is  of  class  either 
^ero  or  unity ;  that  it  is  of  class  unity,  if  only  one  incongruent  value  of  z  correspond  to 
_given  values  of  u  and  v ;  and  that  it  is  of  class  zero,  if  more  than  one  incongruent  value 
of  z  correspond  to  given  values  of  u  and  v.  (Humbert,  Giinther.) 

Ex.  3.  If  between  two  uniform  analytical  functions  P  and  Q,  which  have  an  isolated 
point  for  their  essential  singularity,  there  exist  an  algebraic  relation,  then,  when  either 
is  regarded  as  the  independent  variable,  the  connectivity  of  the  Eiemann's  surface  for 
the  representation  of  the  other  is  not  greater  than  three.  (Pi card.) 


244.]  appell's  factorial  functions  531 

244.  We  now  pass  to  the  consideration  of  another  class  of  functions 
associated  with  a  Riemann's  surface. 

The  classes  of  pseudo-periodic  functions,  which  have  been  discussed, 
originally  occurred  in  connection  with  the  functions  that  are  doubly-periodic 
functions  of  the  first  kind;  and  it  may,  therefore,  be  expected  that,  in  a 
discussion  of  functions  which  are  multiply-periodic,  similar  pseudo-periodic 
functions  will  occur. 

These  functions,  in  particular  such  as  are  the  generalisation  of  doubly- 
periodic  functions  of  the  second  kind,  have  been  considered  in  great  detail  by 
Appell*;  they  may  be  called  factorial  functions  f. 

But  the  essential  difference  between  the  former  classes  of  functions  and 
the  present  class  is  that  now  the  argument  of  the  function  is  a  variable  of 
position  on  the  Riemann's  surface  and  not,  as  before,  an  integral  of  the  first 
kind.  It  is  only  in  subsequent  developments  of  the  theory  of  these  functions 
that  the  corresponding  modification  of  argument  takes  place ;  and  a  factorial 
function  then  becomes  a  pseudo-periodic  function  of  those  integrals  of  the 
first  kind. 

We  consider  a  Riemann's  surface  of  connectivity  2p  +  1,  reduced  to  simple 
connectivity  b}^  2p  cross-cuts  taken,  as  in  §  181,  to  be  a^,  bi,  c^  +  a^,  ho,..., 
Cp  +  ap,  hp.  The  functions  already  considered  are  such  that  their  values 
at  points  on  opposite  edges  of  a  cross-cut  differ  by  additive  constants, 
which  are  integral  linear  combinations  of  the  cross-cut  constants,  necessarily 
zero  for  the  portions  c  in  the  case  of  all  the  functions.  The  values  of  the 
constants  for  the  cuts  a  and  the  cuts  h  depend  upon  the  character  of  the 
functions ;  they  are  simultaneously  zero  only  when  the  function  is  a  uniform 
function  of  position  on  the  Riemann's  surface,  that  is,  is  a  rational  function  of 
w  and  z  when  the  surface  is  associated  with  the  fundamental  equation 

F{w,z)  =  Q. 

A  factorial  function  is  defined  as  a  uniform  function  of  position  on  the 
resolved  Riemann's  surface,  finite  at  the  branch-points  no  one  of  which  is 
at  infinity;  all  its  infinities  are  accidental  singularities,  so  that  it  has  no 
logarithmic  infinities  :  and  at  two  (practically  coincident)  points  on  opposite 
edges  of  a  cross-cut  the  quotient  of  its  values  is  independent  of  the  point, 
being  a  factor  (or  multiplier)  that  is  the  same  along  the  cut  for  all  parts 
which  can  be  reached  without  crossing  another  cut. 

*  "  Snr  les  integrales  des  fonctions  a  multiplicateurs..."  (Mem.  Cour.),  Acta.  Math.,  t.  xiii, 
(1890),  174  pp.  This  volume  is  prefaced  by  an  interesting  report,  due  to  Hermite,  on  Appell's 
memoir. 

They  are  also  discussed  in  Neumann's  Abel'schenJPunctionen,  pp.  273 — 278 ;  in  Briot's  Theorie 
des  fonctions  Aheliennes;  in  a  memoir  by  Appell,  Liouville,  3™^  Ser.,  t.  ix,  (1883),  pp.  5 — 24;  and 
they  occur  in  a  memoir  by  Prym,  Crelle,  t.  Ixx,  (1869),  pp.  354 — 362. 

t  Fonctions  a  multiplicateurs,  by  Appell. 

34—2 


4' 


532 


APPELL  S 


[244. 


Then /or  any  portion  c  the  factor  is  unity,  for  any  cut  a  it  is  the  same  along 
its  whole  length,  and  for  any  cut  b  it  is  the  same  along  its  whole  length. 

In  order  to  consider  the  effect  of  passage  over  another  cross-cut  on  the  con- 
stant factor,  we  take  the  figures  of  |§  196, 
230.     Where  a,-  and  hr  intersect  we  have 

F  {z,)  ^mrF  {z,),     F  {z,)  =  m;  F  {z^) ; 

F{z,)  =  nr'F{z,),     F{z,)  =  nrF{z,); 

where  m^,  m/ ;  n^,  n/  ;  are  the  constants 
for  the  portions  of  the  cuts  a^  and  br. 
From  these  equations  it  follows  that 

F(z,)  =  nrm;  F(z^),  Kg.  85. 

and  also  =nrm.rF{z^, 

so  that  nrnir  =  nr'mr. 

Again,  where  Cr+i  cuts  br,  we  have 

F  {z:)  =  n,;F  (z,),     F  (z:)  =  n^F  (z,), 

so  that,  as  F{z^)  =  F{z^)  when  the  points  are^infinitely  close  together,  we 
have 

F{z,)  =  '^F{z,), 


or  the  multiplier  (say  /J^+j)  for  c^+j  is  Z^+j  = 


whence 


=         =  ir-\-\ . 

nir      nr 


Now  tti  is  met  only  by  b^  and  by  no  cut  c  :  so  that  m-^  =■  m/.  Hence  n-^  =  ti/, 
and  therefore  U^l.  Hence  m^_  =  m^ ;  n^,  =  n^;  and jtherefore  ^3=1;  and  so 
on,  so  that 

the  results  necessary  to  establish  the  proposition.  • 

We  shall  therefore  take  the  factor  along  a^  to  be  nir,  and  the  factor  along 
br  to  be  nr,  for  r=l,...,p:  and,,  by  reference  to  §  196,  the  function  at  the 
positive  edge  is  equal  to  the  function  at  the  negative  edge  multiplied  by  the 
factor  of  the  cut. 

Before  passing  on  to  obtain  expressions  for  factorial  functions  in  terms 
of  functions  already  known,  we  may  shew  that  all  factorial  functions  with 
assigned  factors  are  of  the  form 

$  {z)  R  {lu,  z), 

where  <I>  {z)  is  a  factorial  function  with  the  assigned  factors,  and  R  (w,  z)  is  a 
function  of  lu  and  z,  uniform  on  the  Riemann's  surface.     For  if  "^  {z)  and 


244.]  FACTORIAL   FUNCTIONS  533 

^  (2)  be  factorial  functions  with  the  same  factors,  then  ^  (z)-^^  (z)  has  its 
factors  unity  at  all  the  cross-cuts,  so  that  it  is  a  uniform  function  of  position 
on  the  surface  and  is  therefore  *  of  the  form  R  (w,  z).  Consequently,  it  is 
sufficient  at  jiresent  to  obtain  some  one  factorial  function  with  assigned  factors 

m-L,  ...,  nip,  ??i,  ...,  Up. 

Let  Wi(z),  tUo,{z),  ...,  Wp{z)  be  the  p  normal  functions  of  the  first  kind 
connected  with  a  Riemann's  surface,  with  their  periods  as  given  in  §  235. 

Let  TTj  {z),  instead  of  trrja  of  §  237,  denote  an  elementary  normal  function 
of  the  third  kind,  having  logarithmic  infinities  at  a-^  and  /3i  such  that,  in  the 
vicinities  of  these  points,  the  respective  expressions  for  tti  {z)  are 

-logiz-a^)  -\-P{z-a;), 
and  +log(^-/3^)+ Q(^_/3^); 

then  the  period  of  ttj  {z)  for  the  cross-cut  a^  is  zero,  and  the  period  for  the 
cross-cut  hr  is 

2{w,(A)-Wr(ai)l 

iov  r=l,1,  ...,  p.     It  therefore  follows  that  <I>i  {z),  where 

4)i  {z)  =  e"  (^^ , 

is  uniform  on  the  resolved  Riemann's  surface  :  it  has  a  single  zero  (of  the  first 
order)  at  (3-1  and  a  single  accidental  singularity  (of  the  first  order)  at  «! ;  its 
factor  for  the  cross-cut  a^  is  unity,  and  its  factor  for  the  cross-cut  hr  is 

g2{i«r(|8i)-Wr(ai)} 

The  function  <I>i  {z)  may  therefore  be  regarded  as  an  element  for  the  repre- 
sentation of  a  factorial  function. 

Let  ^  {z)  be  a  factorial  function  on  the  Riemann's  surface  with  given 
multipliers  m  and  n;  and  let  it  have  a  number  q  of  zeros  /Sj,  ^^,  ...,  /3g,  each 
of  the  first  order,  and  the  same  number  q  of  simple  accidental  singularities 
«!,  0L2,  ...,  ofg,  each  of  the  first  order,  and  no  others.  Then  ^'  {z)l^{z)  has  2g' 
accidental  singularities ;  in  the  vicinity  of  the  q  points  jB,  it  is  of  the  form 

and  in  the  vicinity  of  the  q  points  a,  it  is  of  the  form 

1 


z—  a 


+  P(z-a); 


^'(z)       ^ 
hence  ___2^^/(.) 

*  It  may  be  pointed  out  that  this  result  is  an  illustration  of  the  remark,  at  the  beginning  of 
§  244,  that  the  factorial  functions  have  a  uniform  function  of  position  on  the  surface  for  their 
argument  and  not  the  integrals  of  the  first  kind,  of  which  that  variable  of  position  is  a  multiply- 
periodic  function. 


534  EXPEESSIONS   FOE  [244. 

is  finite  in  the  vicinity  of  all  the  singularities  of  ^     .  .     Thus 

s  =  l 

has  no  logarithmic  infinities  on  the  surface :  neither  log  <l>  (z)  nor  any  one 
of  the  functions  tt  (z)  has  infinities  of  any  other  kind ;  and  therefore  the 
foregoing  function  is  finite  everywhere  on  the  surface.  It  is  thus  an  integral 
of  the  first  kind ;  so  it  is  expressible  in  the  form 

2Xi  Wi  (0)  +  2X2^/2  (2)  +  ...  +  2\pWp  (z)  +  constant. 

Hence  (^(z)  =  Ae'='  '^=^ 

where  J.  is  a  constant. 

The  function  represented  by  the  right-hand  side  evidently  has  the  q 
points  /S  as  simple  zeros  and  the  g  points  a  as  simple  accidental  infinities, 
and  no  others.  Higher  order  of  a  zero  or  an  infinity  is  permitted  by  repeti- 
tions in  the  respective  assigned  series. 

In  order  that  it  may  acquire  the  factor  m^  on  passing  from  the  negative 
edge  to  the  positive  edge  of  the  cross-cut  a,.,  we  have 

and  that  it  may  acquire  the  factor  n^  in  passing  from  the  negative  edge  to 
the  positive  edge  of  the  cross-cut  br,  Ave  have 

a  p 

The  former  equations  determine  the  constants  X,,.  in  the  form 

1    , 

\r=Tr--^ogmr, 

for  r  =  1,  2,  ...,  p  ;  the  latter  equations  then  give 

q  1         ^ 

X    [Wr  (/3,)  -  Wr  (Ot,)}  =  i  log  tl^  "  9— ■    ^    (Bjcr  log  m^), 
S=l  ■"'''■  k  =  l 

for  r  =  1,  2,  ...,  p. 

Apparently,  X^  is  determinate  save  as  to  an  additive  integer,  say  31^ ;  and 
the  value  of -^log?v  is  determinate  save  as  to  an  additive  quantity,  say  Nriri, 
where  JS^r  is  an  integer.  The  left-hand  side  of  the  derived  set  of  equations 
being  definite,  these  integers  JSf^  and  Mr  must  be  subject  to  the  equations 

k=l 


244.]  FACTORIAL   FUNCTIONS  536 

for  r=  1,  2,  ...,  p;  and  therefore,  equating  the  real  parts  (§  235),  we  have 

I  M^^pkr  =  0, 

A:  =  l 

SO  that  ^    2  MkMrpkr  =  0, 

k  =  l  r=l 

which,  by  §  235,  can  be  satisfied  only  if  all  the  integers  Mr  vanish  and  there- 
fore also  the  integers  Nr. 

Hence  when  the  foregoing  equations  connecting  the  quantities  a,  13,  log  n, 
log  m  are  satisfied,  as  they  must  be,  for  one  set  of  values  of  log  n  and  log  m, 
that  set  may  be  taken  as  the  definite  set  of  values ;  and  the  only  way  in 
which  variation  can  enter  is  through  the  multiplicity  in  value  of  the  functions 
Wi,  ..,,  Wp,  which  may  be  supposed  definitely  assigned. 

The  expression  for  the  function  O  {z)  is  therefore 

7  1      P 

2  7rs(z)+—,  2  {wAz)iosm.} 

Ae'=' 
the  q  zeros  /9  and  the  q  simple  poles  a  being  subject  to  the  equations 

S  \Wr  (/3,)  -  w,  (a^)}  =  \  log  ??,.  -  ^^— ^  2  {B^r  log  m^fc). 

Corollary  I.  The  function  $  {z)  is  a  rational  function  of  position  on 
the  surface,  that  is,  of  w  and  z,  if  all  the  factors  n  and  m  be  unity.  Such  a 
function  has  been  proved  (§  194)  to  have  as  many  infinities  as  zeros;  and 
therefore  integers  N-l, ... ,  Np  ,  i//,  ...,  MJ  exist  such  that,  between  the  zeros  and 
the  infinities  of  a  rational  function  of  w  and  z,  the  p  equations 

2    [Wr  (^s)  -  Wr  («,)}  =  iriN;  -    I    M^Bkr, 
s=l  k=\ 

for  r  =  1,  2,  ...,  p,  subsist*. 

The  function  O  (z)  then  corresponds  to  a  rational  function,  when  regarded 
as  a  product  of  simple  factors,  in  the  same  way  as  the  expression  (§  241) 
in  terms  of  normal  elementary  functions  of  the  second  kind  corresponds 
to  the  function,  when  regarded  as  a  sum  of  simple  fractions. 

Corollary  II.  Every  factorial  function  has  as  many  zeros  as  it  has 
infinities. 

For  if  a  special  function  <^  (z),  with  the  given  factors  and  possessing  q  zeros 
and  q  infinities,  be  formed,  every  other  function  with  those  factors  is  included 
in  the  form 

F{z)=^(z)R(w,  z), 

where  R  (w,  z)  is  a  rational  function  of  w  and  z.     But  R  iw,  z)  has  as  many 
zeros  as  it  has  infinities ;  and  therefore  the  property  holds  of  F  {z). 

*  Neumann,  p.  275. 


536  FACTORIAL   FUNCTIONS  [244. 

Further,  it  is  easy  to  see  that  the  equations  of  relation  between  the  zeros, 
the  infinities  and  the  multipliers  are  satisfied  for  F{z).  For  among  the  zeros 
and  the  infinities  of  '^■{z),  the  relations 

q  1        J' 

S    [Wr  (/3s)  -  Wr  (tts)}  =  i  log  llr  -  9— •    S   {B^r  log  m,,) 
S  =  l  '^^''  Jc  =  l 

are  satisfied ;  and  among  the  zeros  and  the  infinities  of  R  (w,  z),  the  relations 

S   Wr  (A')  -  Wr  (a/)  =  iriN;  -   I   {Bi,r  M„') 
s=l  it=l 

are  satisfied,  where  i\V  and  the  coefficients  M'  are  integers.  Hence,  among 
the  zeros  and  the  infinities  of  F  (z),  the  relations 

1      ^ 

t  \wr  (zero)  -Wr{cc)]  =  ^  (log  Ur  +  Nr  27ri) - K— •  2  [Bj,r  (log mjfc  +  2 Wtti)} 

are  satisfied,  giving  the  same  multipliers  n^  and  w^  as  for  the  special  function 
<E>(4 

Corollary  III.  It  is  possible  to  have  factorial  functions  without  zeros 
and  therefore  without  infinities:  hut  the  multipliers  cannot  he  arbitrarily 
assigned. 

Such  a  function  is  evidently  given  by  ^ 

derived  from  <I>  {z)  by  dropping  from  the  exponential  the  terms  dependent 
upon  the  functions  tt  (2;).  The  relations  between  the  factors  are  easily 
obtained. 

Note.     The  effect  of  the  p  relations 

q  1      *^  • 

2    [Wr  (/3s)  -  Wr  (Gs)]  =  h  log  ?^r  "  ^T""    ^   (Bkr  log  mfc), 

subsisting  between  the  factors,  the  zeros  and  the  infinities  of  the  factorial 
function,  varies  according  to  the  magnitude  of  q. 

If  q  be  equal  to  or  be  greater  than  p,  it  is  evident  that  all  the  infinities  a 
and  ^— j3  of  the  zeros  ^  can  be  assumed  at  will  and  that  the  above  relations 
determine  the  p  remaining  zeros.  The  function  therefore  involves  2q—p 
arbitrary  elements,  in  addition  to  the  unessential  constant  A. 

In  particular,  when  q  is  equal  to  p,  the  infinities  a  can  be  chosen  at  will 
and  the  zeros  /3  are  then  determined  by  the  relations.  It  therefore  appears 
that  a  factorial  function,  which  has  only  p  infinities,  is  determined  hy  its 
infinities  and  its  cross-cut  factors. 

When  q  is  greater  than  p,  say  =  p+  r,  then  the  q  infinities  and  r  zeros 
may  be  chosen  at  will.  By  assigning  various  sets  of  r  zeros  with  a  given  set 
of  infinities,  various  functions  ^1  (z),  ^.^  (z),  . . .  will  be  obtained  all  having  the 


244.]  BIRATIONAL   TRANSFORMATION  537 

same  infinities  and  the  same  cross-cut  factors.  Let  s  such  functions  have 
been  obtained ;  consider  the  function 

it  will  evidently  have  the  assigned  infinities  and  the  assigned  cross-cut 
factors.  Then  s—1  ratios  of  the  quantities  /jl  can  be  chosen  so  as  to  cause 
<t>  (z)  to  acquire  s  —  1  arbitrary  zeros.  The  greatest  number  of  arbitrary 
zeros  that  can  be  assigned  to  a  function  is  r,  which  is  therefore  the  greatest 
value  of  s  —  1.  Hence  it  follows  that  ?'  +  1  linearly  independent  factorial 
functions  <I>i {z),  ...,  ^r+i  (2)  exist,  having  assigned  cross-cut  factors  and  p  +  r 
assigned  infinities ;  and  every  other  factorial  function  with  those  infinities  and 
cross-cut  factors  can  he  expressed  in  the  form 

/^i^i  {z)  +  ^lf^■2  {z)  +  ...+  /i,-+i^r+i {z), 
tuhere  /jl^,  ...,  /x^+i  are  constants  whose  ratios  can  he  tised  to  assign  r  arhitrary 
zeros  to  the  function. 

These  factorial  functions  are  used  by  Appell  to  construct  new  classes  of  functions  in  a 
manner  similar  to  that  in  which  Riemann  constructs  the  Abelian  transcendents.  Their 
properties  are  developed  on  the  basis  of  algebraic  functions ;  but  as  only  the  introduction 
to  the  theory  can  be  given  here,  recoiirse  must  be  had  to  Appell's  interesting  memoir, 
already  cited.     See  also  Baker's  Abelian  Functions,  ch.  xiv. 

BiRATioNAL  Transformation. 

245.  It  has  already  been  pointed  out  (§  193)  that,  if  w'  denote  any 
arbitrary  rational  function  on  a  Riemann's  surface  (say  S)  associated  with  the 
relation  f{iv,  z)  =  0,  then  w'  satisfies  an  equation  f  (w',  z)  =  0.  Similarly, 
if  /  denote  another  rational  function  on  S,  the  elimination  of  w  and  z 
between  the  three  algebraical  equations 

w'  =  R^  (w,  z),     z  =  R2  (w,  z),    f(w,  z)  =  0, 
leads  to  another  equation  F  {iv  ,  z)  =  0. 

Relations  such  as  these,  which  express  new  variables  w'  and  z'  as  rational 
functions  of  old  variables  w  and  z,  are  called  transformations:  sometimes 
rational  transformations.  Transformations  exist  between  two  sets  of  variables, 
each  set  being  regarded  as  a  pair  of  independent  variables:  with  such 
transformations  (which  include  the  well-known  Cremona  transformations)  we 
are  not  specially  concerned,  seeing  that  our  variables  w  and  z  are  connected 
by  a  permanent  equation  f{tu,z)  =  0.  We  have  to  deal*  with  rational 
transformations  between  two  equations  such  as 

f{w,z)  =  0,     F(w',z')  =  0. 

*  The  difference  is  the  same  as  the  difference  between  the  rational  transformations  of  a  plane 
and  the  rational  transformations  of  a  curve  in  the  plane;  the  former  give  rise  to  the  latter, 
though  not  to  the  whole  of  the  latter. 


538  TRANSFORMATION    OF  .  [245. 

The  degree  of  F  in  w'  and  /,  when  the  orders  of  w'  and  z'  are  known,  can 
be  inferred.  Suppose  that  /=  0  is  of  degree  m  in  lu  and  of  degree  nm  z; 
and  let  the  orders  of  tv'  and  z',  defined  by  the  equations 

w'  =  Ri  (lu,  z),     z  =  i^o  {Wy  z), 

be  m'  and  ?i'  respectively.  There  are  m  positions  on  S  which  correspond  to 
any  given  value  of  w' ;  each  such  position  gives  one  value  of  z' ;  and  therefore 
there  are  m  values  of  z'  for  any  given  value  of  w'.  Similarly,  there  are  7i' 
values  of  w'  for  any  given  value  of  /.     Accordingly,  the  equation 

F(tu',z')  =  0 

is  of  degree  n'  in  w',  and  of  degree  w'  in  /. 

As  the  two  rational  functions  of  position  on  S,  represented  by  w'  and  z', 
are  quite  unrestricted,  it  follows  that  we  can  obtain  an  unlimited  number  of 
transformations  of  the  equation  f=0.  We  assume  that  /'  is  an  irreducible 
polynomial,  that  is,  ^cannot  be  resolved  into  factors  rational  in  w  and  z,  so  that 
/=  0  is  an  irreducible  equation ;  and  we  find  that  the  equation  F  =  0,  arising 
out  of  any  rational  transformation  of /=0,  is  such  that  the  polynomial  F 
either  is  irreducible  or  is  some  poiver  of  an  irreducible  polynomial.  For  let 
TTq  and  Z^  denote  any  values  of  w  and  z,  which  satisfy  ^  =  0 ;  they  arise 
through  some  position  iv^,  Zq  on  S:  and  let  Wj  and  Z^^  denote  any  other 
values  of  w'  and  z,  which  satisfy  F  =  0;  they  arise  through  some  position  Wj, 
Zi  on  S.  (In  each  case,  there  may  be  more  than  one  position.)  We  can  pass 
from  Wq,  Zo  to  w^,  z^  on  ^  by  a  continuous  path,  which  avoids  all  the  branch- 
poirlts,  and  which  does  not  pass  through  any  infinity  of  w'  or  z' ;  during  the 
passage  the  values  Wq  and  Z^  change  continuously  into  W^  and  Z^^.  Hence 
when  we  have  constructed  the  Riemann's  surface  (say  S')  associated  with 
F  =0,  and  take  account  of  the  dependence  of  w'  and  /  upon  w  and  z  that 
leads  to  F=0,  it  follows  that,  on  this  new  surface  S',  a  continuous  path  exists 
which  joins  the  position  Wq,  Z^  to  the  position  TTj,  Z^.  Also  these  positions 
are  any  positions  on  8',  because  the  values  W^,  Z,^',  Wj,  Z^;  are  any  values 
that  satisfy  ^  =  0;  hence  (§  176,  Ex.  5,  Cor.  II.)  F  either  is  an  irreducible 
polynomial  or,  if  reducible,  is  some  power  of  an  irreducible  polynomial. 

Consider  any  position  w',  z'  on  -8'.     To  the  value  of  iv ,  there  correspond 
m!  positions  on  8,  say 

and  to  the  value  of  /,  there  correspond  ri  positions  on  *S^,  say 

/Si,  61;  A>  ^2;   •••;  Ai'>  ^ft'- 

Then  as  w\  z  constitute  a  position  on  8\  it  follows  that  one  (or  more  than 
one)  position  on  8  must  be  common  to  the  two  sets.  First,  let  only  one 
position  be  common  to  the  two  sets.  In  that  case,  the  simultaneous  values 
of  w'  and  /  (which  determine  a  position  on  >S")  determine  a  single  position  on 


245.]  ALGEBRAIC   EQUATIONS  539 

S,  that  is,  give  w  and  z  uniquely.  Now  w  and  z,  as  functions  of  position 
on  S',  at"e  manifestly  not  transcendental;  it  has  just  been  proved  that  they 
are  uniquely  determined  by  iv'  and  /;  and  they  consequently  are  rational 
functions  of  position  on  8',  that  is,  we  have 

lu  =  S,  (w',  /),  z  =  S,  {w\  /),  F  (w',  z')  =  0, 
where  S-^  and  S2  are  rational  functions.  Moreover  in  this  case,  to  a  general 
value  of  z',  there  correspond  n'  different  positions  on  S;  each  of  these 
determines  a  value  of  tv',  so  that  there  are  n  values  of  lu' ;  these  values  are 
all  different,  for  taking  the  single  value  of  z  and  the  various  values  of  w' 
in  turn,  the  n'  positions  in  the  second  set  must  be  exhausted,  and  no  first 
set  has  more  than  one  point  common  with  the  second  set.  Hence  to  a 
value  of  /,  there  correspond  n  different  values  of  w'.  Also  F  is  either  an 
irreducible  polynomial  of  degree  n'  in  tu',  or  it  is  a  power  of  some  irreducible 
polynomial ;  in  the  present  case,  therefore,  F  is.  irreducible.  Accordingly, 
the  surfaces  8  and  8'  associated  with  the  two  equations 

f{w,z)  =  0,     F(iv',z')  =  0 
are  such  that  each  position  on  one  determines  one  (and  only  one)  position 
on  the  other ;  and  the  variables  of  each  position  are  expressible  rationally  in 
terms  of  the  other,  in  forms 

w'  =  -Ri  (w,  z)\  w  =  8-^  (w',  z')] 

z'  =  R2  (w,  z)}  ■  z  =  82  (w',  z')\ 

where  H^,  Ro,  8^,  8.2  are  rational  functions.  Such  a  transformation  is  called 
birational. 

Next,  let  I  of  the  positions  on  8  be  common  to  the  two  sets,  which  give 
the  values  of  w'  and  /  respectively.  To  the  position  a^,  a^  in  the  w'-set, 
there  corresponds  a  definite  value  of  z' ;  as  I  positions  on  8  arise  through 
given  values  of  w'  and  z',  it  follows  that  other  I  —  1  positions  in  the  ly'-set 
give  the  same  value  of  /.  Let  these  be  Cg,  ag;  ... ;  oli,  ai;  so  that  no  other 
position  in  the  w'-set  gives  that  value  of  /.  Take  now  some  other  position 
a^+i,  a^+i;  it  gives  a  definite  value  to  /,  and  there  are  other  ^—  1  positions, 
say  a;_,.2,  a^^-a ;  ...;  a^i,  aazV  which  give  that  value.  Proceeding  in  this  way, 
we  see  that  m  must  be  a  multiple  of  I,  say  in'  =  'ni"l ;  and  that  the  one  value 
of  w ,  which  gives  m'  positions  on  8,  gives  rise  to  m"  values  of  /,  each 
of  them  repeated  I  times.  Dealing  similarly  with  the  ^'-set,  we  see  that  n 
is  a  multiple  of  I,  say  n  =  n"l ;  and  that  the  one  value  of  /,  which  gives 
n'  positions  on  8,  gives  rise  to  n"  values  of  w ,  each  of  them  repeated  I  times. 
In  this  case,  F  is  the  lih.  power  of  an  irreducible  polynomial  F^  {w',  z'),  of 
degree  n"  in  w  and  degree  m"  in  / ;  and  the  surfaces  8  and  8'  associated 

with  the  two  equations 

f{w,z)  =  0,     F,(tv,z')  =  0 

are  such  that  to  one  position  on  8,  there  corresponds  only  a  single  position 
on  8' :  while  to  one  position  on  ;S^',  there  correspond  I  positions  on  8.     The 


540  RATIONAL   TRANSFORMATIONS  [245. 

transformation  from  f  to  F^  is  rational ;  it  is  not  rational  fi^om  F^  to  /;  that 
is,  the  transformation  is  not  birational. 

Sometimes  the  results  are  expressed  in  geometrical  language  by  saying 
that,  in  the  former  case,  there  is  a  (1,  1)  correspondence  between  the  curves 
/=  0,  F  =  0:  and  in  the  latter  case  a  (1,  I)  correspondence  between  the 
curves /=  0,  Fi  =  0.  The  whole  subject  of  rational  transformation  is  involved 
in  the  theory  of  correspondence  between  curves. 

JS^ote  1.  It  has  been  proved  that,  in  the  equation  F(w',  z)  =  0  obtained 
by  eliminating  w  and  z  between 

w'  =  i^i  (w,  z),     z'  =  R.2  {iv,  z),    f{w,  z)  —  0, 

the  polynomial  F  either  is  irreducible  or  it  is  some  power  of  an  irreducible 
polynomial.  Now  when  the  values  iv'  =  R^,  z'  =  R.j,  are  substituted  in  F—0, 
the  result  is  to  give  an  equation  in  w,  z  only;  so  that,  F(Ri,  R^)  must  have 
f{iv,  z)  as  a  factor.  It  is  not  possible  to  prove  that  i^  is  a  power  of /(w,  z), 
because  this  is  not  always  the  case.     As  one  example  of  this  remark,  let 

tv  =  T,{W,Z),     z=T,(W,Z), 

be  substitutions,  which  leave  w'  and  z'  unchanged  in  form,  that  is,  give 

vj'  =  R,  (  W,  Z),     z'=R,{  W,  Z) ; 

the  equation  F  (w',  z')  =  0  will  be  obtained  in  association  with  the  relation 
/(Ti,  Tz)  =  0;  and  therefore  F  will  contain  f{T-^{w,  z),  T^iw,  z)]  as  a  factor. 
Thus  when  we  substitute  for  w'  and  /  in  F,  the  resulting  expression  may  be 
divisible  by  factors  other  than  f{iu,  z). 

Note  2.  When  ^  is  a  power  of  a  pol3rnomial,  some  special  process  of  the 
elimination  indicated  on  p.  537  may  lead,  not  to  F,  but  immediately  to  the 
polynomial.  The  explanation  is  that  the  eliminant  then  obtained  is  not  of 
the  proper  degree  in  w  and  z'  as  required  on  p.  538.  As  a  trivial  example, 
we  see  that  the  equations 

w^  +  z^  =  1,     w'  =  w^,     z'  =  z^. 

lead  at  once  to  w'  +  z'  —  \,  whereas  F{w',  z)  is  {w'  +  z'  —  ly. 

Ex.  1.     Consider  the  transformation  of  the  equation  ^^  +  ^^  =  1  by  the  relations 

Jcvf  =  wz,         %z'  =  aw  +  cz. 

As  regards  the  degrees  of  vJ  and  2',  each  of  these  variables  is  infinite  only  when  \z\^ 
and  so  |  w|,  is  infinite.  There  are  three  such  positions  on  the  surface;  at  each  of  them, 
w'  is  infinite  of  the  second  order,  and  z  of  the  first  order  ;  so  that  wi',  the  degree  of  w\ 
is  6 ;  and  w',  the  degree  of  z\  is  3.  Accordingly,  the  equation  between  w  and  0'  must  be 
of  degree  3  in  to'  and  degree  6  in  z . 

We  have 

{aw  —  cs)^  =  4  (s'^  -  kaciv')  =  iA^, 
say ;  so  that 

aw  =  z'  +  di.,         cz  =  z'  —  ^. 


245.]  EXAMPLES  541 

Substituting  in  w^  +  z^=l,  we  have 

=  (c^  +  a3)  {4z''^  —  3kaciv')  z'  +  (c^-  a^)  A  {'iz'^  —  kacw'), 
and  therefore 

(c3  _  a3)2  {iz'^-kacivy  (s'^  -  kacio')  -  {a^c^  -  {c^  +  a^)  (4s'2  -  Zkacw')  /}2=0, 

which  is  the  equation  F{;w',  z')  =  0.    Manifestly  i^is  irreducible,  so  that  the  transformation 

is  birational ;  in  fact, 

,     a^c^  —  {<?  +  a^)  (42'3  —  ^kacw')  z' 

(c^  —  a^)  (42'2  —  kacw')         ' 

,     a^c^  -  (c^  +  a^)  (4:z'^  -  Skacw') z' 

^0  =  z  ^ ^ — . 

{(^  —  a^)  {4z'^  —  kacw') 

Further,  the  original  relations,  which  express  tv'  and  /  in  terms  of  the  variables 
^0  and  z,  are  unchanged  in  form  when  the  latter  are  subjected  to  the  substitution 

aw  =  cZ,         cz  =  a  W ; 

and  therefore,  when  the  values  of  ^o'  and  z'  are  substituted  in  F{w',  z')  =  Q,  it  is  to 
be  expected  that  the  resulting  equation  will  give  rise,  not  merely  to  -w^+a^— 1  =  0,  but 
also  to  its  transformation  by  the  foregoing  substitution.     We  have 

4  (2'2  —  kacw')  =  {aw  —  cs)^, 

42'2  —  kacio'   =  a^w'^  +  acwz  +  c'^z'^, 

Az"^  —  Zkacw'  =  a^  lo^  —  acwz  +  <?z^^ 

22'  =axo-\-cz, 

so  that  i^=0,  on  multiplication  by  4,  becomes 

(C3_a3)2  (a3^i;3_c323)2_|2a3c3_(c3  +  a3)  (a3w3  +  c323)}2  =  0, 

that  is, 

-4c3a3  {%ifi-\-^  —  \)  {a^w^-\-c^z^  —  a^(?)  =  (^. 
The  equation 

t„3  +  j3_l  =  0 

is  the  original  equation;  when  subjected  to  the  substitution  a^o  —  cZ,  cz  =  aW,  the  other 
factor  in  F  is  obtained,  thus  verifying  the  inference. 

Bx.  2.  Discuss  in  a  similar  manner  the  transformation  of  (i)  the  equation  Xio+fiz  — 1  =  0 
by  the  relations  kw'  =  wz,  2z'  =  aw-\-cz;  (ii)  the  equation  w^-\-z^-3awz=l  by  the  relations 
w'  =  w^,  z'=z^. 

Ex.  3.     Consider  the  transformation  of  the  equation 

aw^  +  bivh  +  z{z:^ +1)^  =  0, 
by  the  relations 

/     s^  + 1  ^/  _  ■^-' 

w    '  2  " 

The  degrees  of  tv'  and  2'  are  equal  to  the  respective  numbers  of  their  infinities.  In  the 
vicinity  of  2=0,  take  z=t^;  then  w  <x.t,  and  w'  <x  t~'^.,  z'  (x:  t~'^\  that  is,  the  point  counts  1 
for  w'  and  2  for  2'.  In  the  vicinity  of  z-=-i,  take  2  =  i  +  ^;  then  wx  ^,  so  that  w'  is  finite 
and  2'  is  zero:  the  point  counts  0  for  w'  and  for  2'.  Likewise  z=—i  (where  iv=0) 
counts  0  for  w'  and  for  2'.  In  the  vicinity  of  2  =  00 ,  take  2=  :7'3 ;  then  w  oc  T^,  and  w'  x  T, 
z'  az  T"^ ;  that  is,  the  point  counts  1  for  w'  and  2  for  2'.  Thus  the  degree  of  w'  is  2 
and  that  of  2'  is  4 ;  the  equation  -between  w'  and  2'  must  be  of  degree  4  in  ^y'  and  degree  2 
in  2'. 


542  INVARIANCE   OF   GENUS   IN  [245. 

Now  we  have 

w'2      z(z^  +  l)'^  ,  z  b 

—r  =  — ^ — 5 — -  =  -a  —  b—=—a  — ,, 
z  vfi  w  z 

so  that 

w"^  +  az'  +  h  =  0 ; 

that  is,  our  equation  F=Q  is 

Because  F  is  reducible,  being  the  square  of  a  polynomial,  the  transformation  is  not 
birational. 

Ex.  4.     Discuss  the  transformation  of  w^-\-z^==\  by  the  relations 

]cw'  =  wz,         '2,z'  =  w  +  z; 
in  effect,  the  case  of  Ex.  1  when  a  =  c,  there  supposed  excluded. 

Ex.  5.  Two  Eiemann's  surfaces  are  so  related  that,  to  each  point  on  either,  there 
corresponds  one  (and  only  one)  point  on  the  other ;  prove  that  the  (bi-uniform)  trans- 
formation between  the  surfaces  is  necessarily  birational  in  its  expression. 

(Picard.) 

246.  We  proceed  to  consider  some  of  the  simpler  properties  of  equations 
(or  Riemann's  surfaces)  which  can  be  birationally  transformed  into  one 
another.  In  the  first  place,  we  have  the  theorem*  that  two  equations  {or 
Riemanns  swfaces),  which  are  birationally  transformable  into  each  other,  are 
of  the  same  genus. 

*      Let  p  denote  the  genus  of  f(tu,  z)  =  0,  and  P  the  genus  of  F(w',  z)  =  0, 
which  are  transformable  into  one  another  by  the  relations 
xv  =  Ri  {w,  z))  w  =  Si  {w,  z')] 

z'=R,{w,z)]'  z  =  So^{w',z')\' 

where  i^^,  R^,  Si,  So  are  rational  functions  of  their  arguments.  It  is  known 
that  there  are  p  linearly  independent  functions  of  the  first  kind  on  the 
Riemann's  surface  S  associated  with  /  =  0 ;  each  of  them  is  everywhere  finite 
on  that  surface.  Denoting  them  hy  Ui,  ... ,  Up  in  their  normal  form,  consider 
any  one  of  them,  say  Ui,  in  the  form 

¥ 

dw 
When  substitution  for  w  and  z  in  terms  of  lu'  and  z'  is  effected  upon  this 
integral,  it  becomes  a  function  of  w'  and  z',  that  is,  it  becomes  a  function  of 
position  on  the  Riemann's  surface  S'  associated  with  F  =  0.  But  though 
its  form  is  changed,  its  value  is  unchanged,  by  mere  transformation  of  its 
variables ;  and  therefore  this  function  of  position  on  S'  is  everywhere  finite 
on  that  surface,  that  is,  it  becomes  a  function  of  the  first  kind  on  S'.  Let 
Vi,  ...,  Vp  denote  the  P  linearly  independent  functions  of  the  first  kind  in 
their  normal  form,  which  belong  to  F  =  0;  then  (§  234)  Ui  is  expressible 
in  terms  of  them  by  an  equation 

Ui  =  Cii^i  -f  C12V0  +  ...  +  C-ipVp  +  c/, 
*  Riemann,  Ges.  Werke,  t.  i,  p.  112. 


246.]  BIRATIONAL   TRANSFOEMATION  543 

where  the  quantities  c  are  constant.     Similarly,  we  have 

U^  =  CnVi  +  a,^V2  +  . . .  +  CrpVp  +  c/,      (r  =  2,...,  p), 
where  also  these  quantities  c  are  constant. 

Now  Ui,  ...,  Up  are  linearly  independent,  so  that  no  relation 

KiUi  +  K2U2  +    •  . .  +  Kp'tlp  =  K 

with  constant  coefficients  can  exist ;  hence  P  must  be  at  least  as  large  as  p, 
for  otherwise  determinantal  elimination  of  the  quantities  v  would  lead  to  the 
forbidden  relation.     Hence  we  have 

_p  ^  P. 

Beginning  .with  the  functions  v-^,  ...,Vp,  and  treating  them  in  the  same 
way  as  li^,  ...,  Up  have  been  treated,  we  similarly  obtain  the  result 

P<i?. 

Combining  the  two  relations,  we  have 

P  =  P, 

that  is,  the  two  surfaces  are  of  the  same  genus. 

Moreover,  the  argument  shews  that  a  function  of  the  first  kind  for  one 
surface  is  transformed  into  a  function  of  the  first  kind  for  the  other  surface ; 
and  that  the  p  normal  functions  of  the  first  kind  for  two  surfaces,  which  are 
birationally  transformable  into  one  another,  are  connected  by  equations  of  the 
form 

Wj.  ^  Cj-{V\  +  C)-2V2  +  . . .  +  CfpVp  +  Cj- , 

for  r=l,2,...,py  the  determinant  of  the  coefficients  Cij  being  different  from 
zero. 

Ex.  Prove  that  a  function  of  the  second  kind  upon  one  of  the  surfaces  is  transformed 
into  a  function,  also  of  the  second  kind,  upon  the  other;  and  likewise  for  functions  of  the 
third  kind ;  the  surfaces  being  birationally  transformable  into  one  another. 

Further,  let  U-^,  ...,  Up  denote  the  p  adjoint  polynomials  of  order  n  —  3 
(where  /  is  of  degree  n  in  w),  which  belong  to  the  p  normal  integrals  of  /  of 
the  first  kind;  and  let  Fi,  ...,  Vp  denote  the  p  adjoint  polynomials  of  order 
n  —2>  .(where  F  is  of  degree  n  in  lu'),  which  belong  to  the  p  normal  integrals 
of  F  of  the  first  kind.     We  have 


■rrr  dz,  Vk  = 

div 


dF       ' 
dw' 


so  that  the  above  p  relations  between  u  and  v  give  p  differential  relations  of 
the  form 

■rpdz  =  ^(c^jV,  +  c^2V2+ ...+c^pVp), 
dw  div' 


544  EQUATIONS   OF   BIKATIONAL    TRANSFORMATION  [246. 

satisfied  in  virtue  of  the  birational  transformation.  Hence,  when  p>l,we 
have 

or  the  ratio  of  two  adjoint  polynomials  of  order  ?i  — 3  for  the  one  surface  is 
transformed  into  the  ratio  of  two  adjoint  polynomials  of  order  n  —  8  for  the 
other  surface,  when  the  transformation  is  birational.     When  p  =  1,  we  merely 

have 

U     .        V     , 

dj_  dF 

dw  dw' 

satisfied  in  connection  with  the  birational  transformation ;  and  when  p  =  0, 
there  is  no  relation. 

The  transformations  of  equations  of  genus  0  or  1  are  to  be  considered 
separately.  It  is  clear  that,  when  two  equations  of  the  same  genus  greater 
than  unity,  are  known  to  be  birationally  transformable  into  one  another,  the 
equations 

Ui     CiiFi  +  ...  +  CipFp 

can  be  used  to  obtain  the  birational  transformation. 

As  a  birational  transformation  conserves  the  genus  of  the  equation  to 
which  it  is  applied,  we  naturally  regard  all  equations,  which  are  birationally 
transformable  into  one  another,  as  belonging  to  the  same  class :  and  we-  have 
to  determine  what  are  the  characteristics  other  than  conserved  genus  upon 
which  the  class  depends. 

To  obtain  these,  take  an  equation  of  genus  p(>  1) ;  (equations  of  genus  0 
and  1  will  be  considered  separately,  from  this  point  of  view  as  well  as  for 
the  reason  above) :  and  on  the  Riemann's  surface  of  the  equation,  take  a 
rational  function  /,  having  /j,  poles  each  of  the  first  order.  Let  the  positions 
of  the  /Lt  poles  be  chosen  quite  arbitrarily,  and  let  their  number  be  >2p  —  2, 
so  that  the  rational  function  is  not  a  special  function  (§  242).  Now  /  contains 
fi-  p  +  1  arbitrary  constants,  which  enter  linearly  (|  240 :  the  number  q 
is  zero,  because  /jb>2p  —  2):  and  therefore  as  the  positions  of  the  poles  are 
arbitrary,  each  of  them  accordingly  being  determined  by  an  arbitrary  quantity, 
it  follows  that  the  total  number  of  arbitrary  constants  in  z'  is 

(fi  -p  +  l)  +  fj.,  =  2fi  -p  +  1. 

Choose  z'  as  the  independent  variable  for  a  transformed  equation.  Since 
the  degree  of  /  on  the  original  surface  is  /x,  being  the  number  of  its 
infinities,  we  know   (|   245)  that  the   degree   of  the   new   equation    in    its 


246.]  CLASS-MODULI  545 

dependent  variable  is  equal  to  ix ;  hence  the  new  Riemann's  surface  is 
/i-sheeted.     As  its  genus  is  equal  to  p,  its  ramification  is  given  by 

n  =  2(/^+i)-l). 

Now  when  the  branch-points  and  the  branchings  of  a  surface,  of  given  genus 
and  given  number  of  sheets,  are  assigned,  the  surface  is  definitely  known  as 
(at  the  utmost)  one  of  a  limited  number  (§  212  :  footnote).  The  corresponding 
equation  is  then  known  (§  193)  so  that,  as  z  is  known,  the  dependent  variable 
can  be  regarded  as  determined  by  the  assignment  of  the  ramification  :  it 
contains  no  independent  arbitrary  element. 

We  have  2^  —  _p  +  1  disposable  constants  by  which  to  meet  the  demands 
of  the  ramification,  which  amount  to  2/Li  4-  2j9  —  2  constants :  hence  there  are 

'ip  -  3, 

=  2yu,  4-  2|)  —  2  —  (2/x  —  j!>  -I- 1),  constants  surviving,  as  undetermined  by  the 
arbitrary  elements  in  z'.  The  transformation  is,  of  course,  definite ;  and 
therefore  these  Zp  —  3  quantities  are  determined  by  the  first  surface. 

It  therefore  follows*  that  the  class  of  equations,  which  are  birationally 
transformable  into  one  another,  are  determined  by  3jw  —  3  quantities ;  they 
are  called  the  class-moduli  of  the  equations. 

In  this  result,  which  is  due  to  Riemann,  one  modification  must  be  made, 
as  pointed  out  by  Klein f.  In  the  course  of  the  proof,  it  was  assumed  that 
all  the  2fj,—p  +  l  disposable  constants  could  be  used  to  determine  2yu,  —  p  -f  1 
quantities  connected  with  the  ramification.  As  will  be  seen,  a  surface  may 
be  transformable  into  itself  by  a  birational  transformation;  and  it  might 
happen,  in  such  a  case,  that  the  transformation  contained  arbitrary  constants. 
If  p  be  the  number  of  these  arbitrary  constants,  then  it  follows  that,  in  the 
transformation  under  our  earlier  consideration,  we  cannot  use  more  than 
2/uL  —  p  +  1  —  p  of  the  arbitrary  constants  in  /  for  the  ramification  of  the 
surface  ;  and  therefore  the  number  of  class-moduli  is 

n  -  (2^  -p  +  l-p) 

=  Sp  —  S  +  p. 

As  a  matter  of  fact,  p  =  0  when  j9  >  1  (§  250) ;  p  =  1  when  p  =  1  (§  248) ; 
p  =^S  when  p  =  0  (§  247) ;  all  of  which  results  will  be  established  later. 

Ex.  1.     Consider  the  equations 

where  Z^  is  a  sextic  function  of  2,  «4  is  a  quartic  and  %  is  a  quadratic  in  i'.  Each  of  the 
equations  is  of  genus  2 ;  and  therefore  it  may  be  expected  that,  if  they  are  birationally 

*  Eiemann,  Ges.  Werke,  p.  113.  It  is  assumed  thioughout  that  each  equation  is  comijletely 
general :  it  may  happen  that,  for  equations  which  are  special  in  form,  the  number  of  class-moduli 
is  less  than  dp  -  3. 

t  Ueher  Riemann's  Tluorie  der  algehraischen  Functionen,  (Leipzig,  Teubner,  1882],  p.  65. 

F.  F.  35 


546  EXAMPLES  OF  [246. 

transformable  into  one  another,  three  (  =  3.2-3)  relations  among  their  constants  will 
be  satisfied. 

Integrals  of  the  first  kind  belonging  to  the  first  equation  are 


I  Z^    2  dz^  I  zZq    ^  dz ; 

and  integrals  of  the  first  kind  belonging  to  the  second  are 

I  (Mo  %)  ~  ^  dz',  j  Z   {U2  Ui)  "  2  dz'. 

As  the  equations  are  to  be  rationally  transformable  into  one  another,  we  have  (§  246) 
Zq~^  dz  —  y {U2tii)~ ^  dz' -\- 8z'  (2t2%)~''^  d^-, 

zZe~^  dz  =  a  (u2Ui)~^  dz  +^z'  {u^Ui)"^ dz', 

and  therefore 

a  +  ^z' 

y  +  ^z' 
Take  w  =  iv'u2K, 

whei'e  A'  is  some  function  of  z ;  then 

ZQ  —  K^u^Ui. 
In  other  words,  the  effect  of  substituting  {a  +  ^z')l{y+bz')  for  z  in  the  sextic  Z^  must  be  to 
give  a  multiple  of  the  sextic  U2Ui ;  so  that,  taking 

where  e  is  a  constant,  we  have 

ZQ{a+l:iz',  y  +  S2')  =  e"^«2«4. 
In  order  that  one  sextic  may  be  transformable  into  another  by  a  substitution 

/  :  l=a  +  /3/  :  y  +  bz', 

they  must  have  their  invariants  the  same  save  as  to  a  factor.  The  invariants  of  a  sextic 
are  of  deo-ree  2,  4,  6,  10  (as  well  as  one  of  degree  1.5,  the  square  of  which  is  expressible  as 
an  integral  function  of  the  others);  denoting  them  for  Z^  by  Li,  I^,  !&,  Jw,  and  for 
^2%  ^^y  ^^2)  Ji^  Jqi  '-^w  respectively,  three  relations  as  required  are 

7^/2-2  =J^J,^-'^] 

In  order  to  find  the  actual  transformations,  we  compare  the  coefficients  in 

ZQ{a+^z',   y+8z')  =  eh(,2th- 

There  are  seven  equations,  each  expressing  some  homogeneous  combination  of  dimensions 
six  in  a,  /3,  y,  S  in  terms  of  e'-^  and  constants.  The  equations  are  equivalent  to  four 
in  virtue  of  the  preceding  three  relations  ;  they  therefore  suffice  for  the  determination 
of  a,  Idj  y,  S.     The  transformation  thus  is 

a  +  3z'  eU2 

y+6z"  {y  +  8z'f 

which  happens  to  be  a  Cremona  transformation. 

Ex.  2.     Consider  the  birational  transformations  of  the  equation 

where   U  is  a  quartic  function  of  z. 


246.]  BIRATIONAL   TRANSFORMATION  547 

Let  H,  $  denote  the  Hessian  and  the  cubicovariant  of  U  respectively :  /,  J  its  quadrin- 
variant  and  its  cubinvariant.     Then 

20.=  ^^- 6^^ 

^z       dz' 


$2  3 

m^^ium- 

-JU^. 

Take  a  new  va] 

fiable  z' 

such 

that 

z'U+H=Q; 

then 

dz'      2$ 
dz~  U^-' 

and  therefore 

9, 

dz 

dz' 

Ui  '  (4£'3-Iz'-J)h' 
Accordingly,  the  integral  of  the  first  kind,  belonging  to  the  equation 

becomes  an  integral  of  the  first  kind,  belonging  to  the  equation 

the  equations  of  (birational)  transformation  being 

U  ,      $ 

The  integral  of  the  first  kind  can  be  transformed  slightly  by  writing 

z'I=JC; 

it  becomes 


^  (4K'-C-i)* 

where 


A  denoting  the  discriminant  of  the  original  quartic,  viz.  A  =  P  —  27J^.     Thus 

2/  dz  _  dC 

so  that  the  integral   of  the   first  kind   associated  with  the  original  equation  becomes 
a  constant  multiple  of  the  integral  of  the  first  kind  belonging  to  the  equation 

with  which  it  is  birationally  related. 

In  order  to  determine  all  the  equations  of  the  same  class  as  w"^  —  U=  0,  it  is  clear  that, 
in  each  case,  their  integral  of  the  first  kind  must  be  a  constant  multiple  of 

dc 


that  is,  the  constant  p  is  the  (sole)  class-modulus.     It  manifestly  is  the  single  absolute 
invariant  possessed  by  the  quartic  ;  and  so  we  infer  the  result  that  the  equations 

w'^=U{z,\\        w'2=F(/,  1), 

ivhere   U  and   V  are  quartic  fimctions  of  z  and  ^,  are  birationally  transformable  into 
one  another,  if  the  quartics  have  equal  absolute  invariants*. 

*  Hermite,   Crelle,  t.  lii,  (1856),  p.  8. 

35—2 


548  EQUATIONS   OF  [246. 

In  particular,  consider  the  equation 

if  it  is  to  belong  to  the  same  class  as  iv^=  U  {z,  1),  its  absolute  invariant  must  be  the  same. 
Now 

accordingly,  the  condition  is  that  c  satisfies  the  equation 

108c  (1-c)*   ~  A" 
As  a  A^ery  special  case,  we  infer  that  the  equations 

are  birationally  trans foi'mable  into  one  another  if 

(H-14a  +  a2)3_(l  +  14c  +  c2)3 
a{l-af  c(l-c)*      ■ 

The  actual  construction  of  the  transformations  is  left  as  an  exercise. 

Ex.  3.     Obtain  Riemann's  theorem  as  to  the  number  of  class-moduli  of  a  class  of 
algebraic  equations  from  a  relation  of  the  type 

V  =  aiUi+ +ap2<p  +  /3, 

which  (§  246)  connects  integrals  of  the  first  kind  associated  with  equations  that  are 
birationally  transformable  into  one  another.  (Riemann.) 

247.  We  proceed  to  the  consideration  of  some  properties  of  equations, 
which  are  of  genus  0  or  1 ;  these  having  been  reserved  for  separate  treat- 
ment. 

As  regards  equations /(w,  ^■)  =  0  of  genus  zero*,  the  fundamental  property 
is  that  each  of  the  variables  can  he  expressed  as  a  rational  function  of  a  single 
parameter :  when  /=  0  is  interpreted  as  a  curve,  it  is  said  to  be  unicursal. 
To  prove  this,  take  a  polynomial  U{tu,  z),  of  degree  m  in  w  and  z ;  and  make 
it  vanish  at  each  of  the  multiple  points  of /=0,  in  such  a  way  that  the 
point  is  of  multiplicity  A.  —  1  for  f7  =  0,  where  X  is  the  multiplicity  of  the 
point  for/=0.  Accordingly,  in  the  intersections  of /=  0,  U=0,  these 
multiple  points  count  for 

2X (X  -  1),  ={n-V) (n  -  2)  -  2pt, 

intersections :  that  is,  in  the  present  case,  they  count  for 

n  (n  -  3)  +  2 

intersections ;  and  therefore  the  remaining  number  of  points  common  to  the 

two  curves  is 

nm  —  [n  (?i  —  3)  -I-  2} 

=  n  (m  —  n  +  3)  —  2. 

*  For  a  full  discussion,  see  Clebsch,  Crelle,  t.  Ixiv,  (1865),  pp.  43—65. 
t  See  p.  524. 


247.]  GENUS   ZERO  549 

Hence  in  must  be  greater*  than  oi—S;  we  take  it  equal  to  n  —  2,  so  that  the 
number  of  other  points  is  n  —  2.  We  therefore  assign  n  —  S  arbitrary  points 
on  /=  0,  and  make  our  polynomial  vanish  at  each  of  them,  so  that  one  point 
is  still  left,  and  therefore  one  arbitrary  element  (it  can  only  be  a  constant)  is 
still  undetermined  in  U.  Take  two  particular  polynomials  of  degree  n  —  2 
satisfying  all  these  conditions,  and  let  them  be  U^^,  U^',  any  other  polynomial 
satisfying  the  conditions  is  of  the  form 

U^  +  fMlL, 

and  the  value  of  /x  will  be  determined  by  making  the  curve  pass  through  one 
other  point  on  /=  0.  Between  /=  0  and  Ui  +  /biUo  =  0,  eliininate  2;  the 
eliminant  is  rational  in  2u  and  /x.  The  roots  of  the  eliminant  are  the  values 
of  w  that  belong  to  the  multiple  points  of  /  in  their  proper  multiplicity,  the 
n  —  3  values  of  lu  that  belong  to  the  assigned  points,  and  one  other.  Removing 
all  the  factors  that  belong  to  the  multiple  points  and  the  assigned  points,  we 
then  have  a  linear  equation  which  is  rational  in  /x :  that  is,  iv  is  expressible 
as  a  rational  function  of  fx.  Similarly,  z  is  expressible  as  a  rational  function 
of  the  same  quantity  fi ;  and  the  proposition  therefore  is  established. 

Moreover,  Ave  have 


f^  =  -Tr 


that  is,  the  argument  //.  of  these  rational  functions  is  expressible  as  a  rational 
function  of  lu  and  2. 

The  degrees  of  the  rational  functions,  which  express  lu  and  z  in  terms  of  /x, 
are  not  greater  than  n.     Let  the  expressions  be 

-v/tj  (/x)  yJTo  ifx) 

i(j  =  — — ^ — 1  2  =      — - • 

4>i  (/^) '  (p2  in) ' 

and  denote  by  ^  (/x)  the  greatest  common  measure  of  cpi  (/x)  and  (^2  (/"•)•     Now 
any  straight  line,  say 

Aiu  +  Bz  +  C  =  0, 

cuts  the  curve  /=  0  in  n  points ;  and  therefore  the  equation 

must  give  n  values  of  /x,  one  for  each  point,  that  is,  it  must  be  of  degree  n. 
Hence  the  degrees  of 

cannot  be  greater  than  n. 

*  This  result  is  in  accordance  with  §  205.     For  U  is  an  adjoint  polynomial ;  the  number  of 
adjoint  polynomials,  which  are  of  degree  71  -  3,  isp,  viz.,  zero  in  the  present  case, 


550  EQUATIONS    OF   GENUS   ZERO  [247. 

Conversely,  if  the  variables  w  and  z  of  an  equation  f=0  are  rationally 
expressible  in  terms  of  an  arbitrary  parameter,  the  equation  is  of  genus  zero. 
For  if  the  genus  were  greater  than  zero,  an  integral  of  the  first  kind,  say 

dw 

would  exist  which  would  be  finite  everywhere  on  the  associated  Riemann's 
surface.  Substituting  for  w  and  z  their  values  in  terms  of  ft,  we  should  have 
the  integral 

R  (/ju)  d/ji 


(where  E  is  a  rational  function)  finite  for  all  values  of  /n — an  impossible 
result.     Hence  the  genus  of  the  equation  must  be  zero. 

Further,  any  curve  (or  equation)  of  genus  zero  can  be  birationally  trans- 
formed into  any  other  curve  {or  equation)  of  genus  zero  by  relations  which 
involve  three  arbitrary  parameters.     Let  one  of  the  equations  be  represented 

by 

w  =  Ri  ifM),      z  =  R2  ilj),     IJ^  =  R  {w,  z) ; 
and  the  other  by 

w'  =  &^{\),      /  =  ^2(^),      \  =  ^{w',z'); 

where  R-^,  R2,  S-^,  Sq,  R,  S  are  rational  fanctions.  In  a  birational  transforma- 
tion, one  set  of  values  of  w  and  z  determines  one  set  of  values  of  w'  and  z', 
and  vice  versa ;  therefore  one  value  of  fi  determines  one  of  A,,  and  vice  versa, 
so  that  the  relation  between  \  and  /i,  is  of  the  form 

_a\  +  b 
^~  c\  +  d' 

where  a,  b,  c,  d  are  arbitrary.  This  relation,  containing  the  three  arbitrary 
parameters  a  :  b  :  c  :  d,  gives  the  birational  transformation 

„  faS  +b\  r>  fci^S  +  b 


^cR  —  aJ'  ^\cR  —  aj 

establishing  the  proposition. 

It  is  an  immediate  corollary  that  any  curve  of  genus  zero  can  be  biration- 
ally transformed  into  itself  by  equations  that  contain  three  arbitrxiry  parameters ; 
thus  the  quantity  p  of  p.  545  is  3  when  p  =  i).  If  desired,  the  three  parameters 
can  be  determined  so  that  any  three  assigned  points  correspond  to  three  other 
assigned  points. 


247.]  SUB-RATIONAL   REPRESENTATION  551 

Note.  It  may  happen,  in  a  particular  instance,  that  the  actual  expressions 
for  w  and  z  in  terms  of  the  parameter  are  obtained  in  a  different  manner,  so 
that 

w  =  R,  (X),     z  =  Ro^  (X), 

but  that  \  is  not  a  rational  function  of  tu  and  z.  Thus  the  possibility  could 
arise  from  the  preceding  result  by  taking 

/jb  =  S  (A,), 

where  *S  is  a  rational  function  of  X  not  of  the  form  j :  we  then  should 

cX  +  a 

only  have  SCX)  equal  to  a  rational  function  of  z.     Such  a  representation, 

which   may  be  called  sub-rational,  is   easily  detected  in  fact,  because  the 

equation 

Aw  +  Bz+  C=0 

then  gives  more  than  n  values  of  A, ;  and  it  can  be  corrected  in  form  by  the 
suitable  inverse  substitution,  which  can  be  obtained  as  follows*. 

Suppose  that,  in  the  expressions 

0,  (X)  e.  (X) 

w  =  — --^ ,         z  =       ,   .  , 

where  the  quantities  6^,  0.,,  ^^i,  -^o  are  polynomials,  s  values  of  A,,  say  Xj,  X2,  . . .,  Xs, 
correspond  to  given  values  of  lu  and  z:  then  the  equations 

E,  (A)  =  d,  (X)  X,  M  -  0,  (AO  Xl  (^)  =  0, 

E,  (A)  =  6,  (A)  X.  M  -  e,  (A,)  X.  (^)  -  0, 

have  the  s  roots  A  =  Aj,  Aj,  ...,  Xg  common.  Also  each  of  these  roots  is  simple 
for  each  of  these  equations ;  because  if  any  one  were  multiple,  say,  Aj  for 
E^  (A)  =  0,  then  it  would  satisfy 

^/(A-)Xi(A0-^x(A0%/(A)  =  O, 
so  that 

^/(A)^^/(A) 

when  A  =  Aj.  The  quantity  Aj  would  then  satisfy  an  algebraic  equation 
the  coefficients  of  which  are  non-parametric  constants — a  result  obviously 
excluded  when  Ai  (and  so  the  other  values  of  Ao,  ...,  A^)  are  parametric.  We 
therefore  can  obtain  the  greatest  common  measure  of  E-^  (A)  and  E2  (A)  in  the 
form 

(A  -  Aj)  (A  -  As)  ...  (A  -  Xs) 

=  \' -  /zi A'~'  +  ^l. X'-' -  .... 

Now  not  all  the  quantities  /n  can  be  absolute  constants :  some  at  any  rate 
must  be  a  function  of  Aj,  say  /tx,.  is  such  a  function.     But  /i,.  is  a  symmetric 

*  Liiroth,  lilath.  Ann.,  t.  ix,  (1876),  pp.  163—165. 


552  EQUATIONS   OF  [247. 

function  of  Xj,  ...,  X^,  so  that  it  does  not  change  its  value  when  X,,  ...,  X^ 
are  substituted  for  Xi ;  hence  it  acquires  only  a  single  value  for  given  values 
of  w  and  z.  Moreover,  to  a  given  value  of  /li,-  correspond  s  values  of  X  ;  if  one 
of  these  be  Xi,  the  others  are  A,,  ...,  X,.,  because  ft  is  a  symmetric  function  of 
Xj,  ...,  Xg.  Hence  to  a  given  value  of  //.;.,  there  correspond  a  single  value 
of  w  and  a  single  value  of  z :  that  is,  when  the  equations 

_^i(X)  ^  O2  (X) 

are  transformed  by  the  relation 

/LL  =  Hr  (X), 
the  result  is  of  the  form 

"^1  (/^)  „      "^2  (fJ^) 


w  = 


<^l  (^)  '  <^2  (^)  ' 


z  =  -j-^ — -  ,         fjb  =  R  (iv,  z). 


Ex.  1.     The  equation* 

(2/2  +  Qxy  +  A'2)2  =  IQxij  {\yx  -  Zx  -Zy  +  4)2 

is  of  genus  0 ;  so  that  x  and  y  are  rationally  expressible  in  terms  of  a  variable  jDarameter. 
To  obtain  their  expressions,  we  notice  that  xy  must  be  a  perfect  square ;  so  that,  writing 

xy  =  6'^,         .r+y  =  ^, 
we  have 

and  therefore 

(^  +  6^)2  =  16^(1  +  ^)2. 

Hence  ^  is  a  perfect  square,  say  6=\^;  then 

xy  =  \^,         a;  +  3/  =  /x=4X— 6X24-4A^ 

Accordingly 

(y_a;)2  =  ;x2_4X4 

=  (4X  -  8X2  4. 4X3)  (4x  _  4X2  +  4X3) 
=  16X2(1-X)2(1-X  +  X2), 

P 

so  that  1  -X+X2  must  be  a  perfect  sqviare.     Take  A  =  7^,  so  that  P- - PQ  +  Q-  is  a  perfect 

square.      This  form  will   be  secured  for  P'-PQ+Q'\  ={P+aQ)  (P+co^Q),  where  «  is 
a  cube  root  of  unity,  by  writing 

P  +  $co  =  (a-«2)2,  P+(^co2  =  (a-«)2, 

SO  that  P=2a  +  a2,  Q=l+2a,  P^- - PQ  +  Q'-  =  {1 +  a  +  a-y-.  which  gives 

_     2a+_a2      l-a2      l+a  +  a2 
^~'^~     Y+2^  '  T+2a'      l+2a     ' 

Also 

^2a  +  a2     2  +  2a  +  a2  +  2a3  +  2a^ 

•^  '^         l+2a  (l+2a)- 

hence 

/'2+a\3  ,  2  +  a 

^  =  nrT2a)'  ^'  =  «l+2-a- 

*  It  is  one  form  of  the  modular  equatiou  in  the  cubic  transformation  in  elliptic  functions ; 
Cayley,  Coll.  Math.  Papers,  t.  ix,  p.  170. 


247.]  GENUS  ZERO        '  553 

It  is  clear  that  the  line 

gives  rise  to  six  values  of  a  ;  it  cuts  the  original  unicursal  sextic  in  six  points  ;  and 
therefore  the  expressions  are  rational,  not  merely  sub-rational.  To  express  a  in  terras 
of  X  and  y,  let 

_     y'^  +  Qxy  +  x^'      _ 

then 

{  l  +  '2a  j 
so  that 

as  may  be  verified  directly  by  substitutuig  the  values  of  x  and  y.     Also 

y_  r    2  +  a 


^-        ia(l+2a)j    ' 
SO  that 

U  X  . 

4y 

=  .r-l-2aA'-2a3, 

from  the  value  of  x ;   and  therefore 


Also 
so  that 
accordingly 
and  therefore 


^-iyr=2a(.r-l)  +  2a(l-a2). 

?7_a(l+2a) 
4j/~      2  +  a      ' 

(^-l).^.  =  2a(.^--l)  +  a(2  +  a)(l 
a2  +  2a ^  +  A-=0. 


From  the  expression  for  — ,   ^ve  have 


^«^+«"-|)-|-«- 


Subtract  this  equation  from  twice  the  preceding  quadratic  ;  and  we  have 


^^ . 


^    u      V     ^y)\         '2y 


4j/ 
which,  on  substituting  \'6xy  for  V^  when  it  occurs,  leads  to 

4.ry-4A-+  U{l-y) 
~~       dixy  —  2y  —  ^x—  U 


554  EQUATIONS   OF  [247. 

Ex.  2.     Shew  that  the  coordiDates  of  the  curve  in  the  preceding  example  can  be 
rationally  expressed  in  terms  of  an  arbitrary  parameter  jS,  by  taking 

/3*- 682  +  4/3 


4^3  -  Qli'i  + 1 

as  a  value  for  X  in  the  investigation.  Obtain  the  relation  between  a  and  /3 :  and  thence 
(or  otherwise)  shew  that  this  representation  is  sub-rational. 

Ex.  3.     Discuss  the  curve  represented  by  the  equations 

Ex.  4.  Obtain  relations  of  birational  transformation  which  transform  the  unicursal 
quartic 

^2y2  _  2_j.^  (^^ -p  ^  fiy-^  _j.  Qj_^2  _|_  2bxy  +  cy"^  =  0 

into  the  circle  x^+y'^  =  l.' 

248.  Some  of  the  simpler  properties  possessed  by  equations  (or  curves) 
of  genus  unity*  can  be  obtained  similarly.  In  the  first  place,  we  have 
Clebsch's  theorem  that  the  variables  can  he  expressed  as  rational  functions 
of  a  parameter  6,  and  of  @^  where  0  is  a  polynomial  of  either  the  third  or 
the  fourth  degree  in  6.  To  establish  this  result,  we  take  an  adjoint  polynomial 
U  of  order  n  —  linw  and  z,  where  /  is  of  order  n ;  we  make  it  vanish  at  each 
of  the  multiple  points  of/  to  the  multiplicity  A,  —  1,  when  X,  is  the  multiplicity 
of  the  multiple  point  of  /;  and  we  make  it  pass  through  n  —  2  arbitrarily 
assigned  points  on  /=  0.  Then  the  number  of  remaining  intersections  of 
U=0  and  /=  0  is 

n  (n  -  2)  -  (n  -2)-tX{\-  1). 
But  (§  240)  we  have 

'^\{\  -  1)  =  (n  -  1) (n  -  2)  -  2p 

=  n  (n  -  3), 

in  this  case ;  and  therefore  the  remaining  number  of  points  of  intersection  is 

n  (n  -  2)  -  (n  -  2)  -  n  (n  -  3),  =  2. 

Let  C/^i  =  0,  U2  =  0,  be  any  two  curves  satisfying  all  the  conditions  of  U,  as 
regards  its  order,  and  its  relations  to  the  multiple  points  of  /,  and  the  n  —  2 
arbitrarily  selected  points  on/;  then 

where  6  is  arbitrary,  is  another  such  curve.  It  cuts/=  0  in  two  points,  other 
than  the  multiple  points  and  the  assigned  ?i  —  2  points ;  hence,  eliminating  z 
between 

u,  +  eu,  =  o,  /=0, 

and  removing  from  the  eliminant  the  factors  that  correspond  to  the  multiple 
points  and  the  assigned  points,  the  remaining  factor  must  give  the  values  of  w 
for  the  two  points,  that  is,  it  is  a  quadratic  in  w.     Hence  we  have 

*  For  a  full  discussion,  see  Clebsch,  Crelle,  t.  Isiv,  (1865),  pp.  210 270. 


248.]  GENUS   UNITY  555 

where  A  and  B  are  rational  functions  of  6,  and  0  contains  no  repeated  factor. 
Similarly,  by  eliminating  w,  we  should  have  z=^  C  +  D^^'^,  where  C  and  D  are 
rational  functions  of  0,  and  ©i  contains  no  repeated  factor.  Substituting 
these  values  of  w  and  z  in  11^+  OUz^O,  /=0,  the  equations  are  to  be 
satisfied;  and  therefore  the  radicals  @^,  ©^^  are  the  same.     Thus  we  have 

Moreover 

B     ' 

the  first  represents  ^  as  a  one-valued  function  of  lu  and  z ;  the  second,  on 
substitution  of  this  value  for  0,  represents  ©^  as  a  one-valued  function  of  w 
and  2.     Hence,  writing 

e  =  z',     ©i  =  w', 
we  have  ■ 

w'-'=%{e)  =  %{z)\ 

results  which  shew  that  the  equations 

/=0,     iv'-  =  %{z), 

are  birationally  related  by  the  equations 

10  =  A  {z')  +  %v'B {z'\     z=C {z')  -f-  iv'D (/). 

Now  when  two  equations  are  birationally  related,  we  know  (§  246)  that  their 
genus  is  the  same  ;  hence  the  genus  of  w'^  =  ©  (/)  must  be  unity.  This  can 
be  the  case  only  if  ©  (z)  is  a  cubic  or  a  quartic  polynomial  in  z  ;  and  there- 
fore @  is  a  polynomial  of  either  the  third  or  the  fourth  degree  in  6.  The 
proposition  is  established. 

Such  curves  (or  equations)  are  called  bicursal  by  Cayley*;  they  also  are 
sometimes  called  elliptic,  because  the  equation  w''^  =  ©  (/)  is  associated  with 
elliptic  functions,  an  association  that  leads  to  another  mode  of  expression, 
as  follows.  If  ©  be  of  the  third  degree  in  d,  a  linear  transformation  of  the 
form  ^  =  a  +  h6  changes  ©  into 

If  ©  be  of  the  fourth  degree  in  6  and  k  be  one  of  its  roots,  a  transformation 

1  1  $2 

0  =  k  +  ^,     ©^  =  X2-' 

leads  to  an  expression  of  the  same  kind,  where  <l>  is  of  the  third  degree  and 
so  can  be  taken  (after  the  above)  as  4i6^  —  gocf)  —  gs-  Moreover,  both  of  these 
transformations  are  birational ;    and  neither    of  them    affects    the    general 

*  Coll.  Math.  Papers,  t.  viii,  p.  181. 


556  ELLIPTIC    FUNCTIONS   AND  [248. 

character  of  the  expressions  for  iv  and  z,  which  accordingly  can  be  taken  in 
the  form 

iv  =  A+  B^^,      z  =  G  +  JJ^^, 
where 

A,  B,G,  D  are  rational  functions  of  <^ ;  and  ^,  <l>2  are  rational  functions  of  w 
and  z.  Now  take  0  =  ^  (a),  where  a  is  a  new  parametric  quantity ;  then 
<I)2  =  ^y  (a),  and  so 

w  =  A  +  Bf',       z=C  +  I)f', 

where  A,  B,  G,  and  D  are  rational  functions  of  ^(a).  In  other  words,  the 
coordinates  are  expressible  as  uniform  doubly -periodic  functions  of  a  single 
parameter  a;  also  ^(a)  and  ^/(a)  are  rational  functions  of  the  coordinates. 

This  form  leads  to  interesting  applications  of  elliptic  functions  to  curves 
of  genus  unity,  in  particular,  to  plane  nocleless  cubics;  these  applications 
must  be  sought  in  treatises  on  elliptic  functions  and  treatises  on  geometry. 

Ex.  Shew  that  if  the  coordinates  of  a  point  on  a  curve  are  expressible  as  uniform 
doubly-periodic  functions  of  a  single  parameter,  the  genus  of  the  curve  cannot  be  greater 
than  unity.     Is  it  necessarily  equal  to  unity,  or  can  it  be  zero  ? 

As  regards  the  degrees  (in  the  parameter)  of  the  various  functions  that 
represent  the  coordinates,  there  is  a  difference  of  form,  according  as  the 
equation  is  of  even  or  of  odd  degree.  Let  R  denote  the  least  common 
multiple  of  the  denominartors  (if  any)  of  the  rational  functions  A,  B,  G,  D; 
and  let 

W  =  5=i ,        z  = 


R       '  R       ' 

where  P,  Q,  R,  S,  T  are  now  rational  polynomials  in  the  parameter  ^. 

If  the  curve  represented  be  of  odd  order  2m  +  1,  the  line 

mu  +  /3z  +  y  =  0 

must  cut  it  in  2m  +  1  points ;  so  that  the  equation 

(aP  +  /3S  +  ryRf  =  (aQ  +  ^Tf  (40^  -  g.,j>  -  g.^) 

must  give  2wi  +  1  values  of  </>.  Hence,  in  the  most  general  case,  the  degrees 
of  P,  S,  R  are  m,  and  the  degrees  of  Q,  T  are  m  —  1.  Thus  a  curve,  of  order 
2??i  +  1  and  genus  1,  is  represented  by 

_(<^,  i)^'^  +  ((^,  1)^-1  #        _((^,  1)"^ +  ((/),  ly^-^^i 

"^  (^TTr  '       ^~  (</^,  1)"^ 

where  <^  =  4<(ff  — g.^(j)—g^;  of  course,  in  particular  instances,  considerable 
simplifications  may  occur. 

If  the  curve  represented  be  of  even  order  2ni,  the  line 

atu  +  /3z  +  y  =  0 


248.]  EQUATIONS   OF   GENUS   UNITY  557 

must  cut  it  in  2m  points ;  hence  the  equation 

{aP  +  ^S  +  yRy  =  (aQ  +  /3Tf  (40^^  -  g.cf,  -  g.^) 

must  give  2m  values  of  cf).  Hence,  in  the  most  general  case,  the  degrees  of 
P,  S,  R  are  m,  and  the  degrees  of  Q,  T  are  m  — 2.  Thus  a  curve,  of  order 
2m  and  genus  1,  is  represented  by 

where  $>  =  40^  —  ^o(/)  —  ^s;  of  course,  in  particular  instances,  considerable 
simplifications  may  occur. 

Ex.  1.     The  sextic  equation 

(j/2  +  Qxy  +  .^2)2  =  1  Qxy  {xij  +  lf 
is  of  genus  1  ;  express  the  A^ariables  x  and  y  rationally  in  terms  of  a  parameter  0  and 

the  appropriate  $^.  (Cay ley.) 

* 

Ex.  2.     Likewise  express  in  that  form  the  variables  of  the  equations  ^ 

y'^  =  {x-ay{x-hf, 
y^={x-af{x-hf, 
y<i=z{x  —  af(x-hy, 
yi  =  {x-a)^x-bf, 
yi={x-af{x-hf, 
y^={x  —  a)'^  (x  —  b)-, 
y^={x  —  af{x-hy{x-  c)^, 
y*'  =  {x  —  of  {x  —  hf{x-  c)3, 
j/3  =  (^  _  a)2  {x  —  6)2  {x  —  c)2, 
respectively :    all  being  of  genus  unity. 

Ex.  3.     Shew  that  the  quartic  equation 

a  (x^  +  ?/2)2  +  i)xy  +  cx{\+y^)  +  dy{\-\- x"^)  =  0 
is  of  genus  unity ;  and  express  its  variables  algebraically  in  terms  of  a  single  parameter. 

Note.  It  may  happen  (as  for  unicursal  equations)  that  expressions  of  the  variables 
in  a  bicursal  equation  have  been  obtained,  which  are  of  the  proper  type  but  are  of  too 
high  degree  :  so  that,  in  particular,  (^  and  ^^  are  no  longer  one-valued  rational  functions 
of  iv  and  z. 

The  representation  of  the  variables  can  be  modified,  so  that  the  new  form  shall  satisfy 
the  conditions  as  to  degree.  A  method  of  modification  is  given  in  the  memoirs  by  Clebsch 
and  by  Cayley,  which  have  already  been  quoted. 

As  regards  birational  transformation  of  equations  of  genus  unity  into 
one  another,  we  infer,  from  Ex.  1,  §  246,  and  from  the  fact  that  such  an 
equation  is  birationally  transformable  into  %v'^  =  @  (z),  that  such  equations 
are  characterised  by  the  possession  of  one  invariant  modulus.  Considering 
then  the  class  of  equations  of  genus  unity  that  is  determined  by  a  modulus, 
we  investigate  the  number  of  birational  transformations  of  one  curve  into 
another  and,  in  particular,  of  one  curve  into  itself 


558  BIRATIONAL   TRANSFORMATIONS    OF  [248. 

We  have  already  seen  that  birational  transformations  exist  between  any 
curve  of  genus  unity  and  the  curve 

where  gigz""^  is  a  measure  of  the  invariant  modulus.  In  the  first  place, 
there  is  an  infinitude  of  birational  transformations  of  this  curve  into  itself 
or,  otherwise  stated,  there  is  a  birational  transformation  of  this  curve  into 
itself  containing  an  arbitrary  parameter.  The  result  can  be  seen  intuitively 
from  the  properties  of  the  plane  nodeless  cubic.  We  take  any  point  on  it, 
say  A,  depending  upon  an  arbitrary  parameter  a,  and  through  A  draw  any 
straight  line  which  will  cut  the  cubic  in  two  other  points,  say  P  and  Q. 
Then  P  and  Q  uniquely  determine  each  other,  that  is,  they  are  birationally 
related ;  the  analytical  expression  of  the  relation  contains  a,  which  is  an 
arbitrary  parameter. 

The  analytical  expressions  can  be  obtained  simply  as  follows.     Any  point  on  the  cubic 
curve  is  given  by  z  =  <p  {a\  w  =  p' (a),  where  a  is  arbitrary,  say  by  p,  p' ;  and  any  line 

through  it  is  given  by 

W-p'  =  m{Z-p). 

Where  it  cuts  the  cubic,  we  have 

4Z^-g2Z-g3  =  {p'  +  m{Z-p)Y 

=  4p^-g2P-93  +  ^^np'  {Z-p)  +  m^Z-p)l 

One  root  is,  of  course,  Z=  p  ;  let  the  other  two  be  z,  s',  so  that  these  are  roots  of  the 
quadratic 


Hence 

and  therefore 

,  which  leads  to 

Also 

which  leads  to 


4:Z-'  +  Z{4:p-m^)  +  m'~p-2mp'  +  4p^-g2  =  0. 
z  +  z'  =  i«i2  -  p,         4zz'=m^p  -  2mp'  +  4p^  - g.^  ■ 

,     ^z^p+z{Ap^-g2)-2wp'-g^p-ig^ 
4.{z-py 

w-p'       z-p' 

=9+i'^"-9) w^w 


The  expressions  for  z'  and  w'  constitute  a  birational  transformation  of  the  given  equation 
into  itself,  the  transformation  involving  an  arbitrary  quantity  a  through  the  functions 
P{a),   P'{a). 

Simple  forms  can  be  obtained  by  assigning  particular  values  to  a.      Thus  writing 
a  =  co",  so  that  P  =  ei,  ^'  =  0,  the  transformation  is  easily  reduced  to 

,       6,2 +  ^2''^  + ^3^1  ,  2^2^  +  6361 

2'  =  ^ ^ 5_i  w  =w -^; 

and  so  for  other  cases. 

Ex.  1.     Indicate  the  relation  of  these  results  to  the  formulae  of  the  addition-theorem 
for  Weierstrass's  elliptic  functions. 


248.]  EQUATIONS   OF   GENUS   UNITY  559 

Ex.  2.     Obtain  birational  transformations  of  the  equations 
(i)     tv^  +  z^  =  l, 
(ii)     w^==(l-z^)(l-cz^), 
suoli  as  to  contain  an  arbitrary  quantity  and  to  transform  the  equations,  each  into  itself. 

Fx.   3.     Prove  that  birational   transformations   of  a   plane   cubic   into   itself  exist 
depending  algebraically  upon  an  arbitrary  parameter  6  in  the  form 

iu'  =  R^  (w,  z,  6),         z'^R^  {iv,  z,  e), 
such  that,  for  a  particular  value  of  6,  the  transformation  becomes  w'  =  iv,  z'  =  z. 

(A]3pell.) 

Returning  now  to   the   consideration   of  two   equations   of  genus   unity 
with  a  common  invariant  modulus,  let  them  have  the  form 

The  former  can  be  birationally  transformed  into  the  curve 

w^  =  4^=5  -  g^z  -  g.,, 
with  the  common  invariant  modulus,  by  relations  of  the  form 
Wi  =  A  {z)  +  wB  (z),         Zi  =  C  (z)  +  ivD  {z)\ 
z  =  Ri  (if  1 ,  Zi),  IV  =  R^  {iVi ,  Zj) 

The  latter  can  be  birationally  transformed  into  the  curve 

W^  =  4.Z-'-g,Z-g,, 
also  with  the  common  invariant  modulus,  by  relations 

w^  =  E(Z)+  WF  (Z),        z^  =  G  (Z)  +WH{Z)\ 
Z  =  8^  (w^,  Z.2),  W  =  S.2  {w.2,  z^) 

Now  the  curves 

w^  =  ^z'-g,z-g„  W'  =  ^Z^^-g,Z-g„ 

(being  one  and  the  same),  can  be  birationally  transformed  into  one  another 
by  relations  that  involve  an  arbitrary  parameter,  say  in  the  form 

w  =  P^{W,  Z,  a),         z  =  P,{^^,  Z,  a)\ 


F  =  Qi  {to,  z,  a),  Z^Q._  (w,  z,  a) 
Then  the  relations 

w^  =^A{z)  +  wB  {z),  z^  =  G  {z)  +  luD  {z)'\ 

iu  =  P,{W,Z,a\  z  =  P,(W,Z,a)    > 

Z=S,(w„z,),  W  =  S,{w„z,)        J 

express  Wj  and  z^  uniquely  in  terms  of  w<2,  z^_,  and  a.      Also  the  relations 

w^  =  E  (Z)  +  WF  (Z),  z,  =  G  (Z)  +  WH  (Z)) 

W=Qj(w,z,a),  Z=Q^{w,  z,a)          \ 

w  =  R2  (w^ ,  z-,),  z=^R,  (wi ,  ^1)            ' 


560  TRANSFORMATION   INTO   ITSELF   OF  [248. 

express  Wa  and  z^  uniquely  in  terms  of  w^,  z^,  and  a.  The  relations  therefore 
express  a  birational  transformation,  and  they  contain  an  arbitrary  constant ; 
hence  we  have  the  theorem:  An  equation,  of  genus  unity,  is  birationally 
related  to  any  other  equation,  of  genus  unity  and  the  same  invariant  modulus, 
by  equations  which  involve  an  arbitrary  constant. 

Also  we  infer,  as  an  immediate  corollary,  that  any  equation  of  genus 
unity  admits  an  infinitude  of  birational  transformations  into  itself;  the 
equations  of  transformation  involve  an  arbitrary  parameter  algebraically. 
Hence,  when  j>  =  l,  the  number  p  of  p.  545  is  unity. 

We  have  seen  that,  in  a  birational  transformation,  an  integral  of  the  first 
kind  belonging  to  one  equation  is  transformed  into  an  integral  of  the 
first  kind  belonging  to  the  other.  When  the  genus  is  unity,  each  equation 
possesses  'only  a  single  integral  of  the  first  kind ;  and  denoting  it  by  v 
for  the  equation  f{w,  z)  =  0,  and  by  u  for  the  transformed  equation 
ty'2  =  4s'^  —  g<2.z'  —  gz,  we  have  (§  246) 

V  =  au  +  b, 

where  a  and  b  are  constants.  But  u  is  the  argument  of  the  doubly-periodic 
functions  in  terms  of  which  w'  and  z  are  expressed.  Hence  v  is  effectively 
that  argument ;  in  other  words,  the  integral  of  the  first  kind  associated  with 
an  equation  of  genus  unity  is  effectively  the  argument  of  the  doubly-periodic 
functions  in  terms  of  which  the  variables  are  expressible. 

lu  connection  with  the  preceding  discussion,  reference  may  be  made  to  Picard*  who 
uses  the  results  to  prove,  among  other  theorems,  that  when  a  differential  equation 
/(w,  i(;')  =  0  has  integrals  w  which  are  uniform  functions  of  z,  these  integrals  are  either 
(i)  doubly-periodic  functions  of  z  ;  or  (ii)  rational  functions  of  e^^,  where  ^  is  a  constant, 
that  is,  are  simjaly-periodic  functions  of  z ;  or  (iii)  rational  functions  of  z.  (See  also,  on 
this  matter,  the  Note  appended  supra  at  the  end  of  the  foregoing  chapter  x.) 

Consider  also,  in  this  connection,  the  birational  transformations  of  the 
equation  f{w,  ^)  =  0  into  itself,  say  into  f{w',  z)  =  0.  After  the  preceding 
result,  we  must  have  some  relation 

V  (w',  z')  =  av  (w,  z)  +  b, 

where  a  and  b  are  constants.  The  integral  of  the  first  kind  connected 
with  an  equation  of  genus  unity  possesses  two. distinct  periods;  let  them 
be  denoted  by  co^,  Wo.  It  is  clear  that  when  v  (iv,  z)  increases  by  a  period, 
then  V  {iv,  z)  also  changes  by  some  period :  and  likewise  for  v  (w,  z), 
when  V  (w',  z')  increases  by  a  period.     Hence  we  have 

\cl>i  +  ficoo  =  acoi,  X'coi  +  X  coo—  awo, 

from  the  first  of  these  results,  X,  fx,  X' ,  jju    being  integers :   and 

&)]  =  a  (pcoi  -t  ao)..),  cOo  =  a  (p'coi  +  a'o).,), 

*  Traite  cVAnahjse,  t.  iii,  (190S),  ch.  iv. 


248.]  AN   EQUATION   OF   GENUS   UNITY  561 

from  the  second,  p,  a,  p',  a   being  integers.     Denoting  Xpf  —  X'fi  by  A  and 
pa'  —  pa  by  A'  the  first  two  equations  give* 

Aa)i  =  a  (fx'coi  —  P'dOo),  ^Wo  —  a  (~  X'coi  +  XcWg). 

When  these  are  compared  with  the  second  two  equations,  we  have 

Lb  u  —  XX,, 

^  &)j  —  ^  0)2  =  pcoi  +  aco2,  —r-  a>j  +  ^co.2  =  p  aii  +  a  0)2- 

Now  the  ratio  o)^ :  w^  is  not  entirely  real  (§  231) ;  hence 

u.  —  iJ'  —  X        ,  \        , 

A  =  P'        -^^""^        ^'  =  P'        A^""' 

and  therefore 

AA'=1. 

Now  A  and  A'  are  integers  :  hence  each  of  them  is  1  or  is  —  1,  that  is, 

A  =  ±  1. 

Also,  let  O,  denote  the  ratio  co.,:  (o^,  so  that  we  have 

fi'n  +  x'  ^^^ 

fiCl  +  X 

that  is, 

/xO-  -f  (\  -fM')n-\'  =  0, 

a  relation  which  either  is  an  identity  or  is  an  equation  satisfied  by  H. 

If  the  relation  is  an  identity,  then 

/x  =  0,         X^  —  0,         X  =  p,'. 

Since  Xpf  —  X'p.  =  ±  1,  we  have  X  =  p  =  ±1  (or  +  i,  but  these  values  are  to  be 
excluded  because  X  and  p!  are  integers) ;  also  a  =X,  a=  pf,  that  is, 

a=  ±1. 
Hence  we  have 

V  {iv\  z')=  ±v  (w,  z)  +  h. 

If  the  relation  is  an  equation,  then  the  value  of  ft  may  not  be  entirely 

real ;  consequently 

(\  -  /)2  +  4  V/x  <  0, 
that  is, 

{X  +  p:y  <  4A, 

so  that  A,  which  is  either  +  1  or  —  1,  must  now  be  -f  1 ;  and  the  possible 
values  ofX  +  p,'  are  0,  1,  —  1.     Also 

^     p,'-X+{{X  +  p.y-4^/\'l^ 

2p,  '  ' 

*  It  is  assumed  that  A  is  not  zero.     If  A  were  zero,  so  that  \'  =  kX,  /x'^k/j,,  where  k  is  real, 
we  should  have  (on  dividing  one  equation  by  the  other] 

Xui  +  fj-ca-i       wi' 
which  would  make  the  ratio  of  the  periods  real ;  this  (§  231)  is  impossible. 

F.  P.  36 


562  EQUATIONS  OF  [248. 

which  is  a  non-parametric  constant ;  so  that  this  case  cannot  occur  if  the 
invariant  modulus  of  the  equation  (which  is  a  transcendental  function  of  II) 
is  parametric.  Even  in  those  cases  where  the  invariant  modulus  has  an 
appropriate  constant  value,  we  have,  on  eliminating  Wj  and  Wo  between 

\o)i  + /ji(02  =  cc(Oi ,  \'aii  +  /x'a).2=  C('(i>2, 

an  equation 

(k  —  a)  {[x  —  a)  —  X'yLt  =  0, 
that  is, 

a^  -a{X  +  fx)+l=  0. 

The  possible  values  of  X  +  /u,'  being  0,  1,  —1,  the  corresponding  values  of 
a  are  given  by 

a^  +  1  =  0,         a^  -  a  +  1  =  0,         a^  +  a  +  1  =  0, 
respectively. 

In  every  instance,  the  constant  a  is  determinate :  and  all  the  conditions 
are  satisfied  when  the  constant  h  is  left  arbitrary.  Moreover  the  relations 
have  arisen  in  connection  with  the  birational  transformation  of  the  curve 
into  itself;  and  we  therefore  infer  the  theorem  that  the  birational  trans- 
formations of  a  curve  of  genus  unity  into  itself  can  be  represented  by 

V  (w',  z')=  ±v  (w,  z)  +  h, 

where  h  is  an  arbitrary  constant :  they  obviously  constitute  the  simple 
infinitude  (p.  560)  of  birational  transformations.  The  cases,  which  corre- 
spond to  a^  -I- 1  =  0,  a^  —  a  -1-  1  =  0,  a^  -F  a  +  1  =  0,  are  each  of  them  extremely 
special. 

249.  As  regards  equations  (or  curves)  of  genus  two,  the  method  adopted 
for  equations  of  genus  zero  or  unity  can  be  applied,  if  we  begin  with  adjoint 
polynomials  of  order  n  —  3  instead  of  with  those  of  order  n  —  2. 

It  is  known  that,  for  equations  of  genus  two,  there  are  two  distinct 
integrals  of  the  first  kind  :  and  each  of  them  determines  an  adjoint  polynomial 
of  order  n  —  3.  Let  these  be  denoted  by  U-^  {w,  z)  and  U^  {w,  z) ;  then  as  each 
of  them  vanishes  to  multiplicity  X,  —  1  at  a  multiple  point  of  /  which  is  of 
multiplicity  A,,  the  polynomial 

^  •  u.  +  eu, 

also  vanishes  to  that  multiplicity,  so  that,  among  the  intersections  ofy"=0 
and  U^  +  6U2  =  0,  such  a  multiple  point  counts  for  \(X  —  1)  intersections. 
Hence  the  number  of  intersections  of/=  0  and  Ui  +  6LL=  0,  other  than  the 
multiple  points  of/,  is 

=  n(n-S)-tX{X-l) 

=  7i{n-:^)-{(n-l)(n-2)-4>] 

in  the  present  case,  that  is,  the  number  is  2.  Eliminate  z  between  the  two 
equations,  and  remove  from  the  resulting  equation  in  w  the  factors  which 


249.]  GENUS  TWO  563 

correspond  to  the  multiple  points  off=  0 ;  the  result  is  a  quadratic  in  w,  the 
coefficients  of  which  are  rational  functions  of  0,  and  the  roots  of  which  are 
the  values  of  w  for  the  remaining  two  points  of  intersection.     They  are 

tv  =  A  +  B(&K 

where  A,  B,  @  are  rational  functions,  @  having  no  repeated  factor.  Proceed- 
ing similarly,  we  find 

z  =  C  +  D@'K 
where  C,  D,  ©'  are  rational  functions ;  and  substituting  in  the  equation  /=  0, 
we  have  ©^  =  ©'- ,  or  the  variables  can  be  represented  in  the  form 

w  =  A+B@i,     2=C+DbK 
where  A,  B,  C,  D,  S  are  rational  functions  of  the  parameter  0.     Moreover 

„  f/i  (w,  z) 


©2  = 


U.2  (.w,  z) ' 
w-  A 


the  first  of  these  expresses  ^  as  a  rational  function  of  w  and  z  ;  the  second,  on 
the  substitution  of  this  expression  for  6,  expresses  also  ©^  ^g  ^  rational 
function  of  tu  and  z.  There  is  therefore  a  birational  transformation  between 
the  curves 

f(w,  z)  =  0,     w'2  =  0  (/) ; 

and  the  curves  are  therefore  of  the  same  genus,  viz.  2,  the  genus  of/.  Hence 
%{z')  is  of  degree  either  five  or  six  (Ex.  2,  §  1'78);  and  therefore  @,  as  a 
polynomial  in  6,  is  of  degree  either  five  or  six.  It  therefore  appears  that 
the  variables  in  an  equation  of  genus  2  are  expressible  as  rational  functions 
of  6  and  of  the  radical  ©-,  ivhere  ©  is  a  polynoinial  in  6  of  the  fifth  or 
the  sixth  degree. 

When  0  is  a  sextic  which  (as  has  been  seen)  has  no  repeated  factor,  it  is 
of  the  form 

{e-a){e-b){9-c){e-d){e-e){e-f). 

Take 

(a  —  b){a  —  c) 


'-a  = 


©2  = 


a  —  c  +  (6  —  c)  </) ' 

1 


a—  c       , 


which  determines  a  birational  transformation ;  then  4>  is  a  constant  multiple 
of  ^(1  —  0)(1  —  K(^)  (1  —  X(jb)  (1  —  fx^)  and,  in  the  circumstances,  can  be 
taken  as  equal  to  this  quantity. 

36—2 


564  EQUATIONS   OF   GENUS   TWO  [249. 

When  @  is  a  quintic  which  (as  has  been  seen)  has  no  repeated  factor, 

it  is  of  the  form 

(d  -a)(0-  h)  {6  -c)(e-  d)  (0  -  e). 
Take 

0  —  a  =  —  (a  —  h)(p, 

which  determines  a  birational  transformation :  then,  as  in  the  preceding  case, 
^  can  be  taken  as  equal  to 

cj,(l-cl>){l-Kcf>)(l-Xcj>)il-fi4>). 

The  representation  of  the  coordinates  in  either  case  becomes 

w  =  E+F(^i,     z^G  +  H^^, 

where  E,  F,  G,  H  are  rational  functions  of  j>,  and 

<!>  =  ,/>  (1  -  c^)  (1  -  «(^)  (1  -  X(^)  (1  - /X(^). 

To  determine  the  degrees  of  these  rational  functions,  let  T  be  the  least 
common  multiple  of  their  denominators  (if  any),  so  that,  say, 

The  line 

aw  -f  /S^'  +  7  =  0 

must  cut  the  curve  in  a  number  of  points  equal  to  its  order,  and  therefore 

the  equation 

{aF  +  /3R  +  ryTf  =  {aQ  +  /SSy  ^ 

must  determine  the  same  number  of  values  of  </>. 

If  the  equation  (or  curve)  /  =  0  be  of  odd  order  2m  +  1,  then  P,  R,  T  may 
be  of  degree  m,  and  Q,  8  may  be  of  degree  m  —  2  ;  so  that  the  representation 
is 

_  ((^,  1)"^  +  {4>,  l)'"-^  O*  ^  (0,  l)"^  +  ((^,  \)rr>-^^h 

If  the  equation  (or  curve)  be  of  even  order  2in,  then  P,  R,  2''  may  be  of 
degree  m,  and  Q,  S  may  be  of  degree  m  —  3 ;  so  that  the  representation  is 

_  (ch,  ly'  +  jcf),  l)"^-''^          ^  ((f>,  iy"  +  (4>,  l)'>^-='# 

'^~       {<t>,ir        '  '         (<^>i)" 

Note  1.     The  three  constants  «:,  A,,  yu,  in 

determine  the  three  class-moduli,  which  (p.  545)  every  equation  of  genus  2 
conserves  as  invariants  under  birational  transformation. 

Note  2.     It  was  proved  (Ex.  1,  §  246)  that  the  equations 


249.]  HYPERELLIPTIC   CURVES  565 

where  u^,  Ui  are  a  quadratic  and  a  quartic,  are  birationally  transformable  into 
one  another,  when  they  have  their  class-moduli  the  same.  Similarly,  the 
equations 

are  birationally  transformable  into  one  another  with  similar  limitations : 
where,  in  each  case,  neither  11.2  nor  Ui  has  a  repeated  factor.     Now 

is  geometrically  interpretable  as  a  quartic  curve :  owing  to  the  forms  of  u^ 
and  u^,  the  curve  has  only  one  double  point*,  as  ought  to  be  the  case, 
because  its  genus  is  2.  Hence  any  plane  curve  of  genus  2  is  birationally 
transformable  into  a  plane  quartic  having  only  one  double  point. 

Note  3.  It  might  be  imagined  that  a  similar  result  would  hold  for  jj  =  3 
and  for  p  >  3  :  but  this  is  not  the  case. 

In  the  first  place,  the  argument  would  not  apply.  It  is  true  that  there 
are  p  adjoint  polynomials  of  order  ?i  — 3,  so  that 

would  be  the  general  equation  of  a  curve  of  order  n  —  3,  vanishing  to  the 
proper  order  at  the  multiple  points  of  /.  But  the  remaining  number  of 
points  of  intersection  is  2p  —  2 ;  and  we  should  then  (if  the  earlier  process 
be  adopted)  have  an  equation  of  degree  '2p  —  2  to  solve,  its  coefficients  being 
rational  functions  of  p  —  1  parameters. 

In  the  second  place,  if  a  curve  of  genus  3  be  birationally  related  to 

lU-  =  /  (1  -  Z)  (1  -  K^Z')  (1  -  K^Z')  (1  -  K,Z')  (1  -  K,Z')  (1  -  K,z), 

it  manifestly  can  have  only  5  invariant  moduli,  to  determine  the  five 
quantities  k.  As  a  curve  of  genus  3  in  general  has  6  (=3.3  —  3)  such 
moduli,  it  follows  that  the  preceding  curve  is  not  general.  This  argument 
applies,  a  fortiori,  when  the  genus  of  a  curve  is  greater  than  3. 

There  are  curves  of  genus  p,  which  are  birationally  related  to 

w-  =  / (1  —  /)  (1  —  K^z) ...  (1  —  K2p--i,z')  ; 

but  they  are  not  general,  for  they  have  only  2p  —  1  moduli  instead  of 
3j9  —  3.     Such  curves  are  often  called  hyperelliptic. 

It  thus  appears  that,  so  far  as  concerns  the  representation  of  the  variables 
of  an  equation  in  a  form  that  is  birational,  there  is  a  fundamental  distinction 
between  the  cases  p  <  3,  p^  3.  It  is  also  found  (though  this  is  beyond  the 
range  of  these  present  investigations)  that  there  is  a  fundamental  distinction 
between  the  properties  of  functions  associated  with  an  equation  of  genus  less 
than  three  and  those  associated  with  an  equation  of  genus  equal  to,  or  greater 
than,  three. 

*  The  double  point  is  at  infinity. 


566  BIRATIONAL   TRANSFORMATION   OF   EQUATIONS  [250. 

250.  We  have  seen  that,  in  the  case  of  an  equation  of  genus  zero,  there 
is  a  triple  infinitude  of  birational  transformations  of  the  equation  into  itself 
(§  247);  and  that,  in  the  case  of  an  equation  of  genus  unity,  there  is  a  single 
infinitude  of  similar  transformations  (§  248):  the  infinitude,  in  each  case, 
arising  through  the  existence  of  arbitrary  constants  in  the  relations  of 
transformation.  For  equations  of  genus  greater  than  unity,  we  have  the 
theorem  that  the  7iumher  of  birational  trarisforniations  of  an  equation  of  genus 
greater  than  unity  into  itself  is  limited.  Schwarz*  first  proved  that  such  a 
birational  transformation  cannot  exist  involving  an  arbitrary  parameter,  so 
that  there  cannot  be  a  continuous  infinitude  of  such  transformations ;  Klein -j- 
first  stated  that  there  could  not  be  a  discrete  infinitude  of  such  transforma- 
tions, that  is,  not  an  infinitude  of  particular  transformations ;  and  Hurwitz]: 
obtained  84  (2>  -  1)  as  the  upper  limit  of  the  number.  The  first  two  of  these 
results  can  be  established  by  the  following  argument,  due  to  Picard§ :  for  the 
third,  reference  can  be  made  to  Hurwitz's  memoir. 

It  was  proved  that,  when  a  birational  transformation  is  effected  upon  an 
equation  of  genus  p,  so  that  it  gives  another  equation  also  of  genus  p,  the 
integrals  of  the  first  order  associated  with  one  equation  are  linearly  express- 
ible in  terms  of  those  associated  with  the  other.  Let  u-^^,  ...,  Up  denote  the 
normal  elementary  integrals  of  the  first  order  associated  with  f{w,  z)  =  Q; 
and  let  u^,  ...,Up'  denote  those  associated  with  f{w'.z')-=0,  a  birational 
transformation  of  /  into  itself.     Then  we  have  p  equations  of  the  form 

where  the  quantities  h  and  h  are  constants  (§  246). 

The  constants  h  depend  only  upon  the  periods.  For  let  the  point  w,  z 
move  on  the  Riemann  surface  from  one  edge  of  the  cross-cut  »«  to  the  same 
point  on  the  opposite  edge  :  then  the  point  w' ,  z'  describes  a  closed  irreducible 
cycle  on  its  surface.  Let  HV.s  denote  the  period  for  w/,  which  is  a  combina- 
tion of  the  periods  of  the  normal  integrals  with  integers  for  coefficients :  we 
therefore  (|  235)  have 

for  all  values  r,  s  ==  1,  2,  . . . ,  |). 

Let  U:i_(w,  z),  U^iw,  z),  ...,  Up{w,  z)  denote  the  adjoint  polynomials  of 
order  n  —  3  arising  through  the  normal  integrals :  then  differentiating  the 
above  relation,  we  have 

duy  =  kridui  +  kroduo  +  ...  +  krpdup , 

*  Grelle,  t.  Ixxxvii,  (1879),  pp.  139—145. 

t  In  a  letter  (dated  1882)  to  Poincare,  quoted  Acta  Math.,  t.  vii,  (1885),  p.  10. 
+  Math.  Ann.,  t.  xli,  (1893),  p.  424;  see  also  a  memoir  by  him,  Math.  Ann.,  t.  xxxii,  (1888), 
pp.  290—308. 

§  Cows  d'Analyse,  t.  ii,  p.  480. 


250.]  OF   GENUS   GREATER   THAN    UNITY  567 

that  IS, 

Ur  (tv',  z') -^  =  \kn U-,  (w,  z)  +  k,.., U^ (w,  z)+  ...+krpUp (w,  z)]  ^. . 
dw'  dtv 

This  holds  for  all  the  values  1,  2,  ...,  p,  and  (by  hypothesis)  j?  >  1.     Taking 
the  relation  for  r=l,  2,  and  dividing,  we  find 

C/g  (w',  z')  _  k^i  f/i  (w,  z)  +  k22  U<2{w,  z)+  ...  +  kip  Up  (w,  z) 
Uj  (w',  z')      kn Ui (w,  z)  +  k-^o, Uoiw,  z)+  ...  +  k^p Up (w,  z) ' 

which,  in  connection  with 

f{w\z)  =  0,     f\w,z)  =  0, 

serves  to  define  the  birational  transformation  of  /  into  itself. 

As  "the  constants  k  depend  only  upon  the  moduli  at  the  cross-cuts  of  the 
Rieraann's  surface,  so  that  they  are  pure  constants  and  are  not  parametric,  it 
follows  that  no  arbitrary  constant  can  occur  in  the  equations  of  the  birational 
transformation.     This  is  Schwarz's  result. 

Any  birational  transformation  of  /'=  0  into  itself  leads  to  a  relation  (or  to 
several  relations)  between  the  adjoint  polynomials  of  the  foregoing  type  ;  and 
such  a  relation  may  be  regarded  as  the  initial  form  of  the  birational  trans- 
formation. In  order  that  the  relations  may  exist,  the  constants  k  must 
satisfy  equations  which  clearly  ax-e  algebraical  in  form.  If  these  equations 
do  not  determine  the  constants  k,  then  one  or  more  of  them  would  be 
arbitrary ;  and  then  the  birational  transformation  would  involve  an  arbitrary 
constant,  contrary  to  Schwarz's  result.  The  constants  k  must  then  be 
determinate,  manifestly  by  a  finite  number  of  algebraical  equations.  Hence 
there  is  a  limited  number  of  solutions ;  accordingly,  as  each  solution  deter- 
mines a  birational  transformation,  there  is  only  a  limited  number  of  birational 
transformations,  distinct  from  one  another.     This  is  Klein's  result. 

The  preceding  argument  is  valid,  only  if  p  ^  2.  The  results  are  known 
not  to  hold  when p  <2. 

For  various  properties  (such  as  the  periodicity  for  repeated  application)  of  the  birational 
transformations  of  an  equation  of  genus  greater  than  unity  into  itself,  see  Hurwitz's 
memoirs  quoted  on  p.  566;   also  Baker's  Abelian  Fimctions,  ch.  xxi. 

251.  The  assignment  by  Riemann  (§  247)  of  a  class  of  equations,  as 
constituted  by  those  of  the  same  genus  birationally  transformable  into 
one  another,  suggests  the  desirability  of  reserving  some  form  of  equation 
of  that  genus  as  a  normal  form  (or  normal  curve).  When  p  =  0,  a  normal 
form  is  superfluous;  when  jo  =  1,  the  normal  form  can  clearly  be  taken  to  be 
the  nodeless  cubic  (§  249) ;  when  p  =  '2,  the  normal  form  can  clearly  be  taken 
to  be  the  uninodal  quartic  (§  250) ;  and  we  therefore  are  concerned  with  the 
cases,  where  p  is  equal  to  3  or  is  greater  than  3, 


568  NORMAL   FORM   OF   EQUATIONS  [251. 

Denoting  the  p  functions,  which  arise  as  the  derivatives  of  the  p  normal 
elementary  integrals  of  the  first  kind,  by  c^j,  ...,  </)^,  and  the  respective 
adjoint  polynomials  of  order  n-S  by  U^,  ...,  Up,  consider  any  linear 
combination 

r=l 

where  the  coefficients  c  are  arbitrary.  Choose  p-S  arbitrary  places  on 
the  surface,  independent  of  one  another,  and  make  the  preceding  combined 
polynomial  vanish  at  each  of  them  :  then  there  are  ^  —  3  relations  established 
among  the  coefficients  c,  and  the  curve 

p 

1   CrUr=0 
r  —  1 

becomes  effectively 

Q,  +  ciQ,  +  ^Qs  =  0, 

where  a  and  /3  are  arbitrary,  and  Q^,  Qo,  Qs  are  definite  adjoint  polynomials 
of  order  n  —  3,  each  of  which  vanishes  at  the  p  —  3  arbitrarily  assigned 
positions  on  the  surface.     Now  let 

,  ^  Qg  Jw,  z)  ,  ^  Qs  (w,  z) 

Qi  {ff^,  z) '  Qi  {w,  z) ' 

These  equations  determine  a  rational  transformation  of  the  surface ; 
to  every  point  on  the  old  surface  corresponds  only  a  single  point  of  the  new 
surface.  In  order  that  the  transformation  may  be  birational,  then  to  any 
point  on  'the  new  surface  must  correspond  only  a  single  point  on  the  old. 
If  this  is  not  the  case,  choose  an  arbitrary  point  on  the  new  surface ;  and 
let  at  least  two  points  on  the  old,  say  w^,  z^\  w^,  z^\  correspond  to  it.     Then 


Now  among  the  curves 


Qs(w„,  zq)     QAf^i,  z,) 

Qi{lUo,Zo)        Qi{Wi,Z,)j 

Q,  +  aQ,  +  I3Q,  =  0 

choose  those  which  pass  through  the  point  w^,  Zq-.  so  that  a  single  relation 
•'is  established  between  a  and  ^ : .  and  there  is  a  single  infinitude  of  such 
curves.  But  on  account  of  the  preceding  relations,  each  such  curve  passes 
through  the  point  w^,  z^;  and  therefore  on  the  original  surface,  the  arbitrary 
point  Wo,  Zo  determines  (on  the  present  hypothesis)  at  least  one  other  point 
U'l,  z,. 

In  general,  this  is  not  the  case :  that  is  to  say,  an  arbitrary  point 
on  a  quite  general  Riemann's  surface  determines  no  other  position  on 
that  surface.  Accordingly,  in  general,  only  a  single  point  on  the  old 
surface  corresponds  to  an  arbitrarily  assigned  point  on  the  new  surface ; 
and  the  transformation  is  then  birational. 


251.]  BIRATIONALLY   TEANSFORMABLE   INTO   ONE   ANOTHER  569 

het  f{w,  z)  =  0,  F(iu',  z')  =  0,  be  the  old  and  the  new  equations  (or 
curves)  respectively;  as  /  is  of  genus  p,  so  F  also  is  of  genus  p.  The 
degree    of  F  is    given   by    the    number    of  zeros   possessed   by  the    linear 

function 

\z'  +  fMiu'  +  V, 
that  is,  by  the  function 

\  Q.2  (w,  z)  +  /xQs  (w,  2)  +  vQ^  (w,  z) 
Qi  iw,  z) 
This  is  a  rational  function  on  the  original  Riemann's  surface :  and  it 
consequently  has  as  many  poles  as  it  has  zeros.  Its  poles  are  the  zeros 
of  Qi  {ic,  z),  which  are  2p  —  2  in  number  (§  241) ;  but  of  these,  p  —  S  are 
common  to  Qi,  Qo,  Qs,  corresponding  to  the  p  —  S  arbitrary  assigned  places 
where  Q^,  Q^,  Qs  vanish,  so  that  these  p  —  3  zeros  of  Q^  are  not  poles  of 
the  function.  There  remain  2^  —  2  —  (p  —  3),  ^ p  +  1,  poles;  and  therefore 
the  degree  of  i^  is  jj  +  1. 

Accordingly,  we  may  take  as  the  normal  equivalent  of  a  quite  general 
equation,  luhich  is  of  genus  p^S,  an  equation  of  degree  p  +  1. 

Ex.  Prove  that,  if  all  the  multiple  points  of  the  normal  equation  are  merely  double 
points,  their  number  is  \'p  {p  —  3). 

Manifestly,  these  arise  as  solutions  of  the  equations 

for  special  (and  not  arbitrary)  positions. 

One  obvious  exception  to  the  general  argument  arises  in  the  case  of 
the  hyperelliptic  eiquations,  of  the  form 

which  are  of  genus  p.     Each  of  the  quantities  Q  is  then  a  polynomial  in  z  of 
degree  ^p  —  1;  and  if 

Ql  +  aQ,  +  /3Q3 

vanishes  for  z  =  Zi,  Z2,  ...,  %_3,  it  vanishes  at  2(p-  3)  places  on  the  surface, 
'  instead  of  only  at  p  —  S.  If  the  relation  between  a  and  /3  is  chosen  so 
as  to  make  the  polynomial  vanish  at  one  other,  say  at  tu  =  t],  z=  ^,  then  it 
vanishes  also  at  the  (distinct)  place  tu  =  -  7],  z  =  ^.  The  preceding  theorem 
therefore  does  not  apply  to  hyperelliptic  equations*. 

252.  One  of  the  most  important  instances  of  birational  transformation 
arises  when  an  algebraical  equation  f{w,  z)  =  0  is  made  to  correspond 
birationally  with  another  algebraical  equation  having  multiple  points  of  only 
the    simplest    character    or,    in    geometrical    language,    when    any    plane 

*  The  theorem  was  first  enunciated  by  Clebsch  u.  Gordan,  Theorie  d.  AbeVschen  Functionen, 
p.  60.  Picard  {Cours  d'Analyse,  t.  ii,  p.  488)  shews  that  the  hyperelhptic  curves  are  the  only 
exception  to  the  theorem. 


570  TRANSFORMATION   INTO   A   CURVE  [252. 

algebraical  curve  is  transformed  birationally  into  another  plane  algebraical 
curve  the  only  multiple  points  of  which  are  double  points  with  distinct 
tangents  *. 

Let  a  denote  any  value  of  z,  and  let  a  denote  one  of  the  roots  of  the 
equation  f  (a,  a)  =  0.  If  a  is  a  simple  root  of  the  equation,  then  we  know 
(Chap.  VIII.)  that  a  branch  of  the  algebraical  function  exists  given  by 

w  —  a  =  ai(z  —  a)  +  a^  {z  —  ctf  -\-  ..., 

where  the  function  on  the  right-hand  side  is  a  uniform  function  of  ^^  —  a,  and 
only  a  finite  number  of  the  initial  coefficients  cti,  a^,  ...  can  be  zero.  If  a  be 
a  root  of  any  multiplicity,  then  the  corresponding  roots  can  be  arranged 
in  systems ;  and  each  system  can  be  expressed  in  the  form 

z-a  =  ^^,         w~a=!^P  P  (^), 

where  P  is  a  regular  function  of  ^  that  does  not  vanish  when  ^  =  0,  and 

is  such  that  the  common  factor  (if  there  be  any  common  factor  other  than 

unity)  of  the  indices  in  ^^  P  (^)  is  not  a  factor  of  q.     Also  p  and  q  are  finite 

positive  integers,  the  point  being  multiple  when  neither  of  them  is  unity. 

We  have  at  once 

p  1 

w  -.  a.  =  (z  -  a)'i  P  {{z  -  a)s}, 

q  1 

z-a^{iu-  a)P  Pi  {(w  -  ayp\ ; 

and  even  if  q  and  p  have  a  common  factor,  still  the  restriction  on  the  form  of 
P  makes  p  the  least  common  multiple  of  the  denominators  in  the  indices 
which  occur  in  the  expansion  of  ^  —  a.  Thus  there  are  q  circulating  values 
of  w  —  a  in  the  vicinity  of  z  —  a]  there  are  p  circulating  values  oi  z  —  a  in 
the  vicinity  of  ■it;  —  a. 

Such  a  system  is  called  a  cycle ;  the  smaller  of  the  integers  p  and  q 
is  called  its  degree ;  the  combination  a,  a  is  called  its  origin.  When  both 
p  and  q  are  unity,  the  cycle  is  called  linear.  The  object  of  the  trans- 
formations is  to  replace  cycles  of  any  degree  by  linear  cycles. 

If  a  be  infinite,  we  replace  w  —  a  by  — ;    likewise,  if  a  be  infinite,  we 

replace  z  —  a  hy  - .  To  take  account  of  all  the  cases,  we  introduce 
homogeneous  variables  X,  Y,  Z,  such  that 

X  Y 

X  =  z  —  a  =  -^ ,  y  =  w  —  a  =  -^; 

*  The  chief  stage  in  the  proposition  is  due  to  Nother,  Math.  Ann.,  t.  ix,  (1876),  pp.  166 — 182. 
See  also  Bertini,  Math.  Ann.,  t.  xliv,  (1894),  pp.  158 — 160;  Jordan,  Cours  d' Analyse,  t.  i,  ch.  v. 
Sections  iii,  iv ;  Appell,  Theorie  des  fonctions  algebriques,  ch.  vi ;  Brill  u.  Nother,  Jaliresh.  d. 
Deutschen  Math.-Vereinigung,  t.  iii,  (1894),  pp.  367 — 402,  where  many  references  are  given. 


252.]  POSSESSING   ONLY   DOUBLE   POINTS  571 

and  we  assume  expansions  of  the  form  • 

where  p,  cr,  t  are  integers;  R,  S,  T  denote  regular  functions  of  t,  which 
do  not  vanish  when  ^=  0,  and  are  such  that  a  factor  (if  any,  other  than  1) 
common  to  all  the  indices  in  any  one  of  the  expansions  for  X,  Y,  Z  is  not 
common  to  all  the  indices  in  any  other  of  those  expansions. 

The  axes  of  the  homogeneous  variables  can  be  transformed  without 
affecting  the  degree  of  the  cycle :  or,  in  other  words,  the  variables  z  and  iv 
can  be  subjected  to  a  homographic  substitution  without  affecting  that 
degree.  For  suppose  such  a  substitution  changes  the  origin  from  a',  a 
to  a,  a:  then 

,        ,        a^iz  —  a)  +  bi  (w  —  a.)  ,       ,         a, (z  —  a)  +  bo (tu  -  a.) 

z  —  a  =  — 7-^ ^ ,    ,    -^  ,         lu  —  a  = 


tts  (5  —  a)  +  63  (w  —  a)  +  1 '  as{z  —  a)  +  63  (w  —  a)  +  1 ' 

so  that  writing  z  —  a  =  ^'i,  w  —  a=  ^p  f  (0>  either  z'  —  a  or  tu'  —  a  begins 
with  a  power  of  ^  equal  to  the  degree  of  the  cycle  and  neither  of  them 
begins  with  a  lower  power.  Accordingly,  we  can  take  linear  combinations 
of  X,  Y,  Z  above,  so  as  to  secure  that  p,  a,  r  are  unequal,  without  affecting 
the  degree  of  the  cycle. 

X  Y 

When  the  cycle  has  a  finite  origin  as  above,  viz.  z  —  a  =  ^ ,  tu  —  a  =  -^ , 

Z  Z 

then  p  > T,  (T>  T\  the  degree  of  the  cycle  is  the  smaller  of  the  two  integers 
p  —  r,  a  —  T.  The  reason  for  the  introduction  of  homogeneous  coordinates  is 
to  treat  cycles  by  one  and  the  same  method,  whether  their  origin  be  in  the 
finite  part  of  the  plane  or  at  infinity.  Accordingly,  we  assume  that  the  three 
integers  p,  cr,  r  are  different  from  one  another  and  are  in  decreasing  order : 
then  o-  —  T  is  the  degree  of  the  cycle.  The  number  p  —  o-  is  the  class  of 
the  cycle. 

When  the  origin  of  a  cycle  is  at  infinity,  the  expression  of  the  branches  in  the  cycle  is 
of  the  form 

■i/=a(,X<l+aiX    1    + , 

when  X  has  infinite  values :  there  are  three  cases,  according  as  y  is  infinite,  finite,  or  zero, 
at  the  origin. 

For  the  first  case,  when  qq  is  not  zero,  if 

_X  _Y 

^~ z'    y~ z' 

we  take 

If  q>p,  the  three  indices  in  decreasing  order  are  q,  q—p,  0:  so  that  q—p  is  the  degree  of 
the  cycle.  If  q—p,  the  degree  depends  upon  the  lowest  index  in  Y—a^^X,  being  equal 
to  that  index  if  less  than  q.  If  g-  <  p,  the  decreasing  order  of  the  indices  is  q,  0,  q  —p  : 
so  that  p  —  q  is,  the  degree  of  the  cycle. 


572  RESOLUTION   OF  [252. 

For  the  second  case,  we  have 

where  l~^\\  as  before,  we  take 

and  therefore 

F-yo^^yiC^  +  H.... 

The  indices,  in  decreasing  order,  are  2"  +  ^,  g,  0:  the  degree  of  the  cycle  is  q. 

For  the  third  case,  we  have 

_k  fc  +  l 

where  ^'  ^  1 ;  we  take 

z-1,      z=cs       r=c«(i3oC'+...)=/^oC'^''+--- 

The  degree  of  the  cycle  is  q. 

Lastly,  y  may  have  infinite  values  for  finite  values  of  x :  the  expression  of  the  cycle 

then  is 

,«  s— 1 

y  =  \x    9  +  Si^      2    +..., 
where  s  >  0.     We  take 

the  degree  of  the  cycle  is  s. 

For  purposes   of  transformation,  we   use   the   birational    transformation 
which  arises  out  of  a  geometrical  relation  used  for  this  purpose  by  Halphen*. 

Given  a  plane  curve  G ;  let  an  arbitrary  conic  %  be  taken  in  its  plane  : 
draw  the  tangent  to  G  at  any  point  p  of  the  curve,  and  let  this  tangent 
be  intersected  at  P  by  the  polar  of  ;p  with  regard  to  S ;  then  P  is  regarded 
as  the  geometrical  transformation  of.j?.  Manifestly,  a  point  p  leads  to 
a  single  point  P ;  to  infer  the  converse,  let  PP'  be  the  polar  of  p  with 
regard  to  the  conic,  P'  denoting  the  point  where  the  line  touches  the 
reciprocal  of  G.  Thus  pPP'  is  a  self-conjugate  triangle ;  ^and  therefore  pP' 
is  the  polar  of  P.  Hence  given  the  locus  of  P,  we  find  p  by  drawing 
the  polar  of  P  with  regard  to  the  conic :  this  will  cut  (7  in  a  number  of 
points:  we  select  as  p*  that  one  of  the  points  such  that  Pp  is  a  tangent 
to  G  at  the  point  f.  Thus  each  point  of  the  locus  of  p  determines  a 
single  point  P,  and  conversely ;  the  analytical  expressions  of  the  geometrical 
relation  constitute  a  birational  transformation,  which  may  be  called  Halphen  s 
transformation. 

Two  forms  arise,  according  as  the  conic  does  not  or  does  cut  in  real  points 
the  curve  to  be  transformed.     The  conic  is  at  our  disposal  and  it  could  be 

*  Liouville,  3™<=  Ser.,  t.  ii,  (1876),  pp.  87—144. 

+  A  limited  number  of  pairs  of  points  can  exist,  for  any  conic  22,  such  that  the  construction 
would  not  discriminate  between  them.  We  do  not  regard  this  as  interfering  with  the  general 
character  of  the  transformation :  its  significance  in  the  result  appears  towards  the  close  of  the 
investigation.  For  the  analytical  relations,  expressing  jj  in  terms  of  P,  see  a  note  by  the  author, 
Messenger  of  Mathematics,  vol.  xxx,  (May,  1900),  pp.  1 — 7;  they  are  not  actually  required  for  the 
succeeding  investigation. 


252.]       ■  A   CYCLE  573 

chosen  so  that  it  does  not  cut  the  curve  to  be  transformed ;  but  the 
transformation  has  to  be  repeated  a  number  of  times,  and  some  of  the 
transformed  curves  might  be  cut  by  the  conic.  On  the  other  hand,  there  is 
a  finite  limit  to  the  number  of  times  the  transformation  is  applied ;  and  we 
may  therefore  assume  that,  if  the  conic  cut  in  real  points  the  curve  or  a 
transformed  curve,  the  tangents  to  the  conic  and  the  curve  are  different  from 
one  another. 

When  the  conic  and  the  curve  do  not  cut,  take  any  point  M  on  the  curve 
and  the  tangent  MM"  to  C;  let  M"M'  be  the  polar  of  M  with  regard  to  t, 
cutting  MM"  in  M" ;  and  let  MM'  be  the  polar  of  M"  with  regard  to  2, 
cutting  M'M"  in  M'.  The  triangle  MM'M"  is  self-conjugate  with  regard 
to  2 ;  when  this  is  chosen  as  the  triangle  of  reference,  the  equation  of 
the  conic  can  be  taken  to  be 

When  the  conic  and  the  curve  cut,  say  in  a  point  M,  then  let  the 
tangent  at  M  to  the  curve  cut  the  conic  in  M  and  M' ;  and  draw  M"M' , 
M"M  the  tangents  to  the  conic  at  these  points.  We  choose  MM'M"  as 
the  triangle  of  reference ;  the  equation  of  the  conic  can  be  taken  to  be 

Y'  +  2ZZ  =  0. 

In  the  former  case,  let  x,  y,  z  denote  the  position  p,  and  X,  T,  Z  that  of 

P.     Then  we  have 

Xx+Yy  +  Zz  =  0, 

because  P  lies  on  the  polar  of  p  with  regard  to  the  conic ;  and 

X,     Y,    Z\  =  0, 

X,     y, 
x,     y',     : 

(where  x  =  dxjdt,  and  so  for  y  and  z),  because  P  lies  on  the  tangent  at  'p  to 
the  original  curve.     Hence 

X  =  y  {xy  -  yx')  —  z  {zx  —  xz')  =  x  {xx  +  yy  +  zz')  —  x'  {jx?  +  y^  +  z")" 
Y  —  z  {yz  —  zy')  —  x  {xy  —yx')  —  y  {xx  +  yy'  +  zz')  —  y'  {x^  +  y'  +  z^) 
Z  =  X  {zx  —  xz)  —  y  {yz'  —  zy')  —  z  {xx  +  yy'  +  zz)  —  z  {x"^  +  y""  +  z'^). 
which  express  X,  F,  Z  in  terms  of  x,  y,  z. 

In  the  latter  case,  we  find  similarly  the  relations 
X  =  y  {xy  —  yx)  —  x  {zx  —  xz')\ 
Y=-x  {yz  -  zy')  -  z  {xy  -  yx)  \ 
Z  =  z  {zx  —  xz')  —  y{yz' —  zy')j 

which  •  express  X,  F,  Z  in  terms  of  x,  y,  z.  These  respective  relations 
constitute  part  of  the  analytical  form  of  Halphen's  transformation,  which  is 
birational  for  the  curve  but  not  birational  for  the  plane. 


574  halphen's  [252. 

Let  any  point  on  the  curve  to  be  transformed  be  denoted  by 

^=r2^(D=r(To+...)i 

where  R,  S,  T  denote  regular  functions  of  ^  in  the  vicinity  of  ^  =  0 :  and  we 
assume  that  the  integers  p,  a,  r  are  distinct  from  one  another. 

When  the  second  transformation  is  the  one  to  be  effected,  we  remember 
that  the  tangent  to  the  curve  is  the  axis  of  y  for  the  triangle  of  reference ; 
accordingly,  we  assume  o-  >  /o  >  t,  so  that  p  —  t  is  the  degree  of  the  cycle,  and 
cr  —  p  is  the  class  of  the  cycle  ;  also  p  —  2o-  +  t  <  0.  The  result  of  the 
transformation  is  *. 

X  =  -  Po^To  (p  -  t)  r^P+^l  +  . . .  =  p/'^P"  +  .  .  .  > 

Y  =  poCToT,  {p-2a  +  t)  ^p+-+-i  +  . . .  =  a-o"r"  +  . , 

Z  =  To^po  (P  -  t)  ^''+'^-'  +...=  To"r"  +  . . 

say.     We  have  a"  >  p"  >  r" ;  also  p"  —  t"  =  p  —  r,  a"  —  p"  =  a  —  p  ;  so  that  the 
degree  and  the  class  of  the  cycle  (if  any)  are  unaltered. 

When  the  transformation  with  regard  to  an  assumed  conic  is  applied 
a  number  of  times,  the  conic  being  assumed  so  as  not  to  cut  the  initial  curve, 
then  it  may  happen  that,  after  a  number  of  transformations,  the  latest  curve 
cuts  the  conic.  The  further  application  of  the  transformation  will  then,  by 
the  foregoing  analysis,  have  no  effect  upon  either  the  degree  or  the  class  of  a 
cycle :  it  therefore  is  ineffective  for  our  purpose  of  further  reduction.  To 
secure  such  further  reduction,  we  should  proceed  to  effect  transformation 
with  regard  to  another  arbitrarily  assumed  conic,  chosen  so  as  not  to  cut  the 
curve. 

When  the  first  transformation  is  the  one  to  be  effected,  assume  that 
p  >  o-  >  T,  so  that  o-  —  T  is  the  degree  of  the  cycle  (if  any)  at  the  point,  and 
p  —  o-  is  its  class.     The  result  of  the  transformation  is 

x  =  ^<'^^—^{p,T,^{T-p)  +  ...]  =  i:^'{p:+...y 

Y=  ^<^+2-i  |^^^^2(^  -a)+...]=^^'  {a:  +  . . .) 
Z=  ^2^+^-1  {croVo(o-  -  t)  +...}  =  r  (To'  +  ...)) 

say.     We  have  p  >  a'  \  also  because  t'  —  a  =  a  —  t,  we  have  t  >  a' ;  there 
are  therefore  three  cases. 

(i)  Let  p'  >t'  >  cr'.  The  degree  of  the  cycle  (if  any)  is  r'  —  cr',  that  is, 
o"  —  T  :  it  is  unaltered  by  the  transformation.  The  class  of  the  cycle  is  p'  —  t', 
which  is  p  —  2(7  +  T,  =  p  -  cr  —  (cr  —  r),  so  that  p  —t'  <  p  —  a;  it  is  decreased 
by  the  transformation.     In  this  case  p  —  2o-  +  t  >  0  ;  also 

p'  — 2t'+c7',      =p  —  2cr-t- t  — (cr  —  t),      <p  — 2cr  +  T. 


252.]  TRANSFORMATION  575 

(ii)  Let  r'  >  p  >  a'.  The  degree  of  the  cycle  (if  any)  is  p  —  a  ,  that  is, 
p  —  (T.  But  t'  >  p',  so  that  p  —  a<  (t  —  t:  hence  p'  —  a  <  a  —  r,  or  the  degree 
is  decreased.  The  class  of  the  cycle  is  t  —  p',  that  is,  a  —  t  —  (p  —  a);  there 
is  no  useful  rule  of  increase  or  decrease  compared  with  the  old  cycle  in 
general,  though  we  note  that  the  new  class  is  less  than  the  old  degree.  In 
this  case  p  —  2cr  +  t  <  0. 

(iii)  Let  t'  =  p',  so  that  p  —  2cr  +  t  =  0,  or,  the  degree  and  the  class  of 
the  cycle  (if  any)  are  equal.     Effect  a  (birational)  transformation 

X'  =  X,       Y'  =  Y,       Z'  ==  aif'  (a  —  t)  X  —  PqTo  {'^  —  p)  ^ 

=  r' (To'" +•..). 

where  r'"  >  p  +  2t—1,  that  is,  r'"  > p  .  Accordingly,  for  the  transformed  cycle, 
we  have  r"'  >  p  >  a-'.  The  degree  of  the  cycle  (if  any)  is  p  —  a' ,  that  is,  p  —  a, 
which  is  equal  to  cr  —  t  :  it  is  unaltered  by  the  double  transformation.     Also 

t'"  -  2p'  +  <t'  >  p  +  2t  -  1  -  2  (p  +  2t  -  1)  +  a-  +  2t  -1  >  a  -  p, 

where  o-  —  p  is  a  negative  quantity.  If  t"  —  2p'  +  o-'  >  0,  another  application 
of  the  transformation  by  the  conic  leads  to  the  first  case  above :  and  the 
class  is  decreased,  while  the  degree  is  unaltered.  If  r"  —  2p'  +  cr'  <  0,  another 
application  of  the  transformation  by  the  conic  leads  to  the  second  case  above  : 
the  degree  is  decreased.  If  r'"  —  2p  +  cr'  =  0,  the  new  class  is  equal  to  the 
new  degree,  each  of  them  equal  to  the  common  value  of  the  old  class  and  the 
old  degree :  the  cycle  must  be  considered  further. 

For  this  last  case,  let  p  —  cr  =  cr  —  r  =  n  :  then 

f-f-(^■■■)■  f=^"(^••■)■ 

or,  remembering  the  source  of  the  homogeneous  coordinates,  and  taking 

^  :   q  :   \  =  y  :  X  :  z. 


we  have 


^  =  pG^(r)> 


where  (r  is  a  regular  function  of  |",  that  does  not  vanish  with  ^ :  and  this 
corresponds   to   the  original   cycle  by  a  birational  transformation.     It  may 

happen  that  the  initial  powers  of  p'  in  G  give  integral  powers  of  ^ ;  let 

^  =  P(l)+f^'^G^i(r), 
where  P  (^)  contains  powers  of  ^  not  higher  than  the  gth  and  not  lower  than 
the  second,  where  7^  >  a  >  0,  and  where  G^  does  not  vanish  with  ^.     To  this 
form,  apply  the  Halphen  transformation  with  respect  to  the  conic 

,  G  {x,  y,  z)  =  0, 


576  RESOLUTION   OF   A   CYCLE  [252. 

taken  arbitrarily  in  the  first  instance.     The  point  X,  Y  which  corresponds  to 
f ,  7]  is  given  by 

F-,  =  ^(X-f)=,-(X-|); 

OX         oy  oz 

where  we  write  x,  y,  z  =  r),  ^,  1  in  the  latter :  thus 

X=a(^,7],v'),     Y=^(^,v,v\ 
where  a  and  /3  are  rational  functions  in  r]'  of  the  first  degree.     Hence 
dY  ,  ^^  +  /3^V  +  /3vV'        ,.        ,     .. 


d'Y 
dX'" 

generally 

dX 
dy 
d| 
dX 
d^ 

—  7  ' 

\i^     '/)     '/    5 

dy 
dt]" 

'/ 

and 

«f  + 

a,^'  +  S 

87 

d'^Y  ?r)" 

Choose  the  conic  C,  so  that 

^,  =1=  0,     a^  +  a^i]  +  a,/  r?"  ^  0, 

for  ^=0,  17  =  0;  then  as  7;<9'+^>  is  the  first  derivative  of  77  with  regard  to  ^ 
that  becomes  infinite  at  ^  =  0,  7;  =  0,  so  F'^'  is  the  first  derivative  of  Y  with 
regard  to  X  that  becomes  infinite  at  the  new  origin  :  that  is,  if  Fq,  X^  be  the 
new  origin,  we  have 

Y-Y,  =  A  (X  -  Xo)  +  (X  -  Xo)'"'^-  G,  {(X  -  Xo)-}. 

Let  the  transformation  be  applied  g-  —  2  times  in  succession ;  we  have,  at  the 
end, 

F'  -  Fo'  =  a,  {X'  -  Xo')  +  a,  (X'  -  Xo')^  +  a,  (X'  -  Xo'f^"  +  .  •  • , 
or  changing  the  axes  by  taking  X'  —  Xq  =  X",  Y'  —  Y^  —  a^  (X'  —  Xq)  =  Y", 
we  have 

F"  =  a,X"^  +  a3X"'^'"^+..., 
where  n  >  a>  0.     This  can  be  represented  in  the  homogeneous  form 
y  =  ^'\     X  =  a^^-"  +  a3^2n+a  ^  _ _ _  ^     ^  ^  1  =  ^o_ 

Applying  now  the  Halphen  transformation  in  its  analytical  form,  we  have 
X  =  -  2na2^-»-^ - (2?2  +  a)  a^l;-''+^-^ -  ..., 
Y^-n^""-^-  ..., 
Z  =      n  ^-^-^  +  2nai  ^'''-'  +  ...; 


252.]  INTO    LINEAR    CYCLES  577 

or  transforming  so  as  to  have  X  +  la^Z,  Y,  Z,  as  the  new  coordinates,  we 
have 

X'"  =-{2n  +  a)  ttst^^+^-i  - . .  .1 

The  three  indices  in  decreasing  order  are  2u  +  a— 1,  2«  —  1,  7i  —  1 ;  the 
degree  of  the  cycle  is  n,  its  class  is  a,  which  is  less  than  n ;  and  therefore  the 
birational  transformations  have  reduced  the  class  below  the  degree. 

Hence  given  a  cycle  of  any  degree  m,  greater  than  unity,  and  of  any  class 
7)i.',  we  can  by  Halphen's  birational  transformation  change  it  into  another 
cycle.  If  m  be  greater  than  in,  the  new  class  is  less  than  m  while  the  new 
degree  is  m :  and  repetition  of  the  transformation  can,  by  the  first  case,  be 
made,  so  long  as  the  class  is  greater  than  the  degree :  and,  by  the  third  case, 
until  the  class  is  less  than  the  degree ;  without  altering  the  degree.  When 
another  repetition  of  the  transformation  is  made,  the  degree  will  (by  the 
second  case)  be  decreased. 

Proceeding  in  this  way,  we  can  make  the  cycle  of  the  first  degree  :  then 
by  the  first  case,  of  the  first  class  also :  that  is,  we  can  make  the  cycle  linear. 
When  this  process  is  applied  to  each  cycle,  the  final  equation  has  only  linear 
cycles :  and  it  is  connected  with  the  initial  equation  by  birational  transformation. 

Further,  it  was  proved  that  a  Halphen  transformation  of  a  linear  cycle  of 
the  form 

w  —  a  ==  A'l  {z  —  a)-\-  ko  {z  —  a)''  +  •  • .  +  ^m  (z  —  a)'"  +  . . . 

leads  to  a  relation 

W-a'=  k,'  (Z  -  a)  +  h'  (Z-a'y+...+  k'„,_,  (Z-  a')^-^  +..., 
where  /c,,„,_i  depends  upon  ^■,^  linearly  and  upon  k^^,  ...,  A'-m^j.  If  therefore 
two  linear  cycles  agree  up  to  (but  not  beyond)  the  nith  order  of  small 
quantities,  the  transformation  replaces  them  by  two  linear  cycles  agreeing  up 
to  only  the  (???,  — l)th  order  of  small  quantities:  and  so  on,  by  successive 
repetitions  of  the  transformation,  until  they  agree  only  in  their  origin,  so  that 
the  tangents  differ.  Now  the  origin  of  a  new  cycle,  engendered  by  trans- 
formation, lies  on  the  tangent  to  the  cycle  which  is  to  be  transformed  :  hence, 
applying  a  Halphen  transformation  once  more,  the  origins  of  the  two  cycles 
are  different.  It  therefore  follows  that  the  cycles  of  any  degree  and  class 
having  a  common  origin  can  be  birationally  transformed  into  linear  cycles, 
each  of  them  with  its  own  origin  distinct  from  that  of  all  the  remainder. 

The  resolution  thus  effected  transforms  every  multiple  point  'of  any 
charaeter  into  an  aggregate  of  simple  points,  and  would  therefore  transform 
an  equation  with  multiple  characteristics  into  an  equation  having  only  simple 
points,  if  new  singularities  were  not  introduced  in  the  process  of  birational 
transformation.  Reverting  to  the  initial  geometrical  exposition  of  the 
birational  transformation,  we  see  that  such  singularities  may  arise  through 
P.  F.  37 


578  BIEATIONAL   TRANSFORMATION  [252. 

exceptional  points  on  the  locus,  as  follows.  From  P,  which  is  uniquely 
obtained  from  the  point  p  on  the  given  curve,  a  number  of  other  tangents 
can  be  drawn  to  that  curve ;  let  q  denote  the  point  of  contact  of  any  one  of 
them.  In  order  that  pq  may  be  the  polar  of  P  with  regard  to  the  conic,  one 
relation  must  be  satisfied  by  x  and  y,  the  coordinates  of  P,  in  addition  to 
the  equation  of  the  curve  on  which  p  lies :  that  is,  there  are  two  algebraic 
equations  satisfied  by  {x,  y),  and  therefore  there  is  a  finite  number  of 
simultaneous  values  satisfying  these  conditions.  Each  such  point  is  a  double 
point  on  the  locus  of  P,  with  distinct  tangents  for  the  branches :  so  that  the 
transformed  curve  thus  possesses  simple  nodes  unrepresented  by  any  multi- 
plicity on  the  original  curve.  But,  in  general,  there  cannot  be  two  points 
such  as  q,  say  q  and  q,  the  tangents  at  which  are  concurrent  with  the  tangent 
at  p ;  for  this  purpose,  some  relation  between  the  coefficients  of  the  curve  and 
the  constants  of  the  conic  would  need  to  be  satisfied — which  is  not  the  case, 
when  the  conic  is  arbitrarily  assumed  and  the  curve  is  not  extremely  special. 
We  therefore  have  the  theorem,  expressed  geometrically : 

A7iy  algebraic  curve  can  he  hirationally  transformed  into  some  algebraic 
carve  the  singularities  of  luhich  are  double  points  with  distinct  tangents. 

The  two  curves  must  be  of  the  same  genus,  and  they  must  have  the  same 
moduli :  the  complexity  of  the  transformation  manifestly  depends  upon  the 
original  equation. 

Ex.     Consider  the  resolution*  of  the  singularity  of 

if  -  "ix^f  +  x^  =  %x°y^  ' 

at  y  =  0,  «  =  0. 

Proceeding  as  in  Chap.  VIII.,  we  find  the  six  branches  of  the  curve  given  by 
where  co^--!,  &^  =  \  :  thus  for  our  homogeneous  coordinates,  we  may  take 

Thvis  p  =  8,  (r=6,  r  =  0.     The  class  of  the  cycle  is  2,  the  degree  is  6. 

Since  p  — 2o-  +  r<0,  Halphen's  transformation,  when  applied,  falls  under  the  second 
case.     The  indices  after  transformation  are 

t'  =2(7+    r-l  =  ll  =  pi'] 

p'  =    p  +  2t  -  1  =    7  =  (Ti>  ; 

0-'=    o-  +  2r-l=    5  =  rJ 
the  class  of  the  cycle  is  4,  the  degree  is  2. 

Since  pj  — 2o-i+ri  >0,  Halphen's  transformation,  when  applied,  falls  under  the  first 
•case.     The  indices  after  transformation  are 

p"=  pi  +  2ri-l  =  20j 
r"=2cri+  ri-l  =  18 
0-"=    o-i  +  2ri-l  =  16j 

*  C.  A.  Scott,  A7ner.  Journ.  Math.,  t.  xiv,  (1892),  p.  318. 


252.]  OF   ALGEBRAIC    CURVES  579 

the  degree  of  the  cycle  is  2,  the  class  is  2.     For  the  cycle,  we  have 

and  therefore 

Y"  =  A^X"^  +  .... 

This  is  an  instance  of  the  second  case,  when  the  class  and  the  degree  are  equal :  the 
actual  form  is 

4  81  47^43       ,„  V-, 

9  4  8 

Three  applications  of  the  Halphen  transformation  will  reduce  the  class  to  unity  and  will 

give  a  cycle  of  the  form 

,,  ,  -.  ,1^ 

y  =a2x''-'  +  a^x  "^+...; 

and  one  more  application  will  give  a  linear  cycle. 

I^ote.  The  preceding  sketch  (§§  245 — 252)  is  intended  only  as  an  intro- 
duction to  the  theory  of  birational  transformation,  the  development  of  which 
really  belongs  to  the  detailed  theory  of  Abelian  functions. 

Moreover,  transformations  which  are  rational  but  not  birational  are 
practically  omitted  from  consideration.  If  further  information  be  <iesired, 
the  various  works  and  memoirs  quoted  in  the  preceding  sections,  and  in 
§  243,  may  be  consulted :  an  ample  account  of  the  general  theory  of  corre- 
spondence will  be  found  in  Baker's  Abelian  Functions. 

The  normal  curve  in  §  251,  adopted  by  Clebsch  and  Gordan,  must  not  be 
supposed  the  only  normal  curve  that  can  be  adopted ;  others  are  discussed  in 
the  places  to  which  references  have  just  been  given. 

Ex.  Prove  that  any  Riemann's  surface  of  genus  p  can  be  birationally  transformed 
into  a  surface  of  2p-  2  sheets,  given  by  an  equation  f{w,  z)  =  0  of  degree  2p  —  2inw:  such 
that  on  this  surface  w  is  an  integral  function  of  z  whose  only  zeros  are  at  the  branch-points 

f  flz  [ ZClZ 

of  the  surface  (Klein's  canonical  form).    Shew  that  on  this  surface  I  —  and  I  —  are  integrals 
^  '  J  w  J  10 

of  the  first  kind. 

Verify  that,  in  general,  if  it  and  v  are  rational  integral  functions  of  z  of  degrees  four 
and  two,  with  no  common  or  repeated  factors,  then 

{io'^  +  uv)^  +  ku^  =  0, 

k  being  a  constant,  is  svich  a  canonical  surface  for  ^  =  3;  and  determine  for  it  three 
independent  integrals  of  the  first  kind.  (Math.  Trip.,  Part  II.,  1899.) 


Supplementary  Notes:  Abel's  Theorem. 

The  results,  which   have   been   outUned   in   the  preceding   chapters   relating  to  an 

algebraical  equation 

f{io,z)  =  0 

and  to  the  functions  of  position  on  the  associated  Riemann's  surface,  are  due  to  the 
development  by  successive  mathematicians  of  those  researches  of  Abel  upon  transcendental 

37—2 


580  ABEL'S  [252. 

functions,  which  completely  revolutionised  both  the  content  and  the  method  of  analysis. 
It  may  therefore  be  deemed  not  out  of  place  to  give  a  brief  exposition  of  Abel's  principal 
result,  commonly  called  Abel's  theorem,  and  a  few  of  its  applications.  The  analysis, 
by  which  that  theorem  was  originally  established,  was  really  only  a  wonderful  exercise 
in  the  integral  calculus ;  it  will  be  shortened  and  simplified  by  using  the  results  obtained 
in  connection  with  Riemanu's  surfaces. 

Abel's  great  memoir  is  2Iemoire  sur  tone  propriete  generale  d'une  classe  tres-etendue 
de  fonctions  transcendantes,  written  in  1826,  published  in  1841,  and  included  in  his 
(Euvres  Completes,  t.  i,  pp.  145 — 211.  The  proof  of  the  main  theorems  is  rearranged 
and  much  simi^lified  by  Rowe,  "  Memoir  on  Abel's  Theorem,"  Phil.  Trans.,  (1881), 
pp.  713 — 750,  whose  e.\position  is  adopted  in  what  follows. 

Other  references  are  to  be  found  in  the  works  quoted  in  §  239,  Note  1. 

I.     We  take  the  equation /'('^-'j  -2^)  =  0  in  the  form  (§  193) 

f{io,  z)  =  w''  +  w'»-i g,(z)+  ...  +  wg n-i  {z)  +  gn  {z)  =  0, 

where  the  functions  g  are  "polynomials  in  z :  and  we  are  concerned  with 
rational  functions  of  position  on  the  Riemann's  surface,  which  (§  193)  can 
be  taken  in  the  form 


bw 


where 


V (iv,  z)  =  w""-^  ko  (z)  +  w'^-2  k,  (z)+  ...+  kn-i  (z\ 

and    the    functions    k^,  kj,    ...,  kn-i  are  rational   functions   of  z  only.     By 
taking 

n 
we  have 

,      U  (w,  z) 

w  = — ^ 


¥ 

dw 


w 


here 


U (w,  z)  =  t(f'--  ho  (z)  +  tv"-'  Jh  {z)+  ...+  K-2  {z), 

and  the  functions  /?o,  hi,  ...,  hn—2  are  rational  functions  oi z  only. 

In  the  preceding  investigations,  we  have  considered  the  special  properties 
of  the  simpler  and  most  typical  forms  of  the  integral 

Iw'dz, 

taking  account,  in  particular,  of  the  effect  upon  its  expression  of  modifying 
the  path  of  variation  from  the  lower  to  the  upper  limit ;  the  integral  being 
a  transcendental  function.  Now  it  is  known  that,  for  quite  simple  equations 
/=  0,  the  sum  of  a  number  of  such  integrals  may  be  expressible  in  simple 
terms,  no  single  integral  itself  being  so  expressible,  when  certain  algebraic 
relations  connect  the  upper  limits.     Thus  for  the  equation 

w''  =  ^z^  -  g-iZ  -  g^, 


252.]  THEOREM  581 

we  have 

^1  dz    r^2  dz    r^3  dz    . 

—  +       — +       —  =  0, 


(or  equal  to  a  period  of  the  integral),  if 


t^l, 

^1, 

1 

w^, 

22, 

1 

Ws, 

Zz, 

1 

=  0 


effectively  giving  the  addition-theorem  in  elliptic  functions.  Similarly,  the 
sum  of  three  elliptic  integrals  (Jacobian)  of  the  second  kind  is  expressible 
as  an  algebraic  function,  and  the  sum  of  three  elliptic  integrals  (Jacobian) 
of  the  third  kind  is  expressible  as  a  logarithmic  function.  We  proceed  to 
that  part  of  Abel's  theorem  which  establishes  the  corresponding  results  for 
the  equation 

f{io.z)  =  0, 

called  the  permanent  equation. 

To  consider  a  sum  of  integrals  of  the  same  form,  we  need  to  have  a 
number  of  upper  limits,  and  the  same  number  of  lower  limits.  The  upper 
limits  will  be  regarded  as  given  by  those  positions  on  the  associated 
Riemann's  surface  at  which 

e  (w,  z)  =  0, 

called  the  conditional  equation.  Manifestly,  6  can  be  made  of  degree 
TO  —  1  in  w  at  most,  by  means  of  the  permanent  equation  :  the  coefficients 
of  the  various  powers  of  -u;  in  ^  are  polynomials  in  z,  and  the  constant 
coefficients  in  these  polynomials  are  completely  arbitrary.  The  actual 
positions,  which  give  the  upper  limits  of  the  integrals,  are  determined  as 
the  simultaneous  roots  of  the  two  equations 

f{w,  z)  =  0,     e  {lu,  z)  =  0. 

Similarly  for  the  lower  limits  of  the  integrals,  which  are  associated  with 
another  conditional  equation.  The  arbitrary  constants  will  be  regarded  as 
parametric  :  when  parametric  variation  of  these  constants  takes  place,  the 
positions  of  the  upper  limits  vary.  If  we  choose,  we  can  associate  variation 
from  the  lower  limit  to  the  upper  limit  with  the  variation ;  but  this  is 
unnecessary,  and  all  that  we  require  is  the  variable  dependence  of  the  limits 
upon  the  parametric  constants. 

To  obtain  the  algebraical  expression  of  these  upper  limits,  we  proceed 
as  follows.  Let  Wi,  Wg,  •••,  Wn  denote  the  n  values  of  w  for  any  value  of  z; 
then  the  eliminant  in  z,  say  E  {z),  is  given  by 

n 

Z=E{z)=  n  0{Wr,z), 

r  =  l 


582  ESTABLISHMENT   OF  [252. 

the  values  of  z  being  given  by  the  roots  of  E{z)  =  ^.  To  obtain  the  value 
of  H)  to  be  associated  with  a  root  of  E  (z)  =  0,  say  z  =  c,  it  is  sufficient  to 
obtain  the  greatest  common  measure  of 

f(tu,  c)  =  0,     6  {w,  c)  =  0. 

The  roots  of  Z=0  may  be  of  two  kinds:  (i),  depending  upon  the  para- 
meters in  6,  and  consequently  varying  with  them ;  (ii),  independent  of 
these  parameters.  If  c  denote  one  of  the  latter  class,  then  the  corre- 
sponding value  of  w,  say  7,  satisfying  f{w,  c)  =  0  cannot  involve  the 
arbitrary  parameters,  so  that  the  position  7,  c  is  independent  of  the 
parameters.  Now  since  E  vanishes  for  z  =  c,  one  of  the  n  quantities 
9{iUr,  z)  vanishes  for  lUr  —  'y,  z  =  c:  that  is,  a  relation  then  exists  among 
the  parametric  constants  in  9  which  is  linear  if  the  parameters  occur 
linearly  in  6.  (We  can,  of  course,  secure  that  the  case  does  not  arise, 
by  taking  all  the  parameters  in  6  initially  independent  of  one  another : 
but  it  sometimes  is  desirable  that  the  case  should  arise,  and  we  therefore 
admit  its  possibility.)  Let  P  {z)  represent  the  product  of  all  the  factors 
in  Z,  which  give  roots  depending  upon  the  parameters  in  6,  and  let  G  {z) 
represent  the  product  of  those  which  give  roots  independent  of  the  para- 
meters, so  that 

Z=G{z)P  {z). 

It  will  be  assumed  that  all  the  roots  of  P  {z)  =  0  are  simple,  so  that  P'  (z) 
and  P  (z)  do  not  vanish  together :  and  that  they  are  /u,  in  number,  say 
iCj,  Wq,  ...,  x^.  Let  the  corresponding  values  of  w  be  y^,  y2,  ••■,  i/i^',  so  that 
the  upper  limits  for  the  integrals  are  (_yi,  ^1),  {i/o,  ^'2),  •••,  (l/i^,  ^ti)- 

The  integral  to  be  considered  is  jtu'  dz,  that  is, 

'U(w,z) 


9/ 
dw 


dz, 


where  w  is  one  of  the  n  branches  of  the  function  defined  hy  f(iu,  z)  =  0:  to  fix 
the  ideas,  let  Wi  denote  this  branch,  so  that  we  also  have  0  {tVi,  z)  =  0.  The 
integral  is  a  function  of  its  upper  limit,  and  therefore-  varies  when  the  limit 
varies,  that  is,  when  the  parameters  vary.  Let  8  denote  small  variations  for 
the  parameters ;  then  the  variations  of  the  roots  of  P{z)  =  0  are  given  by 

P'{z)dz  +  8P{z)  =  0, 

and  therefore 

az        p,  ^^^ 

hE{z) 
-  ~.G{z)P'{zy 


252.]  Abel's  theoeem  583 

because    E  (z)  ^  C  (z)  P  (z),   and    0  (z)    does    not    involve    the    parameters. 

Now 

E(z)=Ud(io,,z), 

r  =  l 

SO  that 

8E(z)=l    -p^Sd(tVr,z). 

In  our  subject  of  integration,  we  are  dealing  with  the  branch  lu^  of  lu;  and 
for  this  branch  6  {w^,  z)  =  Q.  Hence  all  the  terms  in  the  summation  vanish 
except  that  for  r'  =  1,  and  we  have 

Thus 

,,  U{tv„z)         1  E{z)     ..,         . 

dj        (J  {z)  F  {z)  V  (Wi ,  z) 

The  right-hand  side  vanishes  if  7i\  be  replaced  by  any  other  of  the  branches 
w;2,  ...,Wn,  because  E (z)  -^  6  {w,.,  z)  is  zero  unless  ?•  =  1 ;  it  therefore  is 
unaffected  when  we  add  to  it  all  these  vanishing  terms,  that  is,  we  have 

Wdz  =  -  ri.    .    -nr.    .    2    ^7 n-r^ X  Od  {Wr,  z). 

C(z)P(z),.^i        df_       9{tu,.,z) 

dlUr 

The  quantity  ^- — ^-^  is  an  integral  symmetric  function  of  Wj,  ...,  Wr_T^,  w^+i, 

...,  Wn,  (and  it  is  an  integral  function  of  z) :  by  means  of  _/=0  it  can  be 
made  a  function  of  z  and  tVr  only,  which  is  polynomial  in  tUr  and  is  rational 
in  z.  Also  U{iUr,  z),  which  is  a  poljoiomial  in  tu,.,  has  rational  functions 
of  z  for  its  coefficients :  let  M  (z)  be  the  least  common  multiple  of  all  the 
denominators  in  this  rational  function,  so  that,  when  we  take 

„  W(tUr,z) 

W  is  polynomial  in  Wy  and  z,  of  degree  not  higher  than  n  —  2  in  Wr.     Hence 

is  a  polynomial  in  lUr,  the  coefficients  being  polynomials  in  z;  by  means 
of  /=  0,  it  can  be  expressed  in  the  form 


584  ESTABLISHMENT   OF  [252. 

where  the  coefficients  K  are  polynomials  in  z.      But  'Wi,  ...,  Wn  are  the  n 
roots  tu  of  f{w,  z)^0,  and  therefore* 

M-l 

r=l  _0/ 

dWr 

so  that 

'^'^^'~     M{z)G{z)P'{z)     ''-'- 
Hence 

1^  M  TC  ( Z    \ 


,:rM{z;)G{z„)F{z.y 

on  taking  the  summation  over  the  fx  roots  of  P  {z)  =  0. 

To  obtain  an  equivalent  for  the  right-hand  side,  we  consider  the  expres- 
sion of 

Kn-i  JZ) 

M(z)C{z)P(z) 

in  partial  fractions.  Let  or^,  ...,  ofg  denote  all  the  roots  of  71/ =0,  any  (or  all) 
of  which  may  be  repeated,  and  assume  that  no  one  of  these  is  a  root  of 
P(z)  =  0;  and  let  Ci,  ...,Cm  denote  all  the  roots  of  C{z),  any  (or  all)  of 
which  also  may  be  repeated.     Then 


M(z)C{z)P(z)  p^^z-ap      k=iZ-Ck 


-h  2 


:iii/(^,)C(^<,)P'(^,)s-^, 
+ , 


where  the  unexpressed  terms  involve  higher  powers  of  {z  —  ap)~\  (z  —  Cjfc)~\ 
respectively  for  each  repeated  root,  and  I  (z)  is  a  polynoinial  in  z,  if  the 
order  of  Kn-i  is  not  less  than  that  of  M  (z)  C  (z)  P  (z).     Thus 

Ap=  coeff.  of in  the  expansion  of  T  in  ascending  powers  of-^  —  a^, 

z       Ctp 

Ck= oi  z-Ck] 

z  -  Ck 

1  ... 

and  the  coefficient  of  -  in  the  expansion  of  T  in  descending  powers  of  z  is 


r=x  '" Ci  '  r=i^M{z.)C{z^)P'{z^y 

so  that  if,  from  the  sum  of  the  coefficients  of  and  (for  all  the 

z-ap  z-Ck 

*  Buinside  and  Panton,  Theory  of  Equations,  (7th  ed.),  vol.  i,  p.  172. 


252.]  Abel's  theorem  585 

values  of  p  and  k),  the  coefficient  of  1/z  be  subtracted,  the  result  is  the 
expression  required  for  iS  (w'dz)^.     We  express  this  in  the  symbolic  form 


@ 


i^n-i  (^) 


M(2)G(z)j     P(z)    ' 

where  ©  denotes*  the  result  of  the  following  operations:  Take  each  factor 
z  —  a  of  M  (z)  C  (z)  in  turn,  obtain  the  coefficient  of  (z  —  a)~^  in  the 
expansion  of 

M{z)C{z)F{z) 

in  ascending  powers  of  z  —  a,  and  form  the  sum  of  all  of  these  coefficients ; 
from  this  sum  subtract  the  coefficient  of  z-^  in  the  expansion  of  the  same 
expression  in  descending  powers  of  z. 

Hence 


2  {w'dz)„  =  % 

<7=1 


=  © 


_M{z)G{z)y  P{z) 


JI{z)  C(z) 

1 

Jl(z)C{z)_ 


11 

C{z)  2 


"      W(tUr,  Z)      E{z) 


df  d{Wr,z) 

dWr 
Wi'Wr,  z)W{Wr,  Z) 


{Wr,  Z) 


r  =  l 


dWr 


6{%Vr,  z) 


The  symbol  ©  is  clearly  distributive.,  from  its  definition:  and  upon  the 
subject  to  which  it  applies,  integration  can  be  effected  with  regard  to 
parameters  in  such  a  way  that  the  symbol  applies  to  the  integrated  form. 
The  inverse  of  the  operation  S  with  regard  to  the  parameters  on  the  right- 
hand  side  implies  integration  with  regard  to  the  variables  on  the  left.;   hence 


iVa'  •'>)    W  (iV,  Z)    , 

^      4dz  -  A 


M{z) 


¥ 


=  © 


M{z)C{z)j 


C(z)  2     — ^^7 — •'  log  6  (wr,  z)  I  , 


dWr 


where  ^  is  a  constant  of  integration,  and  where  we  now  write  w  for  w-^ 
in  the  subject  of  integration,  Wi  having  been  any  branch  of  the  quantity  w 
defined  by  f{iu,  z)  =  0.  To  evaluate  the  right-hand  side,  only  algebraic 
expansions  are  necessary;  and  the  result  will  be  some  function  of  the 
parameters.      These    parameters    are   connected   by  the   equation  P  (x)  =  0 

*  The  symbol,  in  this  significance,  is  due  to  Boole,  Phil.  Trans.  (1857),  p.  751  ;  the  various 
coefficients  are  manifestly  the  Cauchy  residues  (§  25,  Ex.  9)  of  the  expression  for  its  poles,  which 
arise  through  the  zeros  of  ill  (z)  C  (z)  and  through  an  infinite  value  of  z. 


586  Abel's  theorem  [252. 

with  the  upper  limit  of  the  integrals :  when  expressed  in  terms  of  them,  the 
right-hand  side  is  clearly  a  logarithmic  and  an  algebraic  function  of  those 
limits,  which  in  special  cases  may  degenerate  to  a  more  simple  form.  The 
constant  A  can,  if  desired,  be  determined  by  taking  another  conditional 
equation,  similar  to  0  (lu,  z)  =  0,  in  order  to  assign  lower  limits  to  the 
integral. 

This  result  is  Abel's  Theorem  in  its  most  general  form.  We  proceed 
to  some  applications,  first  recapitulating  the  significance  of  the  various 
symbols.  The  fundamental  equation  is  f{w,  z)  =  0,  of  degree  n  in  w, 
having  the  coefficient  of  w''^  equal  to  unity,  and  polynomials  in  z  for  the 
coefficients  of  the  remaining  powers  of  iv.  The  conditional  equation  is 
6  (w,  z)  =  0,  which  (by  means  of  /=  0)  is  taken  of  degree  not  higher  than 
7?.  —  1  in  iu\  the  coefficients  of  the  various  powers  of  %u  are  polynomials 
in  z,  having  arbitrary  parameters  for  coefficients  of  powers  of  z.  The 
result  of  eliminating  w  between /=0,  ^  =  0,  is  obtained;  G {z)  represents 
the  aggregate  of  factors,  corresponding  to  roots  that  are  independent  of  the 
parameters;  the  roots,  that  depend  upon  the  parameters,  are  taken,  in 
conjunction  with  the  appropriate  values  of  iv,  to  be  the  upper  limits  of 
the  integral.  The  quantity  W  {iv,  z)  is  a  polynomial  in  iv  and  in  z,  of  degree 
not  higher  than  ?i  — 2  in  iv.  The  quantity  M  {z)  is  a  polynomial  in  z\ 
it  may  be  a  constant ;  if  it  is  variable,  no  one  of  its  roots  may  be  one  of  the 
quantities  x^,  ...,  x^  belonging  to  the  upper  limits  of  the  integrals.  And 
lastly,  the  symbol  @  requires  the  various  algebraic  operations  specified 
in  its  definition,  connected  with  the  roots  of  if  (^)  =  0  and  of  G  {z)  =  0 
as  well  as  with  an  expansion  in  descending  powers  of  z. 

Usually  we  have  G  {z)  =  l;  but  even  when  it  is  different  from  unity, 
its  roots  frequently  contribute  only  zero  terms  to  the  final  sum  on  the  right- 
hand  side. 

Note.  The  preceding  proof  dispenses  with  many  of  the  properties  of  functions  of 
position  on  a  Kiemann's  surface  that  have  already  been  estabhslied  ;  the  main  reason 
why  such  a  proof  is  given  is,  that  some  notion  of  Abel's  theorem  may  be  obtained  on  the 
lines  solely  of  Abel's  analysis.  We  shall,  however,  in  the  proof  of  other  results,  use  more 
freely  the  properties  of  functions  of  position  to  which  reference  has  just  been  made. 

Another  method*  of  obtaining  the  result  is  to  consider  the  integral 
taken  over  the  Riemann's  surface,  where  /  denotes 

'  "-N  g  W  {VJ,  Z)  ^_ 

M(z^S 
ow 

If  is  left  as  an  exei'cise  :  it  follows  the  lines  of  §  §  230 — 238. 

*  This  is  Neumann's  method,  Vorlesungen  ilber  Eiemann's  Theorie  der  AbeVsehen  Integrale, 
pp.  285—303. 


252.]  EXAMPLES  587 

Ex.  1.     Let  the  permanent  equation  be 

and  take 

aw  +  hz-\-c=0 

as  the  conditional  equation.     There  are  three  quantities  .rj,  a'2,  x^  given  as  the  roots  of 

a?{Az^-g^z-g^)-{hz  +  cf  =  0. 

Now  take  the  integral  of  the  first  kind  associated  with  the  equation ;  it  is 

^dz 


"We  have  M{z)  =  \,  C{z)  =  l,  W{;w,  z)  =  \;  and  the  roots  of  the  permanent  equation  can  be 
taken  as  w\  —vf.     Hence  in  the  operation  0  we  have  only  to  obtain  the  coefficient  of 

-  in  a  descending  expansion,  so  that 

z 

s    /"'^o-fis  d  1  \ 

2    / A=-Cx\-\og{aw->rhz-\-G) \Qg{-aw-{-hz-\-c)\ 

=  lj        ''<'  -..  \y^  w  J 


o-=l 

~    1  ,      hz-\-c-\-aw 

=  -  (7i  -  log  -. . 

r  10         oz  +  c  —  aw 

When  the  quantity  on  the  right-hand  side  is  expanded  in  descending  powers  of  z,  the  first 

ttZ  1 

term  is  — - ,  so  that  no  term  in  -  exists.     Accordingly 
2z^  ^ 

3     /■  '"-V  dz 


where  ^  is  a  constant  independent  of  the  arbitrary  constants  in  the  conditional  equation, 
and  therefore  determinable  by  assigning  special  values  to  those  constants.  Taking  a=0, 
6  =  0,  the  three  values  oi  x  become  each  infinite,  so  that 

3    /"-Vri^     ^ 
2  -  =  0. 

Now  let  x^  =  p  {u^),  for  a=  1,  2,  3 ;  the  last  equation  can  be  written 

i<l  +  i;2  +  M3  =  0- 

Also  y^  =  p'  {u^) ;  so  that,  as  the  equation 

holds  for  cr  =  l,  2,  3,  we  have 

F(«i),     F(«iX     1     =0. 

P'("3),        ^i.lh),       1 

provided  u-^^  +  U2-\-u^  =  0.     This  is  a  known  form  of  the  addition-theorem. 
Again,  we  have 

Also,  from  the  equation  whose  roots  are  ^j,  x^,  a's,  we  have 

_  6^  _  1  (y\-yj^ 
X1+X2+X3— -—p,— 


4a-     4  \x-^  -  x% 


588  EXAMPLES   OF  [252. 

that  is, 


4  i^(«a)-p(«2) 
another  form  of  the  addition-theorem. 


[zdz 
Ex.  2.     Consider,  in  connection  with  the  same  equations,  the  integral  I  —  .     "We  have 


3    [^(^zdz     „         n   z^      hz-\-c  +  aw 

2    I B=  -  6i  —  log  , 

^^^J         ic  -IV     °bz  +  c-aiv 

in  the  same  way.     Now,  for  the  descending  expansions,  we  have 
,      /bz  +  c  +  awX  2az-^ 

^"gU+^-^w^^'°^"^^''^"^ — 6 — 

1 T  +  ... 

2a3'2- 


so  that 


and  therefore 


=  log(-l)+— +..., 


^   2 ,      bz  +  c  +  avj      h 

w  —  log  i-- = i^  . 

-vj       oz-\-c  —  aw    2a' 


3      /"*V^c^0       ^  6 


B  being  a  constant   independent   of  a,   b,   c.     By  taking   b  =  0,   c  =  0,  and  a  not  zero, 
we  have  e^,  62,  e^  as  simultaneous  values;  hence 

Now  in  the  integral   I  ''-^,  let  s  =  p  (?i),  ic  =  p'  {u) ;  it  becomes 

P{v)du=-((u). 


Also  ?fi,  2*2 J  ^*3  corresponding  to  x-^,  x.2,  ^'3,  are  such  that  ?<^  +  z<2+%  =  0;  e^,  62;  ^3  ^I'e 
possible  values  of  x^,  X2,  x^,  and  we  can  take  ±imi,  ± ^2,  ±0)3  as  possible  values  of  u^,  u^^  u^. 
Choosing  them  so  that  the  sum  shall  still  be  zero,  take  coi,  —0)2,  0)3  as  the  values;  then 

n«,)-f(»,)+a%)+fM+f(«,)-n»,)=-i^g;|^f(^. 

But      ■ 

C(«2)='7  +  V  =  CK)  +  a«3);     • 

and  therefore 

when  iij  +  ?t2  +  M3  =  0:  in  accord  with  Ex.  3,  §  131. 

Ex.  3.     Consider,  also  in  connection  with  the  same  equations  as  in  Ex.  1,  the  integral 

r      dz 


252.] 


ABELS   THEOREM 


589 


where  g  is  a  constant.     Denote  by  y,  —  y,  the  two  vakies  of  iv  {ox  z=g:  so  that,  \i  g  =  <^  (a), 
then  y  —  p'  (a).     We  have 


3    r^ 


dz 


(7=e 


1    1 1  1      hz  +  c  +  aw 

—  [Qty . 

_z-gjiv     °  bz  +  c-aw' 


{z-g)w 
as  before.     In  effecting  the  operations  for  6,  we  note  that 


1 


hz  +  c  +  au' 


Ci  -  log^ 
-2V        bz  +  c  —  aw 

from  Ex.  1 ;  and  therefore,  a  fortiori, 


=  0 


^         1        ,      hz-\-c  +  avj     „ 

Ci  -. r —  log -, =0 : 

-{z  —  g)zo        bz+c-aw 

consequently,  we  shall  have  only  the  coefficient  of to  retain.     Thus 

S     f^<T       dz                  1  bg  +  c  +  ay' 

a=iJ       (2-S')«'            y  bg  +  c~ay' 

where  C  is  a  constant  independent  of  a,  b,  c.  To  determine  (7,  let  a  =  0,  &=0;  then 
CO  is  the  value  of  Xx,  xr^,  ^3 :  and  we  have 


x^{^-9)'i<^     y^"*6(/  +  f  +  ay 

Pia)  , 

-P'(a), 

P{^'-l), 

rK), 

1      , 

= OO- 

P  («2)> 

rw, 

P{a), 

P'M , 

p  (^'1), 

F(«i), 

P{U2), 

F(«2), 

where  x^,  x<i,  x^  =  p  {u^,  p  {u^,  p  {u^,  and  Wj  +  2«2  +  %  =  0. 

Ex.  4.     Prove  that,  with  appropriate  limitations  upon  the  paths  of  the  variables, 

dz       _    [V       dz 


[^       dz  fy       dz  {^  _dz 

jo(l_23)l       J0(l_s3)t      ;0(l_s3)t' 


0  (1-23)3       J   0   {\-z^Y      J    0(] 

provided  x^  +  y^  =  l. 

Ex.  5.  Obtain  the  addition-theorem  for  the  Jacobian  elliptic  functions  in  the  same 
way  as  that  theorem  for  the  Weierstrassian  elliptic  functions  was  obtained  in  Example  1. 

Obtain  also  in  this  way  the  known  theorems  for  the  (Jacobian)  second  elliptic  integral 
and  for  the  (Jacobian)  third  ellijjtic  integral. 

Ex.  6.  Obtain  the  addition-theorem  for  the  doubly-periodic  functions  associated  with 
the  equation 

vfi-\-z^-^aicz  =  l.  (Dixon.) 

Ex.  7.     Two  dependent  variables  it  and  v  are  determined  by  the  (algebraical)  equations 
F {u,  V,  z)—0,         G  {u,  V,  z)  =  0, 
and  a  series  of  values  of  z  are  determined  by  the  association  of  an  algebraical  equation 

ir(u,  V,  z)  =  0 
with  F=  0,  0  =  0:  say  Xi,  x-i,  ...,  x  .     Prove  that 


2 

o-=l 


'^  Tr(M,  v,z) 


M{z) 


dz 


=  e 


W  log  B 


+ constant, 


590 


APPLICATIONS   OF 


[252. 


where  W  is  an  integral  polynomial  in  its  arguments,  M  is  any  polynomial  in  z,  and 
the  summation  on  the  right-hand  side  extends  over  all  the  simultaneous  m-  and  ■y-roots  of 
F=^0,  G  =  0. 

Ex.  8.     Using  the  result  of  Ex.  7,  and  taking 


as  a  special  case,  jjrove  that 


1-22, 

«2  = 

--1- 

F'z\ 

Si,       Ci, 

du 

1 

=  0, 

§2 )       C2 , 

a-2, 

1 

S3,       C3, 

d?., 

1 

Si,       C4, 

di, 

1 

4 
2 

r=l 
V 

E{u,) 

1  +  kHiS-is^Si 

4 
cr=l 

4 
2 

o-=l 

(where  Sj,  Cj,  c^j  ,=sn  %,  en  Wj,  dn  «i,  and  so  for  the  others),  if 

Ui  +  ?(2  +  '^h  +  %  =  0. 
In  the  same  case,  viz.  when  Ui  +  u.^  +  u^  +  tti^O,  shew  that 


and  obtain  an  expression  for 


II.  Reverting  to  the  general  theorem,  it  is  clear  that,  when  the 
operations  on  the  right-hand  side  have  been  completed,  and  the  various 
combinations  of  the  parameters  have  been  expressed  as  functions  of  the 
variables  in  the  upper  limits  of  the  integrals,  the  expression  can,  at  the 
utmost,  involve  only  logarithmic  and  algebraic  functions.  Logarithmic 
functions  will  occur  in  connection  with  each  separate  factor  of  M  {z),  and 
no  other  part  of  the  operation  0  will  give  rise  to  them ;  so  that,  if  the 
right-hand  side  is  to  be  free  from  logarithmic  functions,  M{z)  must  be 
unity  (that  is,  there  are  no  factors),  and  conversely.  Algebraic  functions 
will  occur  when  the  coefficient  of  z~'^  in  the  descending  expansion  in 
powers  of  z  is  different  from  zero ;  the  usual  (but  not  necessary)  form  in 
which  the  coefficient  is  zero  arises  when  the  expansion  begins  with  a 
power  2^"'^+'^',  where  [x  is  a  positive  commensurable  quantity. 

We  proceed  to  apply  the  theorem  to  the  integrals  which  give  rise  to 
functions  of  the  first  kind,  functions  of  the  second  kind,  and  functions  of 
the  third  kind,  respectively  upon  the  Niemann's  surface. 

Integrals  of  the  first  kind.  In  this  case,  we  have  M  (z)  =  1 ;  and 
W{w,z)  is  then  an  adjoint  polynomial  of  order  not  greater  than  n  —  S 
in  w  (§§  205,  234).  Because  M{z)  is  unity,  there  is  no  logarithmic  term 
resulting  from  the  operation  @. 

The  integral,  being  of  the  first  kind,  is  everywhere  finite :  so  that,  when 
I  ^  I  is  very  large,  we  have 


df  z" 


dw 


+ 


252.]  Abel's  theorem  591 

where  A^  is  determinate  only  save  as  regards  multiples  of  the  periods  of  the 
integi'al,  and  k  >0,  being  a  positive  integer  unless  z=  qo  is  a  branch-point. 
Hence,  in  the  vicinity  of  ^^  =  oo  ,  we  have 

W  (w,  z)  _-  kA-^ 

Moreover,  log  Q  (w,  z),  when  expanded  in  descending  powers  of  z,  contains  no 
term  zf  having  a  positive  integer  for  its  index  p.     Consequently,  in 

W{iu,  z)  ^      ar        \ 
(7i — ^^  log  e  (w,  z) 

no  term  in  -  occurs,  for  any  of  the  branches  w ;  and  therefore 

.  ^  of 


Hence 


^M   W  (w,  z) 


dz  =  K, 


dw 

where  K  is   a   constant   of  integration   independent   of  the   parameters  in 
e  (w,  z). 

The  constant  K  can  be  obtained  by  taking  the  sum  of  the  integrals 
as  determined  by  a  different  set  of  parameters  in  6  which  (as  ^  is  a 
constant  independent  of  their  value)  may  be  made  as  particular  as  we 
please.  Moreover,  we  recall  the  property  that  an  integral  of  the  first 
kind  upon  a  Riemann's  surface  is  not  entirely  determinate,  its  expression 
being  subject  to  additive  integral  multiples  of  its  periods:  and  we  have 
the  theorem  that  the  sum  of  the  values  of  an  integral  of  the  first  kind 
at  the  positions  on  a  Riemann's  surface,  tvhere  a  polynovnal  6  (w,  z) 
vanishes,  remains  unaltered  no  matter  hoiu  the  parameters  in  that  poly- 
nomial are  changed:  subject  always  to  modification  by  additive  multiples 
of  the  periods.  The  theorem  is  also  expressed  sometimes  as  follows :  If 
u  (w,  z)  denote  an  integral  of  the  first  kind,  then 

1  v{w^,z„.)=  S  u{y<T,Xa), 

o-  =  l  o-  =  l 

tvhere  u\,  z^;  ... ;  ly^,  z^;  are  the  zeros,  and  y^,  x^;  ...;  y^,  x^;  are  the  infini- 
ties of  any  function  <p  that  is  rational  on  the  Riemann's  surface ;    and  the  ' 
moduli  of  the  congruence  are  the  periods. 

Ex.  1.     Obtain  the  result  in  the  latter  form  by  considering  the  integral 

d 
u  (tv,  z)  -J-  {log  (p  (tv,  z)}  dz 

upon  the  Kiemann's  surface,  (^  denoting  the  rational  function. 


Ji'-lf^  ' 


592  '  EXAMPLES   OF  [252. 

Ex.  2.  The  preceding  result  can  be  otherwise  obtained  :  one  method  is  indicated 
in  the  preceding  example :  another  is  as  follows,  based  upon  a  theorem  due  to  Jacobi*, 
which  will  be  proved  in  deducing  the  result. 

We  have  /  (?t',  0)  =  0  ;  and  therefore 

O'lO  oz 

so  that 

,       dz       div 

dio  dz 

Also  with  the  variation  of  the  parameters  in  6  {w,  z)  =  0,  we  have 

^—div  +  ;^dz+86  =  0, 

CIO  oz 

and  therefore 

d^- 

Ji 

say.     Accordingly,  u  denoting  the  integral  of  the  first  kind, 

du=  U  {ii;  z)  d^ 

_U8d 
~    J    ' 

so  that  this  is  the  element  of  the  integral  to  be  considered,  the  sum  being  taken  for 
the  simultaneous  roots  of 

f{w,z)=0,         6(w,z)  =  0. 

Take  /  to  be  a  polynomial  in  vj  and  z,  and  let  n  be  the  degree  of  terms  of  the  highest 
order :  also  assume  /  to  be  quite  general.  Similarly  take  5  to  be  a  polynomial,  and  let  m 
be  the  degree  of  terms  of  the  highest  order ;  m  can  be  taken  as  less  than  n,  but  that  does 
not  prove  important  for  the  present  purpose.  Let  Z=0  denote  the  result  of  eliminating 
IV  between  /=0  and  ^  =  0,  so  that  Z  is  of  degree  m?i  in  z ;  likewise  let  W  denote  the  result 
of  eliminating  z  between  /=0  and  ^  =  0,  so  that  W  is  of  degree  mn  in  w.  Let  the  roots  of 
Z=0  be  Xi,  ...,  x^nl  those  of  ^-"=0  be  ?/i,  ...,  y,„„ ;  and  let  the  simultaneous  roots 
of  /=0,  0  =  0  be 

which  will  be  called  the  congruous  roots.  The  other  combinations  will  be  called  the 
non-congruous  roots. 

By  the  known  theory  of  elimination,  we  have 

Z=  Af+  B6,  W=  Cf+  D6, 

where  the  orders  in  w  and  z  are : 

A,    m-1  in  w    ,  mn -n  in  z  , 

5,    n  —  1  in  w     ,  17171  — m  iia  z, 

G,     mn—n  in  w,  on  —  l  in  z     , 

JD,    mn  -  m  in  «•,  n  —  \\nz     ; 

and  therefore  A,  =AD-BC,  is  a  polynomial  whose  highest  power  in  ic  alone  is  mn-l, 
whose  highest  power  in  z  alone  is  mn—1,  and  whose  highest  terms  in  w  and  z  combined 
are  of  order  27nn-m-7i,  that  is,  A  is  a  polynomial  of  order  imn-m  —  n.      Now  the 

*  Ges.  Werke,  t.  iii,  p.  292. 


252.] 


ABEL'S   THEOEEM 


593 


congruous  roots  make  /  and  6  vanish ;  the  non -congruous  roots  do  not  make  /  and  d 
vanish,  while  they  do  make  Z  and  W  vanish;  accordingly,  the  non-congruous  roots  make 
A  vanish. 


We  have 


CIV  OW       ,  OW  OW  010 


dZ 


=  a¥-,b'' 


-4--r' 


dz      "82         dz   , 

when  congruous  roots  are  substituted  after  differentiation  :   so  that,  for  any  such  con- 
gruous pair, 


Z'W'  = 


and  therefore 
whence 


OW        ctv  OW         div 


oz         dz 


=  A/, 


rr  h6      UA8d 
1^  M.   /UA8d\ 


where  fi  =  77in.  But  A  =  0  for  the  non-congruous  roots,  and  therefore  the  expression 
Ca86-^{Z'  W)  vanishes  for  non-congruous  roots,  assuming  (as  we  may)  that  Z  and  W  do 
not  possess  equal  roots ;  hence,  adding  these  vanishing  terms,  we  have 

2  dii^=  2     2    [yrTy7) 


M   M  /UA8e\ 

=   22    (  vr-jT^j    , 


say. 


Consider  an  expression    -rpr^,  where  6  is  a  polynomial  in  w  and  z  of  dimensions  not 

greater  than  2??i?i-3:  and  express  it  in  partial  fractions.      As  the  denominator  is  of 
the  form    WZ,  that  is,  is 


we  have 


Clearly 


Now 


9 
WZ'- 


n  (w-^j)  n  (z-Xi), 
i=i  t=i 

2 h    2    — ^—+   2     2 


a 


-.%■ 


lj=l{z-Xi)(w-i/j)' 


Cii  — 


WZ' 


IJ.  IX 

2     2   C'ij 
•  i=l  i=l 

1  G 

is  manifestly  the  coefficient  of  —  in  the  expansion  of  rn-y  in  descending  powers  of  w  and  z. 

As  e  is  of  dimensions  2??^7^-3,  while    W  is  of  dimensions  mn  and  likewise  Z,  the  first 
term  in  that  expansion  is  of  dimensions  -  3,  so  that  there  is-no  term  in  — .    Consequently 


IJ.    IJ. 

2    2  Cij=0, 
i=l  i=l 


F.  F. 


38 


594  APPLICATIONS   OF  [252. 

that  is,  ^ 

This  leads  at  once  to  the  result.      For  the  most  general  integral  of  the  first  kind,   * 
U  is  an  adjoint  polynomial  of  order  n  —  3;  so  that   UaSO  is  a  polynomial  of  order 

{71  —  3)  +  {27nn  —  m  —  n)-\-  m, 
that  is,  of  order  2mn  —  3.     Consequently 

M    M   /UAd6\ 

and  therefore  2  (itio-  =  0, 

(7=1 

or  2  «    is  a  constant,  independent  of  the  parameters  in  6  and  therefore  independent  of 

<T=1 

the  upper  limits  of  the  integrals :  which  is  the  proposition  to  be  established. 

Note.     Jacobi's  theorem,  indicated  at  the  beginning  of  this  proof,  is  an  immediate 
inference  from  the  preceding  analysis,  viz. 


where  F  is  any  polynomial  of  order  not  greater  than  to  +  to-3. 

Integrals  of  the  second  kind.  We  shall  take  only  the  normal  elementary 
integral  of  the  second  kind,  and  denote  by  e  =  a,  w  =  a  its  one  infinity  on  the 
surface.     Then  (§  208)  the  subject  of  integration  is 

U{w,  z) 

{z-afp 
ow 

where  ?7  is  a  polynomial  of  degree  n  —  \,  which  vanishes  at  the  n  —  1  points 
other  than  lu  =  a  given  by  ^r  =  a,  and  also  vanishes  to  order  A,  —  1  at  a  branch- 
point of/  which  is  of  order  A,.     Then 


>    I      '  .dz  =  S 

At 

(z  -  of 


<r  =  l\  ._  „V2^/ 


1 


_  {z-af\r=\\      _^ 

dw  V       dWr 


s  \ES:!^l^\oge{wr,z)\  +K, 


where  K  is  an  arbitrary  constant,  independent  of  the  parameters  in  6  (w,  z). 
In  order  to  effect  the  operations  in  @,  we  require,  first,  the  coefiicient  of 

in  the  expansion  of 

js  —  a  ^ 

{z-a)-r=i       _0J_  , 

dWr 

in  ascending  powers  of  z  —  a:    and  secondly,  the  coefficient  of  -  in  the  ex- 

1        "    U(w.-  z) 
pansionof  _____  ^^—^  log^  K,  .), 

dWr 

in  descending  powers  of  z. 


252.]  Abel's  theorem  595 

The  latter  is  manifestly  the  coefficient  of  z  in  the  expansion  of 

Z  — ^. —  log  e  {Wr,  z) 

dWr 

in  descending  powers.  Now  our  integral,  being  finite  everywhere  on  the 
surface  except  for  the  place  where  ^  =  a,  is  finite  at  z  =  qo  ,  and  therefore 

U  (Wr,  Z)        K2       K3 

when  I  ^  I  is  very  large :    (there  is  no  term  —  because  the  integral  has  no 

logarithmic  infinity).  Moreover  log  6  (wr,  z),  expanded  in  descending  powers, 
has  no  positive  powers  of  z.  Hence  the  coefficient  of  z  in  the  preceding 
expansion  is  zero ;  and  therefore  the  part  in  0  contributed  by  the  coefficient 

of  -  is  zero. 

z 

To  obtain  the  coefficient  of  — in   the  expansion,  it  is  sufficient  to 

0  ^  ct 

obtain  the  coefficient  of  z  —  a  in  the  expansion  of 

2  \  log  e  (lUr,  Z). 

dWr 

Take  z  —  a  =  ^,  lUr  =  a,-  +  v^  (say  w  =  cc  +  v);  then,  as  in  §  208, 


U(w,  z)=  U(a,  a)+  ^ 


da 

KHLU) 


a/-  e  («,  a) ' 

da 


¥  (¥]'  t^(«.  ")  1  §/■' 

diu       [         \daj  j  da. 

loge  (w,  z)  =  loge  (a,  a)  +  ^.^^.l^^^. 
°     ^    '    ^         o     \  J    /     df  6  {a,  a)  d  (a,  a) 

da 

The   points  z  =  a,  w  —  a^,  «3,  ...,  «„  are  neither  algebraic  nor  logarithmic 
infinities  for  the  integral ;    hence  (§  208) 

U{ar,  a)  =  0,     {r  =  2,...,n), 

38—2 


596  ABEL'S  THEOREM  AND  [252. 

•raic  infinity,  and 
0,     {r=%...,n), 


which  conditions  exclude  an  algebraic  infinity,  and 


9  (a,.,  a) 

which  conditions  exclude  a  logarithmic  infinity,  at  any  of  the  n  —  1  places. 

The  point  z  =  a,  w  =  a^  is  an  algebraic  but  not  a  logarithmic  infinity ; 
hence 


U(a„a)'^V'daJ  _d(f,  U) 


df         d{ai,  a)       3(ai,  a) ' 

which  excludes  the  logarithmic  infinity  for  the  integral.  The  algebraic 
infinity  at  Qj,  a  for  the  integral  is  clearly 

U(a,,  a)     1 
df       z  —  a' 

The  coefficient  of  ^  in  the  foregoing  expansion  is 

U{a„a)        1        dif,e) 
df        d{a^,  a)  3 («!,«)' 

Hence 

^    f-<r    U(tv,  z)    ^^^     ^_  U(a„  a)        1        d  (f,  6) 

The  right-hand  side  is  a  rational  function  of  the  parameters  in  6  and  there- 
fore is  an  algebraic  function  of  the  variables  x.  Hence  the  sum  of  the  values 
of  a  normal  elementary  integral  of  the  second  kind  at  the  -positions  on  a 
Riemanns  surface,  where  a  polynomial  vanishes,  is  an  algebraic  function 
of  the  variables  of  those  positions. 

The  result  can  also  be  enunciated  in  the  form :   Let  E  {w,  z)  denote  a 

normal  elementary  function  of  the  second  kind,  having  its  sole  simple  infinity 

A 
at  the  position  z  =  a,  iv  =  a,  of  the  form  — ^ — .     Also  let  lu^,  z^\  . . .  ;  w^,  z^; 

z  —  a 

denote  the  zeros:  and  i/i,  x-^;  ...  ;  y^,  x^  denote  the  infinities  of  a  function  (f>, 

rational  on  the  Riemann  surface:  then 

^  E{w„,z,)-  ^  E{ya,x^)  =  -A     -^^ — ^ 

The  equality  is  subject  to  additive  multiples  of  the  periods  of  the  integral. 

Ex.  1.     lu  the  preceding  investigation,  the  assumption  is  made  that  a  is  finite,  so  that 
?=ao  is  a  point  where  the  integral  is  finite. 

Obtain  the  corresponding  result  when  2  =  oc  is  the  sole  simple  infinity  of  a  normal 
elementary  integral  of  the  second  kind. 


252.]  THE   ELEMENTARY   INTEGRALS  597 

Ex.  2.  The  line  aw 4-/32  + 7  =  0  touches  the  curve  f{^o,  z)=0,  and  U  {w,  z)  is  an 
adjoint  polynomial  of  order  n  —  2,  which  vanishes  at.  each  of  the  remaining  n  —  2  points 
•where  the  line  meets  the  curve.     Obtain  an  expression  for 

M       f'^a-      U  {W,  Z)       dz 

where  the  upper  limits  of  the  integrals  are  the  points  of  intersection  oif{w,  z)  =  0  with  any 
other  curve  6  {w,z)=0;  also  an  expression  for 

U  {w,  z)       dz 


<r-li       {aW  +  ^Z  +  y)\^' 

dw 
where  k  is  an  integer  greater  than  unity. 

Integrals  of  the  third  kind.     We  shall  take  only  the  normal  elementary 

integral  of  the  third  kind ;  and  we  shall  denote  by  z  =  c,  w  =  'yi;  2  =  c,  w  =  y^', 

its  two  logarithmic  infinities  on  the  surface.     Then  (§  211,  Ex.  2)  the  subject 

of  integration  is 

V(w,z) 

where  V(iu,z)  is  an  adjoint  polynomial  of  order  ??  —  2,  which  vanishes  at 
the  n  —  2  points,  other  than  the  infinities  of  the  integral.  Moreover  as 
V  is  of  order  less  than  w  —  1  in  w,  we  have,  by  a  known  theorem  in  partial 
fractions, 

«     V  {Wr,  Z)  _ 

so  that,  in  the  present  case, 

say,  thus  verifying  the  property  proved  in  §  211. 
We  have 


{z-c) 


<r=l    I  /_         „x   9/ 


1 


z  —  c 
dw  dw. 


2    ^ log^(w^,  ^r), 


where  L  is  an  arbitrary  constant,  independent  of  the  parameters  in  6  (w,  z). 

In  the  same  manner  as  for  the  normal  elementary  integrals  of  the  second 
kind,  we  have 

Ci S  . — -  log  6  (wr,  z)=0: 

z  ^  —  C  r  =  l  Of 

dWr 


598  APPLICATIONS  OF  [252. 

and  therefore  the  right-hand  side  is 

S — ^ — \oge{yr,c). 

r=l         0/^ 

In  this  sum  all  the  terms  vanish  except  for  r  =  1,  ?^  =  2  :  it  thus  becomes 

2 ^ —  log  e  (7,,  c) 


and  therefore 

dw 

Proceeding  as  before,  this  result  may  be  enunciated  as  follows :  Let 
IliaCw,  z)  denote  a  normal  elementary  function  of  the  third  kind,  having  its  two 
logarithmic  infinities  at  z  =  c,  w  =  y^ ;  z  =  c,  tu  =  jo',  of  the  form 

C  -C 

+ , :  + , 


z  —  c  z  —  c 

respectively.     Also  let  (f)(w,z)  denote  a  7^ational  function  of  position  on  the 

Riemann's  surface,  having  its  zeros  at  w^,  Zj]  ;  w^j^,  z^',  and  its  poles  at 

2/1 }  ^1 5  * J  Vi*- '  '^M •     -^  lien 

i  u,,{w.,z.)-  i  n,,(y.,.^.)=aiog|^^. 

The  equality  is  subject  to  additive  multiples  of  the  periods  of  the  integral. 

Ex.  Let  U{w,z)  be  an  adjoint  polj'nomial  of  order  %— 2,  vanishing  at  n  —  'i  of  the 
points  where  the  (non -tangent)  line  aw-'t-^z-\-y  =  0  meets  the  curve.  Obtain  an  expression 
for 

^a-      U  {lO,  Z)       dz 


2  

o-=li      a^  +  /32  +  y  9/"' 

where  the  upper  limits  of  the  integrals  are  the  points  of  intersection  of  f  {u;  s)  =  0  with  any 
other  curve  6  (w,  s)  =  0.  (This  is  more  general  than  the  case  in  the  text,  where  z  is  taken 
to  have  the  same  value  at  the  two  points.) 

III.     Abel's  theorem  in  general,  and  the  special  examples  in  particular, 
shew  that  the  sum  of  a  number  of  integrals 

I  {w^,  z„),     =1    "    " dl, 

can  be  expressed  as  a  logarithmic  and  algebraic  function,  say  G,  so  that 

<T  =  1 


252.]  Abel's  theorem  599 

and  therefore 

cr  =  l  <r=p+l 

say,  where  p  is  the  number  of  parameters  in  6  (tv,  z).  As  there  are  fx  roots  of 
the  equation  F{z)  =  0,  or  fi  simultaneous  sets  of  roots  of 

f{iv,  z)  =  0,     d  {w,  z)  =  0, 

which  depend  upon  the  parameters,  it  follows  that  the  parameters  can  be 
regarded  as  functions  of  p  of  these  roots,  and  that  the  remaining  im  —  p  roots 
can  then  be  regarded  as  functions  of  the  p  roots.  At  this  stage,  these  p  roots 
may  now  be  considered  as  arbitrary  quantities  ;  and  therefore  the  sum  of  a 
number  of  integrals  with  arbitrary  upper  limits  can  be  expressed  as  a  sum 
(with  an  additive  function  G)  of  a  number  of  integrals  with  other  upper 
limits,  which  are  algebraic  functions  of  those  arbitrary  quantities.  The 
question  then  arises :  what  is  the  smallest  number  of  integrals  in  the  second 
set  as  a  sura  of  which  any  number  of  integrals  can  be  expressed  ? 

This  number  is  equal  to  p,  the  genus  of  the  permanent  equation.  Abel 
originally  obtained  the  number,  though  not  in  this  special  form  :  the  investi- 
gation is,  however,  a  long  one. 

The  result  can  be  more  briefly  established  by  using  the  properties  of 
functions  on  a  Riemann's  surface  which  have  already  been  proved.  It  will 
suffice  to  establish  the  result  for  a  sum  of  p  +  1  integrals  with  arbitrary 
upper'  limits ;  for,  granting  the  result  for  this  case,  then  a  sum  of  p  +  2, 
=  (^p-j- 1)4- 1^  integi-als  can  be.  transformed  into  a  sum  of  ^  +  1  integrals  by 
transforming  the  sum  of  any  p  +  l  in  the  p  +  2  into  a  sum  of  p  integrals ; 
and  so  for  the  sum  of  a  greater  number.  There  are  two  cases,  according  as 
the  equation  is  hjrperelliptic,  or  is  not  hyperelliptic. 

When  the  equation  is  hyperelliptic,  we  take  it  in  the  form 

Any  other  uniform  function  on  the  Riemann  surface  can  be  made  of  only  the 

first  degree  in  w;    and  we  therefore  take  the  conditional  equation  in  the 

form 

d(w,  z)=tv-®  =  0. 

The  upper  limits  of  the  integrals  satisfy  the  equation 

Let  ^1,  ...,  %4-i  be  ^  + 1  arbitrary  quantities,  taken  to  be  upper  limits  for  an 

integral   fdl :    and  choose  S  to  be  a  polynomial  in  z  of  degree  p  having 

arbitrary  coefficients.     The  most  general  form  of  @  contains  p  +  1  of  these 

coefficients;    they  therefore   can   be   chosen   so  that  z-^,  ...,  Zp^^^  satisfy  the 

equation 

7       _  (h)2  =  0 


600  ABEL'S  [252. 

and  they  then  are  determinate.  The  equation  is  of  degree  2p  +  1,  and 
consequently  it  has  other  p  roots,  say,  %,  ...,  ccj^:  these  are  functions  of  the 
constants  in  ©  and  therefore  are  functions  of  z^,  ...,  z^^-^,  being  manifestly 
algebraical  functions.  Now  applying  Abel's  theorem  to  the  integral  ^dl,  we 
have 

S  / (w,7,  ^a)  +  2  / (2/«,  x^  =  G, 

so  that 

(7  =  1  K  =  l 

or  the  sum  of  ^j  +  1  integrals  is  expressible  as  a  sum  (with  an  additive 
function  G)  of  p  integrals.  The  result  is  therefore  proved  for  the  hyper- 
elliptic  case. 

When  the  equation  is  not  hyperelliptic,  we  take  it  in  the  customary  form 

f{w,z)  =  ^, 

where  the  degree  n  off  in  w  is  equal  to  or  is  greater  than  3.    The  conditional 

equation 

e{w,z)  =  0      ' 

is  obtained  by  taking  6(w,  z)  as  the  most  general  adjoint  polynomial  of  order 
n  —  2 :  that  is,  6  (w,  z)  is  to  vanish  to  order  A,  —  1  at  a  multiple  point  of  / 
which  is  of  order  X.  Now  in  order  that  a  point  on  a  curve  may  be  multiple 
of  order  X  —  1,  a  number  of  conditions  equal  to  ^\{X—  1)  must  be  satisfied; 
hence  the  number  of  conditions  to  be  satisfied,  in  order  to  make  6  {w,  z)  an 
adjoint  polynomial,  is 

=  l{n-l){n-2)-p, 

by  §  240.  As  ^  is  a  polynomial  of  order  n  —  2,  the  number  of  terms  it 
contains  is  ^{n  —  l)n:  and  therefore  the  number  of  disposable  constants  in 
the  equation  6^  =  0  is  ■!•(■??-  l)n  —  l.  Some  of  these  are  used  to  make  the 
polynomial  adjoint ;  hence  the  number  of  disposable  constants  in  the  adjoint 
polynomial  of  order  ?;  —  2  is  . 

hAn-\)n-l-[^{n-l){n-2)-p] 

=  n  +p  —  2, 

in  all.  As  n^3,  it  follows  that  7i+p  —  2^p  +  l.  If  the  number  =^  +  1, 
choose  the  disposable  arbitrary  constants  so  that  6  shall  vanish  at  the  p  +  I 
positions  Wj,  2,;  ...;  w^+j,  %+i;  given  as  assigned  upper  limits  for  the 
integral.  If  the  number  is  >^  +  l,  choose  ^  +  1  of  the  constants  so  that  6 
shall  vanish  at  the  assigned  p  +  1  positions  Wg,  z^,  and  the  remaining  ?i  —  3  of 
the  constants,  so  that  6  shall  vanish  at  n  —  S  definite  positions  a^,  a^;  ...; 


252.]  THEOREM  GOI 

dn-z,  cin-s',  on  the  surface,  to  be  regarded  as  fixed  positions  defined  by  pure 
constants. 

In  the  intersections  of  /=  0,  ^  =  0,  a  multiple  point  of  order  \  on/,  being 
of  order  X  —  1  on  ^,  counts  for  X  (X  —  1)  points ;  hence  the  number  of  points, 
other  than  the  multiple  points,  given  by  ^  =  0  on  the  Riemann's  surface  is 

n(n-2)-l\(\-l) 

=  n  {n  -  2)  -  {(».  -  1 )  {n  -^)-2p\ 

=  n  +  2p  —  2 

=  {)i+p-2)+p. 

Of  these,  we  have  the  7i  +  p  —  2,  given  by 

Wg,  2^ff,   for  cr  =  l,    ...,p  +  l, 

a,.,  a,.,  for  o'  =  1,  ...,  n  —  S, 

the  latter  set  not  occurring  if  ?i  =  3  ;  let  the  remaining  p  be 

y^,  x^,  for  K  =  l,  ...,  p: 

so  that  the  upper  limits  for  our  integral  are  the  sets  w^,  z^;  a,.,  a^;  y^,  ^k- 
Applying  Abel's  theorem  to  the  integral  Jdl  with  these  as  limits,  we  have 

p+l  »-3  p 

2  I(iUg,  2„)+  %  I{ar,  «,.)+  S  I(y^,  x^)  =  0. 

<7  =  1  )•  =  !  K  =  l 

Now  each  term  in  the  second  sum  is  a  pure  constant,  so  that,  when  the  sum 
is  transferred  to  the  right-hand  side,  it  can  be  absorbed  into  the  constant  of 
integration  that  occurs  in  G  ;  hence 

p+i  p^ 

S  /(w<r,  2„)  =  -  1  I(y^,  x^)  +  Gi, 

a=l  K=l 

thus  proving  the  result  for  the  case  which  is  not  hyperelliptic. 

This  result,  that  the  sum  of  any  number  of  integrals  can  be  expressed  as  a  sum 
of  p  integrals,  has  of  course  an  entirely  different  significance  from  the  result  that  jo  linearly 
independent  integrals  of  the  first  kind  exist  upon  a  Riemann's  surface.  The  two  results 
can  be  combined,  as  in  §  239,  so  as  to  lead  again  to  the  inversion-problem ;  this 
discussion,  however,  is  a  fundamental  part  of  the  theory  of  Abelian  functions,  and  is 
beyond  our  present  range. 


CHAPTER  XIX. 

CONFORMAL  REPRESENTATION:  INTRODUCTORY. 

253.  In  §  9  it  was  proved  that  a  functional  relation  between  two 
complex  variables  W  and  z  can  be  represented  geometrically  as  a  copy  of 
part  of  the  5^-plane  made  on  part  of  the  w^-plane.  At  various  stages  in  the 
theory  of  functions,  particularly  in  connection  with  their  developments  in  the 
vicinity  of  critical  points,  considerable  use  has  been  made  of  the  geometrical 
representation  of  the  analytical  relation ;  but  it  has  been  used  in  such  a  way 
that,  when  the  equations  of  transformation  define  multiform  functions,  the 
branches  of  the  function  used  are  uniform  in  the  represented  areas. 

The  characteristic  property  of  the  copy  is  that  angles  are  preserved,  and 
that  no  change  is  made  in  the  relative  positions  and  (save  as  to  a  uniform 
magnification)  no  change  is  made  in  the  relative  distances  of  points  that  lie 
in  the  immediate  vicinity  of  a  given  point  in  the  ^^-plane.  The  leading 
feature  of  this  property  is  maintained  over  the  whole  copy  for  every  small 
element  of  area :  but  the  magnification,  which  is  uniform  for  each  element, 
is  not  uniform  over  the  whole  of  the  copy. 

Two  planes  or  parts  of  two  planes,  thus  related,  have  been  said  to  be 
conformally  represented,  each  upon  the  other. 

Now  conformal  representation  of  this  character  is  essential  to  the  consti- 
tution of  a  geographical  map,  made  as  perfect  as  possible :  and  a  question 
is  thus  suggested  whether  the  foregoing  functional  relation  is  substantially 
the  only  form  that  leads  to  what  may  be  called  geographical  similarity.  In 
this  form,  the  question  raises  a  converse  more  general  than  is  implied  by 
the  conveise  of  the  functional  relation,  inasmuch  as  it  implies  the  possibility 
that  the  property  can  be  associated  wath  curved  surfaces  and  not  merely 
with  planes.  But  a  little  consideration  will  shew  that  the  generalisation  is 
a  priori  not  unjustifiable,  because,  except  at  singular  points,  the  elements  of 
the  curved  surface  can,  in  this  regard,  be  treated  as  elements  of  successive 
planes.  We  therefore  have*  to  determine  the  most  general  form  of  analytical 
relation  between  parts  of  two  surfaces  tuhich  establishes  the  'projjerty  of 
conformal  similarity  between  the  elements  of  the  surfaces. 

*  The  following  investigation  is  due  to  Gauss :  for  references,  see  p.  611,  note. 


253.]  CONFORMAL   EEPRESENTATION  603 

Let  X,  y,  z  be  the  cooi'dinates  of  a  point  R  of  one  surface  with  t,  u  for  its 
parameters,  so  that  x,  y,  z  can  be  expressed  in  terms  of  t,  u\  and  let  X,  F,  Z 
be  the  coordinates  of  an  associated  point  R'  of  the  other  surface  with  T,  U 
for  its  parameters,  so  that  X,  Y,  Z  can  be  expressed  in  terms  of  T,  U.  Then 
the  analytical  problem  presented  is  the  determination  of  the  most  general 
relations  which,  by  expressing  T  and  U  in  terms  of  t  and  u,  establish  the 
conformal  similarity  of  the  surfaces. 

Suppose  that  G  and  H  are  any  points  on  the  first  surface  in  the  imme- 
diate proximity  of  R,  and  that  G'  and  H'  are  the  corresponding  points  on 
the  second  surface  in  the  immediate  proximity  of  R':  then  the  conformal 
similarity  requires,  and  is  established  by,  the  conditions :  (i),  that  the  ratio 
of  an  arc  RG  to  the  corresponding  arc  R'G'  is  the  same  for  all  infinitesimal 
arcs  conterminous  in  R  and  R'  respectively;  and,  (ii),  that  the  inclination 
of  any  two  directions  RG  and  RH  is  the  same  as  the  inclination  of  the 
corresponding  directions  R'G'  and  R'H'.  Let  the  coordinates  of  G  and  of  H 
relative  to  R  be  dx,  dy,  dz  and  ^x,  Sy,  Sz  respectively;  and  those  of  G'  and 
of  H'  relative  to  R'  be  dX,  dY,  dZ  and  SX,  SF,  hZ  respectively.  Let  ds 
denote  the  length  of  RG  and  dS  that  of  R'G' ;  let  m  be  the  magnification  of 
ds  into  dS,  so  that 

dS  =  mds, 

a  relation  which  holds  for  every  coiTesponding  pair  of  infinitesimal  arcs 
at  R  and  R'. 

By  the  expressions  of  x,  y,  z  in  terms  of  t  and  u,  we  have  equations  of 
the  form 

dx  =  adt  +  a'du,     dy  =  bdt  +  b'du,     dz  =  cdt  +  c'du, 

where  the  quantities  a,  b,  c,  a',  b',  c  are  finite.  Let  there  be  some  relations, 
which  must  evidently  be  equivalent  to  two  independent  algebraical  equa- 
tions, expressing  T  and  JJ  as  functions  of  t  and  ii ;  then  we  have  equations 
of  the  form 

dX  =  Adt  -}-  A'du,     dY=  Bdt  +  B'du,     dZ  =  Cdt  +  C'du, 

where  the  quantities  A,  B,  C,  A',  B',  C  are  finite  and  are  dependent  partly 
upon  the  known  equations  of  the  surface  and  partly  upon  the  unknown 
equations  of  relation  between  T,   IT  and  t,  u.     Then 

ds-  =  (a^  +  6=  +  c^  df  +  2  {aa  +  bb'  +  cc) dtdu  +  (a'^  +  b'^  +  c'^) du\ 

dS'  =  {A-'  +  B'+  C)  df  +  2(AA'+  BB'  +  CC)  dtdu  +  (A"  +  B''  +  C)  du\ 

Since  the  magnification  is  to  be  the  same  for  all  corresponding  arcs, 
it  must  be  independent  of  particular  relations  between  dt  and  du ;   and 

therefore 

A^'  +  B'  +  C  _  AA'  4-BB'  +  CC'  ^  A'-'  +  B'^  +  C'^ 
a^  +  b^  +  c^   ~     aa'+bb'  +  cc'  a^  +  b'^  +  c'^   ' 

each  of  these  fractions  being  equal  to  m^  ,- 


6G4  CONFORMAL   REPRESENTATION  [253. 

Again,  since  the  inclinations  of  the  two  directions  RG,  RH;  and  RG', 
RH';  are  given  by 

dsSs  cos  GRH 

=  (a-  +  6^  +  C')  dtht  +  {aa'  +  hh'  +  cc)  {dthii  +  Ztdu)  +  (a  ^  +  &'=  +  c"")  duSu, 

dSSS  cos  G'R'R' 

={A'+B'  +  C')dm  +  (AA'+BB'  +  CG'){dtSu+Udu)+{A''+B''  +  C'')duSu, 

we  have,  in  consequence  of  the  preceding  relations, 

nMsSs  cos  GRH=dSSS  cos  G'R'H'. 

But  dS  =  7nds,  BS  =  niBs ;  and  therefore  the  angle  GRH  is  equal  to  the 
angle  G'R'H'.  It  thus  appears  that  the  two  conditions,  which  make  the 
magnification  at  R  the  same  in  all  directions,  are  sufficient  to  make  the 
inclinations  of  corresponding  arcs  the  same ;  and  therefore  they  are  two 
equations  to  determine  relations  which  establish  the  conformal  similarity 
of  the  two  surfaces. 

These  two  equations  are  the  conditions  that  the  ratio  dS/ds  may  be 
independent  of  relations  between  dt  and  du;  it  is  therefore  sufficient,  for 
the  present  purpose,  to  assign  the  conditions  that  dS/ds  be  independent  of 
values  (or  the  ratio)  of  differential  elements  dt  and  du. 

Now  ds'-^  is  essentially  positive  and  it  is  a  real  quadratic  homogeneous 
function  of  these  elements ;  hence,  when  resolved  into  factors  linear  in  the 
differential  elements,  it  takes  the  form 

ds'^  =  7?  {dp  +  idq)  (dp  —  idq), 

where  n  is  a  finite  and  real  function  of  t  and  u,  and  dp,  dq  are  real  linear 
combinations  of  ctt  and  du.     Similarly,  we  have 

dS'^  =  N{dP+  idQ)  (dP  -  idQ), 

where,  again,  iV"  is  a  finite  and  real  function  of  t  and  u  or  of  T  and  U,  and 
dP,  dQ  are  real  linear  combinations  of  dt  and  du  or  of  dT  and  d  U.     Thus 

„  _  N  (dP  +  idQ)  (dP  -  idQ) 
n    (dp  +  idq)  (dp  —  idq) 

It  has  been  seen  that  the  value  of  m  is  to  be  independent  of  the  values  and. 
of  the  ratio  of  the  differential  elements. 

Now  taking 

.     aa'  +  bb'  +  cc'  ,      a'^+ft'^+c'^ 

a'  +  Jf  +  c-  ^       a2  +  6^  +  c" 

so  that  6  and  </>  are,  by  the  two  equations  of  condition,  the  same  for  ds  and 
dS,  and  denoting  by  t/t  the  real  quantity  (<^  —  6")^,  we  have 

ds^  =  (a'  +  b^  +  C-)  {dt  +  du  (6  +  ?»}  [dt  +  du  (6  -  ?»}, 

and  dS-^=  (A""  +  5^  +  C^)  [dt  +  du(e  +  ;»}  [dt  +  du (9  -  ?»}. 


253.]  OF   SURFACES  605 

Then,  except  as  to  factors  which  do  not  involve  infinitesimals,  the  factors  of 
cZi'-  and  of  dS^  are  the  same.  Hence,  except  as  to  the  former  factors,  the 
numerator  of  the  fraction  for  m^  is,  qua  function  of  the  infinitesimal 
elements,  substantially  the  same  as  the  denominator;    and  therefore  either 

.,       dP  +  idQ      .dP-idQ         „   .,  ..       .      .  . 

(a)         , . ,     and    , r^-  are  finite  quantities  snnultaneouslv  ; 

dp  +  idq  dp  —  idq  ^  -^ 


or 


.^,,       dP  +  idQ      ,dP-idQ        .   .,  .  .       .      .  . 

(p)       --. r-^  and  r^ r^  are  finite  quantities  simultaneously. 

dp  —  idq  dp  +  idq  -^ 

Either  of  these  pairs  of  conditions  ensures  the  required  form  of  in,  and  so 
ensures  the  conformal  similarity  of  the  surfaces. 

Ex.     Shew  that  both  p  and  q  satisfy  the  partial  difierential  equation 

Consider  {a)  first.  Since  {dP  +  idQ)/(dp  +  idq)  is  a  finite  quantity,  the 
differentials  dP  +  idQ  and  dp  +  idq  vanish  together,  and  therefore  the  quan- 
tities P  4-  iQ  and  p  +  iq  are  constant  together.  Now  P  and  Q  are  functions 
of 'the  variables  which  enter  into  the  expressions  for_p  and  q;  hence  P  +  iQ 
and  p  +  iq,  in  themselves  variable  quantities,  can  be  constant  together  only  if 

P  +  iQ=fip  +  iq), 
where /denotes  some  functional  form.     This  equation  implies  two  independent 
relations,  because  the  real  parts,  and  the  coefficients  of  the  imaginary  parts, 
on  the  two  sides  of  the  equation  must  separately  be  equal  to  one  another ; 
and  from  these  two  relations  we  infer  that 

where  /  {p  —  iq)  is  the  function  which  results  from  changing  i  into  —  i 
throughout /(p  +  iq)  and  is  equal  to  f(p  —  iq),  if  i  enter  into /only  through 
its  occurrence  in  p  +  iq.     From  this  equation,  it  follows  that 

dP  -  idQ 

dp  -  idq 
is  finite  ;  and  therefore  a  necessary  and  sufficient  condition  for  the  satisfaction 
of  (a)  is  that  P,  Q  and  p,  q  be  connected  by  an  equation  of  the  form 

P  +  iQ=f(p+iq)- 
Moreover,  the  function  /  is  arbitrary  so  far  as  required  by  the  preceding 
analysis;    and   so   the   conditions   will    be   satisfied,   either  if  special   forms 
of  /  be   assumed   or  if  other  (not  inconsistent)  conditions  be  assigned   so 
as  to  determine  the  form  of  the  function. 

Next,  consider  (/3).     We  easily  see  that  similar  reasoning  leads  to  the 

conclusion  that  the  conditions  are  satisfied,  when  P,  Q  and  p,  q  are  connected 

by  an  equation  of  the  form 

p  +  iQ  =  g(p-  iq)  ; 


606  gauss's  [253. 

and  similar  inferences  as  to  the  use  of  the  undetermined  functional  form  of  g 
may  be  drawn.     Hence  we  have  the  theorem  : — 

Parts  of  two  surfaces  may  he  made  to  correspond,  point  by  point,  in 
such  a  way  that  their  elements  are  similar  to  one  another,  by  assigning 
any  relation  between  their  parameter's,  of  either  of  the  for-ms 

P  +  iQ=f{p  +  iq),     P  +  iQ^g(p-iq); 

and  every  such  correspondence  between  two  given  surfaces  is  obtained  by  the 
assignment  of  the  proper  functional  form  in  one  or  other  of  these  equations. 

Ex.  In  establishing  this  conformal  representation,  only  small  quantities  of  the  first 
order  are  taken  into  account.  Sketch  a  method  whereby  it  would  be  possible  to  evaluate, 
to  a  higher  order  of  small  quantities,  the  magnitude 

dS'     ds' 

where  dlS,  dS'  are  two  small  conterminous   arcs  on  one  surface,  and   ds,  ds'  are   the 
corresponding  small  conterminous  arcs  on  the  other  surface.  (Voss.) 

254.  Suppose  now  that  there  is  a  third  surface,  any  point  on  which 
is  determined  by  parameters  \  and  /ju ;  then  it  will  have  conformal  similarity' 
to  the  first  surface,  if  there  be  any  functional  relation  of  the  form 

\  +  i/jb  =  h(p  +  iq). 

But  if  h~'^  be  the  inverse  of  the  function  h,  then  we  have  a  relation 

P  +  iQ=f{h-'(X  +  ifM)] 

=  F{\  +  ifi), 

which  is  the  necessary  and  sufficient  condition  for  the  conformal  similarity 
of  the  second  and  the  third  surfaces. 

This  similarity  to  one  another  of  two  surfaces,  each  of  which  can  be  made 
to  correspond  to  a  third  surface  so  as  to  be  conformally  similar  to  it,  is  an 
immediate  inference  from  the  geometry.  It  has  an  important  bearing,  in 
the  following  manner.  If  the  third  surface  be  one  of  simple  form,  so  that  its 
parameters  are  easily  obtainable,  there  will  be  a  convenience  in  making  it 
correspond  to  one  of  the  first  two  surfaces  so  as  to  have  conformal  similarity, 
and  then  in  making  the  second  of  the  given  surfaces  correspond,  in  conformal 
similarity,  to  the  third  surface  which  has  already  been  made  conformally 
similar  to  the  first  of  them. 

Now  the  simplest  of  all  surfaces,  from  the  point  of  view  of  parametric 
expression  of  points  lying  on  it,  is  the  plane  :  the  parameters  are  taken  to 
be  the  Cartesian  coordinates  of  the  point.  Hence,  in  order  to  map  out  two 
surfaces  so  that  they  may  be  conformally  similar,  it  is  sufficient  to  map 
out  a  plane  in  conformal  similarity  to  one  of  them  and  then  to  map  out 
the  other  in  conformal  similarity  to  the  mapped  plane :    that  is  to  say,  we 


254.]  THEOREM  607 

may,  without  loss  of  generality,  make  one  of  the  surfaces  a  plane,  and  all 
that  is  then  necessary  is  the  determination  of  a  law  of  conformation. 

We  therefore  take  P  =  X,  Q=Y,N=1:  and  then 

P  +  iQ  =  X  +  iY=Z, 

where  Z  is  the  complex  variable  of  a  point  in  the  plane ;  and  the  equations 
which  establish  the  conformation  of  the  surface  with  the  plane  are 

ds'  =  n  (dp-  +  dq^)  • 

X  +  i7=f(p  +  iq) 

m^n  =f  (p  +  iq)f'  (p  -  iq) 

where  f I  (p  —  iq)  is  the  form  of  f(p  +  iq)  when,  in  the  latter,  the  sign  of  i  is 
changed  throughout. 

As  yet,  only  the  form  P  +  iQ  =  f(p+  iq)  has  been  taken  into  account. 
It  is  sufficient  for  our  present  purpose,  in  regard  to  the  alternative  form 
P  +  iQ  =  g{p  —  iq),  to  note  that,  by  the  introduction  of  a  plane  as  an  inter- 
mediate surface,  there  is  no  essential  divStinction  between  the  cases*.  For 
as  P  =  X,  Q  =  Y,  we  have 

X-YiY=g  {p-iq), 

and  therefore  X  —  iY=  g-i^{p  +  iq), 

which  maps  out  the  surface  on  the  plane  in  a  copy,  differing  from  the  copy 
determined  by 

X  ^iY  =  g,{p  +  iq), 

only  in  being  a  reflexion  of  that  former  copy  in  the  axis  of  X.     It  is  therefore 
sufficient  to  consider  only  the  general  relation 

X  +  iY=f{p-^iq). 

Ex.  We  have  an  immediate  proof  that  the  form  of  relation  between  two  planes,  as 
considered  in  §  9,  is  the  most  general  form  possible.  For  in  the  case  in  which  the 
second  surface  is  a  plane,  we  have  ds^  =  dx''-  +  dif,  so  that  n  =  l,  p  =  x,  q=y:  hence  the 
most  general  law  is 

X+iY=f{x  +  iy), 

that  is,  w—f{z), 

in  the  earlier  notation.     Some  illustrations  arising  out  of  particular  forms  of  the  function 

/will  be  considered  later  (§  257). 

255.  In  the  case  of  a  surface  of  revolution,  it  is  convenient  to  take  <^ 
as  the  orientation  of  a  meridian  through  any  point,  that  is,  the  longitude  of 
the  point,  a  as  the  distance  along  the  meridian  from  the  pole,  and  q  as 
the  perpendicular  distance  from  the  axis ;  there  will  then  be  some  relation 
between  cr  and  q,  equivalent  to  the  equation  of  the  meridian  curve.     Then 

ds"^  —  da-  +  q-d^'^ 

^  q^  (d(f>' +  d0% 

*  A  discussion  is  given  by  Gauss,  Ges.  Werke,  t.  iv,  pp.  211 — 216,  of  the  corresponding  result 
when  neither  of  the  surfaces  is  plane. 


608  '  CONFOEMAL  REPRESENTATION  OF  [255._ 

where  dd  ■=  — ,  so  that  ^  is  a  function  of  only  one  variable,  the  parameter  of 

the  point  regarded  as  a  point  on  the  meridian  curve.  Here  n=q";  and  so 
the  relation,  which  establishes  the  law  of  conformation  between  the  plane 
and  the  surface  in  the  most  general  form,  is 

*'  +  iy  =/(<^  +  *'^) ; 

and  the  magnification  m  js  given  by 

m^ q^  =f  ((b  +  ie)f,'  (0  -  id). 

Evidently  the  lines  on  the  plane,  which  correspond  to  meridians  of 
lono'itude,  are  given  by  the  elimination  of  6,  and  the  lines  on  the  plane, 
which  correspond  to  parallels  of  latitude,  are  given  by  the  elimination  of  <p, 
between  the  equations 

2w  =/{(!)  + id) +f,{6-ie)) 
2uj  =  f{cj>  +  ie) -/,{<!> -ie)\- 

Ex.  1.  A  plane  map  is  made  of  a  surface  of  revolution  so  that  the  meridians  and  the 
parallels  of  latitude  are  circles.  Shew  that,  if  (r,  a)  be  the  polar  coordinates  of  a  point  on 
the  map  determined  by  the  point  {B,  (p)  on  the  surface,  then 

?^^  =  -  2ac  {ae^"^  cos  2  {c^  +g)  +  b  cos  {g  +  h)}, 
r 

?^  =      2ac  {ae^'^  sin  2  {c^  +g)  +  b  sin  [g  +  h)], 

where  a,  6,  c^g,  h  are  constants. 

Prove  also  that  the  centres  of  all  the  meridians  lie  on  one  straight  line  and  that  the 
centres  of  all  the  parallels  of  latitude  lie  on  a  perpendicular  straight  line.        (Lagrange.) 

Ex.  2.     Prove  that,  in  a  plane  map  of  a  surface  of  revolution,  the  curvature  of  a 

meridian  at  a  point  <9  is  ;^  (  —  ) ,  and  the  curvature  of  a  parallel  of  latitude  at  a  point  <h 
^  dd  \mqj 

is  ( — y     Hence  shew  that,  if  the  meridians  and  the  parallels  of  latitude  become 

dfji  \>nqj 

circles  on  the  plane  map  given  by 

z=f{<t>+ie), 

the  function /and  the  conjugate  function /j  must  satisfy  the  relation 

{f,^  +  ie}=-{fu<t^-i^}, 
where  {/,  fj.}  is  the  Schwarzian  derivative.  (Lagrange.) 

Ex.  3.     On  the  surface  of  revolution,  let 

■v//-=  —  4i  jm'^gda; 
where  m,  q,  a  have  the  significations  in  the  text ;  shew  that  ^  and  ^  satisfy  the  equation 

where  %,  02  are  the  conjugate  complexes  x  +  iy  in  the  plane.  (Korkine.) 


256.]  SURFACES  OF  REVOLUTION  609 

256.  The  surfaces  of  revolution  which  occur  most  frequently  in  this 
connection  are  the  sphere  and  the  prolate  spheroid. 

In  the  case  of  the  sphere,  the  natural  parameter  of  a  point  on  a  great- 
circle  meridian  is  the  latitude  X.  We  then  have  da  =  ad\,  where  a  is  the 
radius  ;  and  q  =  a  cos  \  so  that 

ds^  =  a^  dX^  +  or  cos^  \  dcfi- 

=  a-cos''\{d(f>^-  +  d'^% 

where  sech  ^  =  cos  \.     Hence  we  have 

X  +  iY=f((j>  +  i^); 

and  the  magnification  m  is  given  by 

ma  cos  \  =  {/'  {(j)  +  ^^)//  (q6  -  i^)}^.     . 

There  are  two  forms  of  /  which  are  of  special  importance  in  representa- 
tions of  spherical  surfaces. 

First,  let/(/Lt)  =  k/x,  where  k  is  a  real  constant ;  then 

X  +  iY=k(cf>  +  i^), 

and  therefore  X  =  k(f),       Y=  k^  =  k  sech~^  (cos  X) ; 

that  is,  the  meridians  and  the  parallels  of  latitude  are  straight  lines, 
necessarily  perpendicular  to  each  other,  because  angles  are  conserved. 
The  meridians  are  equidistant  from  one  another ;  the  distance  between 
two  parallels  of  latitude,  lying  on  the  same  side  of  the  equator  and 
having  a  given  difference  of  latitude,  increases  from  the  equator.  We 
have  /'  ((f)  +  i^)  =  k  =//  (<^  —  i^) ;   and  therefore 

m  =  ~  sec  A,, 
a 

or  the  map  is  uniformly  magnified  along  a  parallel  of  latitude  with  a 
magnification  which  increases  very  rapidly  towards  the  pole.  This  map  is 
known  as  3Iercator's  Projection. 

Secondly,  let/(yu,)  =  ke^'''^,  where  k  and  c  are  real  constants ;  then 

'    X  +  iY=  ke^"  <*+^^'  =  ke-"'^  (cos  c^  +  i  sin  c0), 

and  therefore  X  =  ke'"^  cos  ccj)  and  F=  ke^"'^  sin  ccf). 

For  the  magnification,  we  have 

/'  ((/)  +  {^)  =  icke^'^  '*+'"^'   and  //  ((/>  -  t^)  =  -  icke-''^  <*-^'^', 

so  that  ma  cos  X.  =  cke~'''^, 

ck     ,,        ,       c^^(l-sinX)4(''-^) 
or  •  m  =  — e  "^    secX  =  —  7^7— — r       ,         . 

F.  F.  39 


610  MAPS  [256. 

The  most  frequent  case  is  that  in  which  c  =  1.     Then  the  meridians  are 
represented  by  the  concurrent  straight  lines 

F=Z  tan  </>; 

the  parallels  of  latitude  are  represented  by  the  concentric  circles 

1  —  sin  A, 


)f  the  circ 
lines ;   and  the  magnification  is 


1  +  sin  X ' 
the  common  centre   of  the   circles  being  the   point   of  concurrence  of  the 


k 


a(l  +  sin  A.) " 

This  map  is  known  as  the  ste7^eographic  projection.     The  South  pole  is  the 
pole  of  projection. 

It  is  convenient  to  take  the  equatorial  plane  for  the  plane  of  z:  the 
direction  which,  in  that  plane,  is  usually  positive  for  the  measurement  of 
longitude,  is  negative  for  ordinary  measurement  of  trigonometrical  angles. 
If  we  project  on  the  equatorial  plane,  we  have 

which  gives  a  stereographic  projection. 

Ex.  1.  Prove  that,  if  x,  y,  z  be  the  coordinates  of  any  point  on  a  sphere  of  radius  a  and 
centre  the  origin,  every  plane  representation  of  the  sphere  is  included  in  the  equation 

for  varying  forms  of  the  function  /. 

Ex.  2.  In  a  stereographic  projection  of  a  sphere,  the  complex  variable  of  a  point 
corresponding  to  a  point  Pon  the  sphere  is  x-\-%y.     Prove  that  the  complex  variable  of  the 

point,  which  corresponds  to  the  point  diametric'ally  opposite  to  P,  is . 

Ex.  3.  Two  conforraal  representations  of  the  surface  of  a  sphere  on  a  plane  are  given 
by  Mercator's  projection  and  a  stereographic  projection.  Find  the  form  of  relation  which 
will  transform  these  projections  into  one  another. 

Ex.  4.  Shew  that  rhumb-lines  (loxodromes)  on  a  sphere  become  straight  lines  in 
Mercator's  projection  and  equiangular  spirals  in  a  stereographic  projection. 

Ex.  5.  A  great  circle  cuts  the  meridian  of  reference  (0  =  0)  in  latitude  a  at  an  angle  a; 
shew  that  the  corresponding  curve  in  the  stereographic  projection  is  the  circle 

{X-\-h  tan  aj'  +  ( F+  h  cot  a  sec  cCf  =  k"^  sec^  a  cosec^  a. 

Ex.  6.     A  small  circle  of  angular  radius  r  on  the  sphere  has  its  centre  in  latitude  c  and 

longitude  a ;   shew  that  the  corresponding  curve  in  the  stereographic  projection  is  the 

circle 

''  „     k cos  c  cos  a\ 2     /  ^^    k  cos  c  sin  a\^  k^  sin^  r 

^  + T---     +  M  + 7-^      =/ r-- — N2- 

cosr  +  smc/       \        cos  ?•  + sine/      (cosr+smc)^ 


256.]  GENERAL  REFERENCES  611 

The  less  frequent  case  is  that  in  which   the   constant   c  is  allowed  to 

remain  in  the  function  for  the  purpose  of  satisfying  some  useful  condition. 

One  such  condition  is  assigned  by  making  the  magnification  the  same  at 

the  points  of  highest  and  of  lowest  latitude  on  the  map.     If  these  latitudes 

be  Xi,  \2,  then 

(1  -  sin  \j)i  "'-^'  _  (1  -  sin  >^)^  '°-i' 

(1  +  sinXi)*'"^"  ~  (1  +  sin  \,)i^<=+'^ ' 
so  that 

1  —  sin  XA      ,      /l  +  sin  X^ 


1      fi-  —  sm  XA      , 

^°Hr3^mxJ  +  i°g 


1  +  sin  \. 


,      ,  1  —  sm  XA      1       / 1  +  sin  X, 

logU— T-;rr  -log 


J  —  sm  Xg/  \1  +  sm  Xg/ 

This  representation  is  used  for  star-maps :   it  has  the  advantage  of  leaving 
the  magnification  almost  symmetrical  with  respect  to  the  centre  of  the  map. 

Ex.     Prove  that  the  magnification  is  a  minimum  at  points  in  latitude  arc  sin  c. 

Shew  that,  if  the  map  be  that  of  a  belt  between  latitudes  30°  and  60°,  the  magnification 
is  a  minimum  in  latitude  45°  40'  50";  and  find  the  ratio  of  the  greatest  and  the  least 
magnifications. 

Note.  Of  the  memoirs  which  treat  of  the  construction  of  maps  of- 
surfaces  as  a  special  question,  the  most  important  are  those  of  Lagrange* 
and  Gauss  f.  Lagrange,  after  stating  the  contributions  of  Lambert  and  of 
Euler,  obtains  a  solution,  which  can  be  applied  to  any  surface  of  revolu- 
tion ;  and  he  makes  important  applications  to  the  sphere  and  the  spheroid. 
Gauss  discusses  the  question  in  a  more  general  manner  and  solves  the 
question  for  the  conformal  representation  of  any  two  surfaces  upon  each 
other,  but  without  giving  a  single  reference  to  Lagrange's  work :  the 
solution  is  worked  out  for  some  particular  problems  and  it  is  applied,  in 
subsequent  memoirsj,  to  geodesy.  Other  papers  which  may  be  consulted 
are  those  of  Bonnet§,  Jacobi||,  Korkinelf,  and  Von  der  Miihll**;  and  there 
is  also  a  treatise  by  Herz"f*f. 

But  after  the  appearance  of  Riemann's  dissertation JJ,  the  question 
ceased  to  have  the  special  application  originally  assigned  to  it;  it  has 
gradually  become  a  part  of  the  theory  of  functions.  The  general  develop- 
ment will  be  discussed  in  the  next  chapter,  the  remainder  of  the  present 

*  Nouv.  Mem.  de  VAcad.  Roy.  de  Berlin,  (1779).  There  are  two  memoirs :  they  occur  in  his 
CGllected  works,  t.  iv,  pp.  635 — 692. 

t  Schumacher's  Astr.  Abh.  (1825) ;  Ges.  Werke,  t.  iv,  pp.  189—216, 

X  Gott.  Abh.,  t.  ii,  (1844),  ib.,  t.  iii,  (1847);  Ges.  Werke,  t.  iv,  pp.  259—340. 

§  Liouville,  t.  xvii,  (1852),  pp.  301—340. 

II    Crelle,  t.  lix,  (1861),  pp.  74—88 ;  Ges.  Werke,  t.  ii,  pp.  399—416. 

Ii  Math.  Ann.,  t.  xxxv,  (1890),  pp.  588—604. 
**  Crelle,  t.  Ixix,  (1868),  pp.  264—285. 

ft  Lehrbuch  der  Landkartenprojectiunen,  (Leipzig,  Teubner,  1885). 

Xt  "  Grundlagen  fiir  eine  allgemeine  Theorie  der  Functionen  einer  veranderlichen  complexen 
Grosse,"  Gottingen.  1851 ;  Ges.  Werke,  pp.  3 — 45,  especially  §  21. 

39—2 


612  •  EXAMPLES  [256. 

chapter  being   devoted   to   some   special   instances   of  functional  relations 
between  lu  and  z  and  their  geometrical  representations. 

The  following  examples  give  the  conformal  representation  of  the  respective  surfaces 
upon  a  plane  or  a  part  of  a  plane. 

Ex.  1.  A  point  on  an  oblate  spheroid  is  determined  by  its  longitude  I  and  its 
geographical  latitude  /x.  Shew  that  the  surface  will  be  conformally  represented  upon  a 
plane  by  the  equation 

for  any  form  of  the  function  /;   where  sech  0  =  cos  ju,  and  e  is  the  eccentricity  of  the 
meridian. 

Also  shew  that,  if  the  function  /  be  taken  in  the  form  f{u)  —  ke^'^,  the  meridians  in 
the  map  are  concurrent  straight  lines,  and  the  parallels  of  latitude  concentric  circles ;  and 
that  the  magnification  is  stationary  at  points  in  geographical  latitude  arc  sin  c.     (Gauss.) 

Eoc.  2.  Let  the  semi-axes  of  an  ellipsoid  be  denoted  by  p,  (p^  —  b^)"^,  (p^-c^)"^,  in 
descending  order  of  magnitude.  Shew  that  the  surface  will  be  conformally  represented 
upon  a  plane  by  the  equation 

V  ,   -TT    V- fz.  /     ,   ■\,l^      Q(u  +  a)e{iv  +  a)] 
-^  \    ^  ^     -     °  e{u-a)e{tv-a)l 

for  any  form  of  the  function  /; 'where  u  and  v  are  expressed  in  terms  of  the  elliptic 
coordinates  p^  and  p2  of  a  point  on  the  surface  by  the  equations 

cHpi'-b^)_        2  l{pl^)_        2- 

p    /c2-62\4 

the  modulus  is  -  (  —^ — ^r, )  ,  the  constant  a  is  given  by 

6  =  c  dn  a, 

and  the  value  of  the  constant  h  is  tn  a  dn  a  -  Z  (a).  (Jacobi.) 

Ex.  3.  The  circular  section  of  an  anchor-ring  by  a  plane  through  the  axis  subtends  an 
angle  tt  —  2e  at  the  centre  of  the  ring,  and  the  position  of  any  point  on  such  a  section  is 
determined  by  I,  the  longitude  of  the  section,  and  by  X,  the  angle  between  the  radius  from 
the  centre  of  the  section  to  the  point  and  the  line  from  the  centre  of  the  section  to  the 
centre  of  the  ring. 

Shew  that,  by  means  of  the  equations 

l  =  2nx, 

tan  ^X  =  cot  Je  tan  (ttt/  tan  e), 

the  surface  of  the  anchor-ring  is  conformally  represented  on  the  area  of  a  rectangle  whose 
sides  are  1  and  cot  e.  (Klein.) 

Ex.  4.  Consider  the  surface  generated,  by  revolution  round  the  axis  of  ?/',  of  the  curve 
whose  equations  are 

.^■'  =  a  sin  t,         i/  =  a  (cos  t  +  log  tan  y), 

sometimes  called  the  tracti'ix. 

The  radius  of  curvature  of  the  curve  is  —  a  cot  t ;  the  length  of  the  normal  intercepted 
between  the  curve  and  the  axis  of  y'  is  a  tan  t.  Hence  the  Gauss  measure  of  curvature  of 
the  surface  of  revolution  is  —  1/a^,  that  is,  the  surface  of  revolution  is  one  of  constant 
negative  curvature.     Surfaces  of  the  same  measure  of  curvature  can  be  deformed  into  one 


256.]  EXAMPLES  613 

anotlier ;  in  particular,  when  the  measure  is  constant,  the  surface  is  applicable  upon  itself 
in  an  infinite  variety  of  ways*. 

The  arc-element  of  the  surface  of  revolution  is  given  by 

=  d^  cot^  t  dt"^  +  or  sin^  t  dcf)'-^ 
Let  a  new  variable  x//'  be  introduced  by  the  relation 


so  that 

then  the  arc-element  is  given  by 

When  we  write 
this  becomes 


, ,           cos  t  , 
a\l/-  = 7—r-  dt, 

^      sin  t 


(/)  =  .r,  >/^  =  ?/, 


ds^  =  —^  {dx^  +  d'lf' 


and  io  the  surface  can  be  represented  conformally  upon  the  x^  y  plane. 

For  the  upper  half  of  the  surface,  corresponding  to  the  positive  part  of  the  x' ,  y'  plane 
of  the  original  curve,  the  range  of  ;;  is  from  tt  to  ^tt  ;  and  therefore  the  range  of  y  is  from 
00  to  1.  The  range  of  x  is  from  0  to  Itv.  The  area  in  the  x,  y  plane  is  a  half -rectangle ;  it 
is  bounded  by  a  line  x=Q>  while  y  ranges  from  co  to  1,  by  a  hne  y  =  \  while  x  ranges  from 

0  to  277,  by  a  line  x=^tt  while  y  ranges  from  1  to  oo .  Thus  the  relations,  between  the 
coordinates  X,  F,  Z  of  the  surface  and  the  coordinates  a;,  y  of  the  part  of  the  plane  upon 
which  the  surface  of  revolution  is  conformally  represented,  are 

„  cos  X  -r^  sin  A' 

X  =  a ,         1  =a , 

V  y 

Z=a  (cos  ?;+log  tan  \t), 
where  y  sin  t=\. 

For  the  lower  half  of  the  surface,  corresponding  to  the  negative  part  of  the  x' ,  y'  plane 
of  the  original  curve,  the  range  of  t  is  from  ^tt  to  0 ;  and  therefore  the  range  of  y  is  from 

1  to  -f  CO  .     The  range  of  x,  as  before  is  from  0  to  27r. 

When  in  the  original  curve,  t  ranges  from  ir  to  Stt,  the  value  of  y'  is  complex ;  the 
corresponding  sheet  of  the  surface  is  imaginary. 

When  t  ranges  from  27r  to  Stt,  there  is  a  real  sheet  of  the  surface  coincident  with  the 
former  real  sheet.     And  so  on,  for  the  successive  7r-intervals  of  the  quantity  t\. 

Ex.  5.  In  the  representation  of  the  surface  of  constant  curvature  given  in  the 
preceding  example,  prove  that  any  geodesic  upon  the  surface  becomes  a  circle  in  the  x,  y 
plane  having  its  centre  on  the  axis  of  x. 

*  See  my  Lectures  on  Differential  Geometry,  §§  211 — 213. 

t  For  a  further  discussion  of  the  surface  and  its  representation  upon  the  2-plane,  see  Darboux's 
TMorie  generale  des  surfaces,  t.  iii,  pp.  394  et  seq. 

From  later  investigations  it  will  appear  that,  by  other  transformations,  the  infinite  strip  in 
the  s-plane  can  be  represented  upon  different  forms  of  areas  in  different  planes,  so  that  any 
number  of  representations  of  the  surface  of  revolution  upon  a  plane  can  be  obtained. 


614  CONFORMAL  REPRESENTATION  [257. 

257.     It  was  pointed  out  (§  254)  that  the  conformation  of  surfaces  is 
obtained  by  a  relation 

and  therefore  that  the  conformation  of  planes  is  obtained  by  a  relation 

w=f{z\ 
whatever  be  the  form  of  the  function  /,  or  by  a  relation 

(/)  {w,  z)  =  0, 
whatever  be  the  form  of  the  function  </>.     Some  examples  of  this  conformal 
representation   of  planes   will   now  be    considered ;    in   each    of  them   the 
representation  is  such  that  one  point  of  one  area  corresponds  to  one  (and 
only  one)  point  of  the  other. 

Ex.  1.     Consider  the  correspondence  of  the  two  planes  represented  by 

{a-h)vfi-  2ztv  +  {a+b)  =  0, 
that  is,  . 

7^         a-\-b 
zz=(a  —  o)w-\ . 

Let  r,  6  be  the  coordinates  of  any  point  in  the  io-plane  :   and  x,  y  the  coordinates  of  any 
point  in  the  2-plane  :  then 


2^=    (a -6) 


r 


2y-. 


'  a  +  b' 

{a  —  b)r 


Hence  the  s-curves,  corresponding  to  circles  in  the  H'-plane  having  the  origin  for  their 
common  centre,  are  confocal  ellipses,  2c  being  the  distance  between  the  foci,  where 
c^=a^  —  b^:  and  the  ^-curves,  corresponding  to  straight  lines  in  the  w-plane  passing 
through  the  origin,  are  the  confocal  hyperbolas,  a  result  to  be  expected,  because  the 
orthogonal  intersections  must  be  maintained. 

Evidently  the  interior  of  a  t«-circle,  of  radius  unity  and  centre  the  origin,  is,  by  the 
above  relation,  transformed  into  the  part  of  the  s-plane  which  lies  outside  the  ellipse 
x'^ja^+y^lb'^=l,  the  i<;-circumference  being  transformed  into  the  s-ellipse. 

Ux.  2.     Discuss  the  correspondences  tvz^=l,  w  +  z^  =  l. 

Ex.  3.     Consider  the  correspondence  implied  by  the  relation 

2K 


,_i  f2K  \  ,       ,  ,      .  ,      , 

k    ^  11)  =  sn  (  —  z\—s,nz,  where  x  +ty  =  2  = 


with  the  usual  notation  of  elliptic  functions.     Taking  vj=X+iY.,  we  have 
^"  2  (X+ {F)  =  sn  {x'  +  iy') 

sn  x'  en  iy'  dn  iy' + sn  iy'  en  x'  dn  x' 
"  1  —  X-2  sn^  x'  sn^  iy' 

Let  y'=  ±hK' :  then  sn  iy' =^  ±  —r=. ,  en  iy'  ■=  a/     ,     ,  dn  iy'=Jl-{-k^  so  that 

,  _  i  •  T^N  _  1  +  ^'       sn  x'  i_  en  x'  dn  x' 

^  ^     +'^  ^~  "^  1  +/[•  sn2.r'  -  ik  1  +/f^^2^' ' 

whence  (l+^)sn^'  cn^Vdn^ 

l+/?rsn2^''  -\^l%v?x" 

and  therefore  Jr2+F2  =  l, 


257.]  OF   PLANES  •  615 

which  is  the  curve  in  the  i/;-plane  corresponding  to  the  hnes  y'  =  ±^K'  in  the  /-plane, 
that  is,  to  the  lines  y=  ±  -r-^  in  the  z-plane. 

When  v=-\ 77-  and  x'  lies  between  K  and  —  K.  that  is,  x  lies  between  \ir  and  —  in-,  then 

T  is  positive  and  X  varies  from  1  to  —  1  ;  so  that  the  actual  curve  corresponding  to  the 
line  y= — ^   is  the  half  of  the  circumference  on  the  positive  side  of  the  axis  of  X. 

ttK' 
Similarly,  the  actual  curve  corresponding  to  the  line  y=  —  -jy^  is  the  half  of  the  circum- 
ference on  the  negative  side  of  that  axis. 

The  curve   hereby  suggested   for   the  ^-plane   is  a  rectangle,  with  sides  ^  =  ±  -gTr, 
?/=  +—  -=.     To  obtain  the  zy-curve  corresponding  to  x=\iv^  that  is,  to  x'=K,  we  have 


4  K 


7     1  /  -ir     •  T7A     cn  iw 


1  cn  tv^ 
so  that  F=0  and  X=Jc-i  -. — ^, . 

an  ly 

Now  y'  varies  from  ^K'  through  0  to   —hK':  hence  X  varies  from  1  to  k^   and  back 
from  ^*  to  1.     Similarly,  the  curve  corresponding  to  x=  —\n, 
that  is,  to  x'  =  —  K,  is  part  of  the  axis  of  X  repeated  from 
—  1  to  —k^  and  back  from  -k^  to  —1. 

Hence  the  area  in  the  w-plane,  corresponding  to  the  rect- 
angle in  the  3-plane,  is  a  circle  of  radius  unity  with  two  diametral 
slits  from  the  circumference  cut  inwards,  each  to  a  distance  k'^ 
from  the  centre. 

The  boundary  of  this  simply  connected  area  is  the  homo- 
logue  of  the  boundary  of  the  ^-rectangle  given  hj  x=  ±W, 

y=  ±^^^-—  :  the  analysis  shews  that  the  two  interiors  corre-  ^^S-  ^"• 

spond*.  And  the  sudden  change  in  the  direction  of  motion  of  the  w-point  at  the  inner 
extremity  of  each  slit,  while  z  moves  continuously  along  a  side  of  the  rectangle,  is  due  to 
the  fact  that  dwjdz  vanishes  there,  so  that  the  inference  of  §  9  cannot  be  made  at  this 
point.     (See  also  Ex.  15.) 

Corollary.  We  pass  at  once  from  the  rectangle  to  a  square,  by  assuming  K'  =  2^ ;  then 
k  =  {j2—  1)2,  and  the  corresponding  modifications  are  easily  made. 

JEx.  4.  Shew  that,  if  z=sn^(^w,k)  where  iv  =  u  +  iv,  then  the  curves  «  =  constant, 
w  =  constant,  are  confocal  Cartesian  ovals  whose  equations  may  be  written  in  the  form 

r^  -  r  dn  (^«,  k)  =  cn  («,  k),        r^  -f  r  dn  {vi,  k')  =  cn  {vi,  k'), 
where  r  and  j'l  denote  the  distances  from  the  foci  z  —  0  and  z  =  l. 

If  J2  denote  the  distance  of  a  point  from  the  third  focus  0=^,  find  the  corresponding 
equations  connecting  r,  r2 ;  and  r-^ ,  ^2 . 

Shew  that  the  curves  to  =  K,v  =  K'  are  circles,  and  that  the  outer  and  the  inner  branches 
of  an  oval  are  given  by  lo  and  2K-u,  or  by  v  and  2K'-v.  (Math.  Trip.,  Part  II.,  1891.) 

*  For  details  of  corresponding  curves  in  the  interiors  of  the  two  areas,  see  Siebeck,  Crelle, 
t.  Ivii,  (1860),  pp.  359—370;  ib.,  t.  lix,  (1861),  pp.  173—184  :  Holzmiiller,  treatise  cited  (p.  2,  note), 
pp.  256—263  :  Cayley,  Gamb.  Phil.  Tram.,  vol.  xiv,  (1889),  pp.  484—494,  Collected  Mathematical 
Papers,  vol.  xiii,  pp.  9 — 19. 


616 


EXAMPLES   OF 


[257. 


Ex.  5.     The  zf-plane  is  conformally  represented  on  the  s-plane  by  the  equation 

C        \\—Wi 

where  h  and  c  are  real  positive  constants. 

Shew  that,  if  an  area  be  chosen  in  the  w-plane  included  within  a  circle,  centre  the 
origin  and  radius  unity,  and  otherwise  hounded  by  two  circles  centres  1  and  —  1  (so  that 
its  whole  boundary  consists  of  four  circular  arcs),  then  the  corresponding  area  in  the 
2-plane  is  a  portion  of  a  ring,  bounded  by  two  circles,  of  radii  c^  and  ce~''  and  centre  the 
origin,  and  by  two  lines  each  passing  from  one  circle  to  the  other. 

Prove  that,  when  the  semi-circles  in  the  w-plane  are  very  small,  so  as  merely  to 
exclude  the  points  1  and  —1  from  the  circular  area  and  boundary,  the  corresponding 
2-figure  is  the  ring  with  a  single  slit  along  the  axis  of  real  quantities  *. 

Ex.  6.     Consider  the  correspondence  implied  by  the  relation 

z  =  c  sin  iv. 
Taking  w=X-{-iY.,  we  have 

X  +  iy  =  c  sin  {X + ^  T) 

=  c  sin  X  cosh  Y+  ic  cos  X  sinh  F, 
so  that  X = c  sin  X  cosh  Y,        y  =  c  cos  X  sinh  Y. 


When  Y  is  constant,  then  z  describes  the  curves 

-  + 


x^ 


r 


=1, 


c^  cosh^  Y  '  c'^  sinh^  Y 
which,  for  different  values  of  Y,  are  confocal  ellipses. 

Now  take   a   rectangle   lying   between   ^^=  +  577,    Y=±'k.      For   all   values   of   X, 
cos  X  is   positive  :    hence   when    Y=  +  X,  y 
is   positive  and  x  varies   from    c  cosh  A    to 
—  c  cosh  X,  that  is,  the  half  of  the  ellipse  on 
the  positive  side  of  the  axis  of  3/  is  covered. 

Let  X=  —  ^TT  :  then 

3/  =  0  and  X—  —c  cosh  Y. 
As    Y  varies  from    +X  through  0  to    —X 
along  the   side   of  the   rectangle,   x   passes 
from  B  to  H  (the  focus)  and  back  from  H 
to  B. 

When  F=  —X,  then  z  describes  the  half  of  the  ellipse  on  the  negative  side  of  the  axis 
of  y  :  when  X=  +^7r,  then  ,?/  =  0,  x  =  c  cosh  Y,  so  that  z  passes  from  A  to  S  (a  focus)  and 
back  from  5  to  J. 

Hence  the  s-curve  corresponding  to  the  contour  of  the  ^(J-rectangle  is  the  ellipse 
with  two  slits  from  the  extremities  of  the  major  axis  each  to  the  nearer  focus  :  the 
analytical  relations  shew  that  the  two  interiors  correspond. 

Ex.  7.     Consider  the  correspondence  implied  by  the  relation 


/2K  . 
■  sn    —  sni " 

V  T 


-(v^ 


From  Ex.  3,  it  follows  that  the   interior   of  a   w-circle,  centre   the   origin  and  radius 
unity,  corresponds  to  the  interior  of  the  ^-rectangle  bounded  by  ^=  +  |^7r,  3/=  ±  ^-^  , 


See  reference,  p.  488,  note. 


257.]  PLANE  CONFOEMAL  REPRESENTATION  617 

provided  two  diametral  slits  be  made  in  the  w-circle  along  tlie  axis  of  a;  to  distances 
1—k^  from  the  circumference  ;  and,  from  Ex.  6.  it  follows  that  the  same  ^-rectangle  is 
transformed  into  the  interior  of  the  z-ellipse 

^     f_ 

where  a=c  cosh  ^-=-  and  6  =  c  sinh  — ^^  ,  provided  two  slits  be  made  in  the  elliptical  area 
along  the  major  axis  from  the  curve  each  to  the  nearer  focus. 

Thus,  by  means  of  the  rectangle,  the  interiors  of  the  slit  w-circle  and  the  slit  j-ellipse 
are  shewn  to  be  conformal  areas. 

But  the  lines  of  the  two  slits  are  conformally  equivalent  by  the  above  equation.     For 
the  slit  on  the  positive  side  of  the  axis  of  x  extends  from  x=c  to  ^  =  ccosh\,  where 

X=--^:^,  and  it  has  been  described  in  both  directions:  we  thus  have 
4A 

z  =  c  cosh  /3, 
where  /3  passes  from  0  to  X  and  back  from  X  to  0.     Hence 

sin~  1  -  =  sin~'i  (cosh  /3)  =  \tt  +  i0, 

so  that  the  corresponding  iv-cnrve  is  given  by 

k    -w=sn    A-1 


f2K0t 
en    — 

■) 

W"^' 

-^ 

Then,  when  /3  assumes  its  values,  w  passes  from  1  to  t^  and  back  from  X-^  to  1,  that  is, 
w  describes  the  circular  slit  on  the  positive  side  of  the  axis  of  X. 

Similarly  for  the  two  slits  on  the  negative  side  of  the  axis  of  real  quantities.  Thus 
the  two  slits  may  be  obliterated  :  and  the  whole  interior  of  the  ^(;-circle  can  be  represented 
on  the  interior  of  the  ^-ellipse. 

From  the  equations  defining  a  and  b,  it  follows  that 

-by     -^' 

in  the  Jacobian  notation  ;  and  c^=a^—  b^. 

Combining  the  results  of  Ex.  1  and  Ex.  7,  we  have  the  theorem*  : — 

The  part  of  the  z-plane,  which  lies  outside  the  ellipse  x'^/a^-\-9/^lb^  =  l,  is  transformed 
into  the  interior  of  a  iv-circle^  ■  of  radius  \inity  and  centre  the  origin^  by  the  relation 

(a-6)w2_22w+(a  +  6)  =  0  ; 

and  the  part  of  the  z-plane,  which  lies  inside  the  same  ellipse,  is  transformed  into  the  interior 
of  the  same  w-circle  by  the  relation 

Jc~'^tv  =  sn\  '^—  sin~i{3(a2_62)~2| 

tvhere  the  Jacobian  constant  q  luhich  determines  the  constants  of  the  elliptic  functions,  is 
given  by        ■ 

fa-b\'^ 
^  =  1^6,)  • 

*  Schwarz,  Ges.  Werke,  t.  ii,  pp.  77,  78,  102—107,  141. 


618  EXAMPLES  OF  [257. 

Ex.  8.  Investigate  the  equations  that  effect  the  conformal  representation  of  the 
annular  region  between  two  confocal  eUipses,  whose  semi-axes  are  ao,  b^  and  %,  6i,  upon 
the  annular  region  between  two  coaxal  circles  whose  limiting  points  are  A  and  B.  Prove 
that,  if  the  circles  cut  BA  produced  in  Pq  ^-nd  P\ ,  then  the  ratio  (ai  +  bi)  :  (%  +  b^)  is  one 
of  the  anharmonic  ratios  of  the  raoge  {BA,  PqPi).  (Math.  Trip.,  Part  II.,  1896.) 

Ex.  9.     Consider  the  correspondence  implied  by  the  relation 

{'W+lfz=A. 

When  w  describes  a  circle,  of  radius  unity  and  centre  the  origin,  then  w  =  e** :  so  that, 
if  r  and  6  be  the  coordinates  of  z,  we  have 

-  (cos  ^ -z  sin  ^)  =  (1  +  6*^)2, 

or  -p  (  cos --i sin  -  j  =  1  +  e****  =  1  +  cos  ^  + 1  sin  0. 

s'r  \       ^  ^J 

Hence  -—  cos  ^  -  1      +  -  sm^  -  =  1, 

a 

that  is,  ?'Cos2-=l,  ^ 

shewing  that  z  then  describes  a  parabola,  having  its  focus  at  the  origin  and  its  latus 
rectum  equal  to  4. 

Take  curves  outside  the  parabola  given  by 
where  yx  is  a  constant  ^  1. 


so  that 
therefore 

so  that 

a  series  of  circles  touching  at  the  point  X=  —  1,  ^^"=0,  and  (for  ^  varying  from  1  to  oo ) 
covering  the  whole  of  the  interior  of  the  -w-circle,  centre  the  origin  and  radius  unity. 

Hence,  by  means  of  the  relation  (w  + 1)20=4,  the  exterior  of  the  £-space  bounded  by 
the  parabola  is  transformed  into  the  interior  of  the  w-space  bounded  by  the  circle. 

Ex.  10.     Shew  that,  if  z  (w**  — 1)2  +  4 w"  =  0,  the  curves  in  the  w-plane  corresponding  to 
the  real  axis  in  the  2-plane  are  three  arcs  of  circles  ;  and  find  the  angles  at  which  they  cut. 

Ex.  11.     Prove  that  the  infinite  half-strip  in  the  2-plane,  bomided  by 

0^^^27r,     0<?/, 
is  represented  upon  the  interior  of  a  2«;-circle  of  radius  unity  by  the  relation 

1  — ^■  cos  \z 


?■= 

=^2sec2-. 

Then 

1 

=  -  COS  \6, 

w-M  = 

2     . 

i 

X-fl: 

=  -cos2i^=-(l+cos 

/x            -          ;x 

e\ 

T- 

1 

sin  6y 

(x-fl- 

w= 


l  +  icos^z' 


257.]  PLANE  CONFORMAL  REPRESENTATION  619 

Ex.  12.     Consider  the  correspondence  implied  by  the  relation 

We  have =co&(hirz^)  =  Gos,(\Trf'^  e^^^), 

\+w  •^ 

HO  that,  if  «'  +  l  =  i2e'\     ^l  =  \'^■r'^  co^  ^6,     v=^7Tr^  s,m\6, 

then  2i2~i  cos  9  —  1  =  cos  u  cosh  y, 

222  ~i  sin  e       =sin  m  sinh  v. 

The  w-ciu'ves,  corresponding  to  the  confocal  parabolas  in  the  s-plane,  are 

(2  cos  e-R)'^      4  sin'^  Q _  pa 
cos^  ti  sin"'^  w 

If  ?i  <  ^TT,  then  2i2-i  cos  9  >  1,  that  is,  R<2  cos  9  ;  while,  if  u  >  ^tt,  we  have  i2>  2  cos  9. 
It  thus  appears  that  the  s-space  lying  within  the  parabola  ?<■= Jtt,  that  is,  r  cos2i^  =  l, 
is  transformed  into  the  interior  of  a  -^y-circle,  centre  the  origin  and  radius  unity,  by  means 
of  the  relation 

By  the  two  relations*  in  Ex.  9  and  Ex.  12,  the  spaces  within  and  without  the  parabola 
are  conformally  represented  on  the  interior  of  a  circle. 

Ex.  13.     Consider  the  relation 

I  — to 

then,  if  s=.^■  +  ^^/  and  w  =  X+iY,  we  have 

x^ty-      j:2  +  (1+7)2     • 

When  10  describes  the  whole  of  the  axis  of  X  from  -  oo  to  +  oo ,  so  that  we  can  take 

A''=tan^,   F=0,  where  0  varies  from  -  ^  to   +  ^,  we  have  .r  =  cos2(^,  y  =  sin20;  and 

z  describes  the  whole  circumference  of  a  circle,  centre  the  origin  and  radius  1.  For 
internal  points  of  this  circle  \—x^-y'^  is  positive:  it  is  equal  to  4F-4-{2'2  +  (l  +  3^)^},  and 
therefore  the  positive  half  of  the  '2<;-plane  is  the  area  conformal  with  the  interior  of  the 
circle,  of  radius  unity  and  centre  the  origin,  in  the  2-plane. 

Ex.  14.  On  the  circle  in  the  s-plane  in  the  preceding  example,  three  points  ABC  are 
taken  as  the  angular  points  of  an  equilateral  triangle.  Circles  BA.,  AC,  CB  are  drawn 
touching  OB,  OA;  OA,  OG;  OC,  OB;  respectively,  where  0  is  the  centre.  Draw  the 
figure  in  the  w-plane  corresponding  to  the  curvilinear  triangle  ABC. 

Ex.  15.     Again,  consider  a  relation 

^z  -  {c\^ 

\z  +  ic) 

„,    ,                               ^     .^^    (^■2  +  y2-c2)2-4c2.z;2  +  4^•c^(c2-^2_^2) 
We  have  X-\-iY= 1  o  ,  , — ,    xo^o ' 

{x^  +  {y  +  cYY 


so  that  X= ^,  ,  ,     ,    ,212 


r= 


(a-2  +  /  -  2CX  -  C2)  (.372  +  3/2  +  2CX  -  C^) 
{^'2  +  (y  +  c)2}2 

4cx{G^  —  x'^  —  'lf) 


{x^  +  {y+c)Y  ' 
Let  x  =  0,  so  that  F=0  ;  then 


*  Schwarz,  Ges.  Werke,  t.  ii,  p.  146. 


620 


EXAMPLES   OF 


[257. 


As  z  passes  from  A  io  B  (where  OA  =  OB  =  c),  then  y  changes  from  -c  to  +c,  and  X 
changes  continuously  from   +  qo  to  0. 

Let  x^+y^-c^  =  0,  so  that   r=0;  then 

X=  ^  ~^f\  .^  =  _  ^Z^  =  _  tan2  U, 
(2c  +  2j/)^         c+y  2  ; 

where  3/  =  c  cos  6.  Hence,  as  z  describes  the  semi-circular 
arc  BCA,  the  angle  6  varies  from  0  to  tt  and  X  changes 
from  0  to  —  00  . 

(The  whole  axis  of  X  is  the  equivalent  of  AOBCA  ;  and 
at  the  w-origin,  corresponding  to  B,  there  is  no  sudden 
change  of  direction  through  ^tt.  The  result  is  apparently 
in   contradiction   to    §  9 :   the  explanation  is   due   to   the 

div 


Fig.  88. 


fact  that  -^-  =  0  at  B.  and  the  inference  of  ^  9  cannot  be  made.     Similarly  for  A,  where 

dz 


is  infinite.     See  also  Ex.  3. 


For  any  point  lying  within  the  ^-semi-circle,  both  x  and  c^  —  x^—y'^  are  positive,  so 
that   Y  is  positive.     Hence  by  the  relation 


the  interior  of  the  s-semi-circle  is  conformally  represented  on  the  positive  half  of  the 
ic-plane. 

It  is  easy  to  infer  that  the  positive  half  of  the  ?(;-plane  is  the  conformal  equivalent  of 
(i)      the  interior  of  the  semi-circle  ACBA  by  the  relation  w-- 


(iii) 


CBDC 
BDAB 


(iv) DAGD 


■+ic 
z  +  c 

Z  —  Gj 

—  ic) 

ST- 


And,  by  combination  with  the  result  of  Ex.  13,  it  follows  that  the  relation 


■-^ic. 


i^ 


■ic 

z  +  ic 


.z^-c^  +  2cz 


conformally  represents  the  interior  of  the  s-semi-circle  ACBA   on  the  interior  of  the 
««;-circle,  radius  unity  and  centre  the  origin. 

Similarly  for  the  other  cases. 


Ux.  16.     Shew  that,  by  the  relation 


z^  +  2z-l 
''z^-2z-l 


the  interior  of  the  circle  |  ??;  |  =  1  is  conformally  represented  on  the  interior  of  a  semi-circle 
in  the  2-plane. 


257.]  CONFORMAL   REPRESENTATION  621 

Ex.  17.  Find  a  figure  in  the  2:-plane,  the  area  of  which  is  conformally  represented  on 
the  positive  half  of  the  'i(;-plane  by 

(i)    ir=z-,  (ii)    ^'^=(^^^\  (iii)    tv=z-^{l-zY. 

Ex.  18.     Consider  the  relation 

tv  =  ae^^ : 

then  jr= ae~^  cos  x,       T=  ae~y  sin  x. 

The  curves  corresponding  to  y  =  constant  are  concentric  circumferences;  those  corre- 
sponding to  .r= constant  are  concurrent  straight  lines. 

As  X  ranges  from  0  to  ^tt,  both  X  and  Y  are  positive ;  for  a  given  value  of  x  between 
these  limits,  each  of  them  ranges  from  0  to  oc ,  as  ?/  ranges  from  co  to  —  cjo  .  As  ^  ranges 
from  \tt  to  TT,  X  is  negative  and  Y  is  positive  ;  for  a  given  value  of  x  between  these 
limits,  -  A'  and   Y  range  from  0  to  oo ,  as  j/  ranges  from  oo  to  —  qc  . 

Hence  the  portion  of  the  2-plane  lying  between  ?/=  —  oo,  y  =  oo,  .r  =  0,  x  =  '77,  that  is, 
a  rectangular  strip  of  finite  breadth  and  infinite  length,  is  conformally  represented  by  the 
relation 

on  the  positive  half  of  the  t<;-plane.  Combining  this  result  with  that  in  Ex.  13,  we  see 
that  the  same  strip  is  conformally  represented  on  the  area  of  a  ^«;-circle,  centre  the  origin 
and  radius  a,  by  means  of  the  relation 

=  aie^. 

w+1 

Ex.  19.  Find  a  portion  of  the  s-plane  that  corresponds,  luider  the  relation  w  =  e*^,  to 
the  interior  of  the  circle 

and  the  portion  of  the  ?<?-plane  that  corresponds  to  the  interior  of  a  circle 

in  the  z-plane,  for  varying  values  of  c. 

Note.  It  may  be  convenient  to  restate  the  various  instances  of  areas  in  the  ^-plane, 
bounded  by  simple  curves,  which  can  be  conformally  represented  on  the  area  of  a  circle 
in  the  -w-plane: 

(i)  The  positive  half  of  the  z-plane;  Ex.  13. 

(ii)  An  infinite  strip  of  finite  breadth;  Ex.  11,  Ex.  18,  Ex.  19. 

(iii)  Area  without  an  elhpse;  Ex.  1. 

(iv)  Area  within  an  ellipse ;  Ex.  7. 

(v)  Area  without  a  parabola;   Ex.  9. 

(vi)  Area  within  a  parabola;  Ex.  10. 

(vii)  Area  within  a  rectangle ;  Ex.  3. 

(viii)  Area  within  a  semi-circle ;  Ex.  16. 

(ix)    As  will  be  seen,  in  §  258,  any  circle  changes  into  itself  by  a  proper  homo- 
graphic  relation.  , 


622 


EXAMPLES   OF 


[257. 


Ex.  20.     Consider  the  correspondence  implied  by  the  relation 


Then  we  have  two  values  of  iv^,  say  w^^,  Wg^,  where 


w-i^=  ■ 


wf  =  - 


1+32 


Fig.  89. 


Let  z  describe  the  axis  of  .r,  so  that  z  =  x. 

When  0  <  ^-^  <  1,  then  iv-^  is  real  and  less  than 
unity  and  w^  is  real  and  greater  than  unity.  Hence 
drawing  a  circle  in  the  w-plane,  centre  the  origin 
and  radius  1,  and  six  lines  as  diameters  making  angles 
of  \'iv  with  one  another,  and  denoting  a  cube  root  of 
1  by  a,  then,  as  z  passes  from  0  to  1  along  the  axis  of  x, 
ivi  passes  from  A  to  0, 

A  to  A'  (at  infinity), 

C  to  0, 

C  to  C"  (at  infinity), 

. E  to  (9, 

E  to  E'  (at  infinity). 

When  1  <^<  00  ,  then  w-^  is  a  real  quantity  changing  continuously  from  0  to  -  1,  and 
1(72^  is  a  real  quantity  changing  continuously  from  -  co  to  —  1.  As  s  passes  from  1  to  qo 
along  the  positive  part  of  the  axis  of  x, 

Wi  passes  from  0  to  F, 


W2 

aiVi 


5'  (at  infinity)  to  5, 

0  to  B, 

D'  (at  infinity)  to  Z), 

0  to  D, 

i^'  (at  infinity)  to  F. 

Hence,  as  z  describes  the  whole  of  the  positive  part  of  the  axis  of  x,  the  branches  of  iv 
describe  the  whole  of  the  three  lines  A'D',  B'E',  C"F'. 

When  X  is  negative,  we  can  take  x=  —  tan^  (^,  so  that  cj)  varies  from  0  to  ^tt.     Then 


t«2 

aWo 


1  —  ^  tan  (^ 


-t^i 


^      1  +  i  tan  (^ 
so  that,  as  z  passes  from  0  to  -  oo  ,  ?»i  describes  the  arc  of  the  circle  from  A  to  i^,  atv^  the 
arc  from  C  to  B,  and  aHv^  the  arc  from  E  to  D.     And  then 

so  that  W2  describes  the  arc  of  the  circle  from  A  to  B,  aw^  the  arc  from  C  to  D,  and  a^Wo 
the  arc  from  E  to  F.  Hence,  as  z  describes  the  whole  of  the  negative  part  of  the  axis  of  x, 
the  branches  of  w  describe  the  whole  of  the  circumference. 

As  z  describes  a  line  parallel  to  the  axis  of  x  and  veiy  near  it  on  the  positive  side,  the 
paths  traced  by  the  branches  are  the  dotted  lines  in  the  figures ;  the  six  divisions,  in 
which  the  symbols  are  placed,  are  the  conformal  representations  by  the  six  branches  of  iv 
of  the  positive  half  of  the  2-plane* 

*  Cayley,  Gavih.  Phil.  Trans. ,  vol.  xiii,  (1880),  pp.  30,  31 ;  Coll.  Math.  Papers,  vol.  xi,  pp.  174,  175. 


257.]  CONFORMAL   REPRESENTATION  623 

Ex.  21.     When  the  variables  are  connected*  by  a  relation 


^-(^o('-') 


where  c^o  is  the  function  which  in  coefficients  is  conjugate  to  0,  then  the  s-circumference, 
centre  the  origin  and  radius  c,  is  transformed  into  the  ■zi>-circumference,  centre  the  origin 
and  radius  c. 

Taking  Wq  and  Zq  as  the  conjugate  variables,  we  have 

c2™  +  2    0(,)       ^^(,;^ 


L'o- 


^«(?)' 


so  that  tviV(j=- 

Now  if  z  describe  the  circumference  of  a  circle,  centre  the  origin  and  radius  c,  we  have 


1       •'O" 

so  that  ?<;?fQ  =  c2, 

shewing  that  lu  describes  the  circumference  of  a  circle,  centre  the  origin  and  radius  c. 

To  determine  whether  the  internal  area  of  the  ^-circumference  corresponds  to  the 
internal  area  of  the  ?(;-circumference,  we  take  zZ(f  =  c^  —  e,  where  e  is  small.     Then 

^0  ( -j  =  <^0  (  2o  +  ^  j  =  00  (^O)  +  \  4>o'  (2o)  ; 

therefore  ^^,^=e^  (i  +  ^J^)  jl-^^H  il-'-^^A 

i  0(2)         0o(2o)J 

so  that  the  interior  of  the  ^-circumference  finds  its  conformal  correspondent  in  the  interior 
or  in  the  exterior  of  the  ^^-circumference  according  as 

«.<or>.fi^  +  .,^, 

4>  (2)  00  (2o) 

taken  along  the  circumference. 

The  simplest  case  is  that  in  which  (^  (2)  is  of  degree  m,  so  that  it  can  be  resolved  into 
m  factors,  say  (li{z)  =  A{z  —  a){z-^)...{z  —  6):  then 


c^\        .    fC''         \   fc 


and 

A  (z-a)iz-l3)...{z-6) 


iu  = 


^«^""Vi-5.)(i-|A..(i-'j. 


*  Cayley,  Crelle,  t.  evil,  (1891),  pp.  262—277;  Coll.  Math.  Papers,  vol.  xiii,  pp.  191—205. 


624 


EXAMPLES   OF 


[257. 


But  the  converse  of  the  result  obtained — that  to  the  ?i'-circumference  there  corresponds 
the  s-circumference — is  not  complete  unless  the  correspondence  is  (1,  1).  Other  ciirves 
which  are  real — they  may  be,  but  are  not  necessarily,  circles — and  imaginary  curves  enter 
into  the  complete  analytical  representation  on  the  2-plane  corresponding  to  the  ^c-circum- 
ference,  of  centre  the  origin  and  radius  c  on  the  -w-plane. 


Ex.  22.     Discuss  the  s-curves  corresponding  to  |  ^(^  |  =  1,  determined  by 


Ex.  23.     Consider  the  relation 


We  have 


W  —  Wq  = 


1-n/22  ' 

4  {z^-z  +  Vf 
^"~Ti      {f-zf    ■ 


(Cayley.) 


27 


(22-2)2  {Zi-Z^f     J 

The  function  on  the  right-hand  side,  being  connected  with  the  expressions  for  the  six 
anharmonic  ratios  of  four  points  in  terms  of  any  one  ratio,  vanishes  for 

^0  ~  -I  ^0 


so  that 

tV—tVo 

Hence,  taking 

we  have 

2^■^^ 


2o 


4  (z  -  So)  (2^0  -  1)  (s  +  ^0  -  1)  {2  (^0  -  1)  - 


0}  (2Zo  -  ^0  + 1)  {2  (^0  -  1)  + 1} 


27 


2_«  -12 


(s^-0)2(2o2-0o; 

^t;  =  J^+^i'',     z=x-\-iy, 
4  2%  (^2^.^2 _  1)  (2a^ -  1)  (^'2+/-2A-){(x2+,y2-.r  +  l)^+/ 


27  (x'2+y2)2(^2+^2_2^-+l)2 

Hence  it  appears  that,  when  1^=0,  so  that  iv  traces  the  axis  of  real  quantities  in  its  own 
plane,  the  s- variable  traces  the  curves 

y  =  0,     ^2+2/2-1=0, 

2.r-l  =  0,     ^2^2^2_2^=:0, 

that  is,  two  straight  lines  and  two  circles  in  its 
own  plane. 

In  order  to  determine  the  parts  of  the  a-plane 
that  correspond  to  the  positive  part  of  the  w-plane, 
it  is  sufficient  to  take  Y  equal  to  a  small  positive 
quantity  and  determine  the  corresponding  sign  of 
y.     Let 

where  Y  (and  therefore  y)  is  small :  then,  to  a  first 
approximation, 

_27 x^{x-\f 


Fig.  90. 


'^      4:{2x-l){x  +  l)(x-2){x''--x  +  lf' 

and  the  sign  of  y.  determines  whether  the  part  on  the  positive  or  negative  side  of  the  axis 
of  X  is  to  be  taken. 

When  X  <-\,  ^  is  negative;  z  lies  below  the  axis  of  x.  When  x  is  in  AO,  so  that 
x>—l<0,ix\s.  positive ;  z  lies  above.  When  x  is  in  OB,  so  that  .i-  >  0  <  ^,  /x  is  negative ; 
z  lies  below.  When  x  is  in  BC,  so  that  .r  >  -I  <  1,  /x  is  positive;  z  lies  above.  When  x  is 
in  CD,  so  that  a'  >  1  <  2,  jn  is  negative;  z  lies  below.     And,  lastly,  when  x  is  beyond  D,  so 


257.]  CONFORMAL  REPRESENTATION  625 

that  ^  >2,  |i  is  positive  and  z  lies  above  the  axis  of  real  quantities.  The  parts  are  indicated 
by  the  shading  in  fig.  90. 

It  is  easy  to  see  that  w  =  0,  for  z  =  P,  Q;  that  w  =  l,  for  z  =  A,  B,  D;  and  that  w  =  cc, 
for  z=0,  C.  The  zero  vahie  of  w  is  of  triple  occurrence  for  each  of  the  points  P  and  Q; 
the  unit  value  and  the  infinite  value  are  of  double  occurrence  for  their  respective  points*. 

Note.  It  is  easy  to  see  that  figures  89  and  90  are  two  diflfereut  stereographic  projections 
of  the  same  configuration  of  lines  on  a  sphere  (§  277,  I.,  w  =  3),  so  that  the  relations  in 
Ex.  20  and  Ex.  23  may  be  regarded  as  equivalent. 

Eo%  24.  Find,  in  the  same  way,  the  curves  in  the  0-plane,  which  are  the  conformal 
representation  of  the  axis  of  X  in  the  w-plane  by  the  relation t 

^~  108  (2* +  2-*- 2)2  • 
Ex.  25V    Shew  that,  by  the  relation 

the  lines,  .:«;  =  constant  in  the  2- plane,  are  transformed  into  a  series  of  confocal  lemniscates 
in  the  ?{>-plane ;  and  that,  by  the  relation 

where  c  is  a  real  positive  constant  greatel'  than  unity,  the  interior  of  a  s-circle,  centre 
the  origin  and  radius  unity,  is  transformed  into  the  interior  of  the  lemniscate  RR'  =  c 
in  the  it^-plane,  where  R  and  R'  are  the  distances  of  a  point  from  the  foci  (1,  0)  and 
(-1,0).  (Weber.) 

258.  The  preceding  examples |  may  be  sufficient  to  indicate  the  kind 
of  coriielation  between  two  planes  or  assigned  portions  of  two  planes,  that  is 
provided  in  the  conformal  representation  determined  by  a  relation  ^  {w,  z)  =  0 
connecting  the  complex  variables  of  the  planes.  We  shall  consider  only  one 
more  instance ;  it  is  at  once  the  simplest  and  functionally  the  most  important 
of  all§.  The  equation,  which  characterises  it,  is  linear  in  both  variables ;  and 
so  it  can  be  brought  into  the  form 

az  +  h 

w  = J  , 

cz  +  a 

where  a,  h,  c,  d  are  constants :  it  is  called  a  homographic  transformation, 
sometimes  a  homographic  or  a  linear  substitution. 

Taking  first  the  more  limited  form 

/^ 

tu  =  - , 

z 

and  writing  lu  =  i^e'®,  z  =  re^^,  fx  =  Pe^^*,  we  have 

Rr^k\  %  +  d  =  27,  that  is,  0  -  7  =  7  -  d, 

*  See  Klein-Fricke,  vol.  i,  p.  70. 

t  See  Klein-Fricke,  vol.  1,  p.  75. 

X  Many  others  will  be  found  in  Holzmuller's  treatise,  already  cited,  which  contains  ample 
references  to  the  literature  of  the  subject. 

§  For  the  succeeding  properties,  see  Klein,  Math.  Ann.,  t.  xiv,  pp.  120 — 124,  ib.,  t.  xxi, 
pp.  170 — 173 ;  Poincare,  Acta  Math.,  t.  i,  pp.  1 — 6 ;  Klein-Fricke,  Elliptische  Modulfunctionen, 
vol.  i,  pp  163  et  seq.  They  are  developed  geometrically  by  Mobius,  Ges.  Werke,  t.  ii,  pp.  189—204, 
205—217,  243—314. 

F.  P.  40 


626  HOMOGRAPHIC  [258. 

and  therefore  the  new  w-locus  will  be  obtained  from  the  old  ^^-locus  by 
turning  the  plane  through  two  right  angles  round  the  line  7  through  the 
origin,  and  inverting  the  displaced  locus  relative  to  the  origin.  The  first 
of  these  processes  is  a  reflexion  in  the  line  <y ;  and  therefore  the  geometrical 
change  represented  by  ■w^  =  /u-  is  a  combination  of  reflexion  and  inversion. 

A  straight  line  not  through  the  origin  and  a  circle  through  the  origin  are 
corresponding  inverses ;  a  circle  not  through  the  origin  inverts  into  another 
circle  not  through  the  origin,  and  it  may  invert  into  itself;  and  so  on. 

Taking  now  the  general  form,  we  have 

a  ad  — he 


w  —  =  — 


c  „  (       d 

o              o 
or  transforming  the  origins  to  the  points  -  and in  the  w-  and  the  ^•-planes 

respectively,  and  denoting ^ —  by  fx,  we  have  WZ  =  fi,  that  is,  the  former 

c 

case.     Hence,  to  find  the  w-locus  which  is  obtained  through  the  transforma- 

d 
tion  of  a  ^-locus  by  the  general  relation,  we  must  transfer  the  origin  to , 

turn  the  plane  through  two  right  angles  round  a  line  through  the  new  origin 

whose  angular  coordinate  is  ^  arg.  f —  j ,  invert  the  locus  in  the  displaced 

position  with  a  constant  of  inversion  equal  to 


and  then  displace  the 

c"     r  ^ 

origin  to  the  point  —  - .     Hence  a  circle  will  be  changed  into  a  circle  by  a 
c 

homographic  transformation  unless  it  be  changed  into  a  straight  line ;  and 
a  straight  line  will  be  changed  into  a  circle  by  a  homographic  transformation 
unless  it  be  changed  into  a  straight  line. 

The  result  can  also  be  obtained  analytically  as  follows ;    the  formulae 
relating  to  the  circle  will  be  useful  subsequently. 

A  circle,  whose  centre  is  the  point  (a,  y8)  and  whose  radius  is  ?•,  can  be 
expressed  in  the  form 

{2 -a-  /3i)  (zo-<^  +  I3i)  =  r\ 
or  zzo  +  6z  +  60Z0  +  7  =  0, 

where  —  6  =  a  —  ^i,  —  dg  =  a  +  ^i,  7  =  66^^  —  rl  Conversely,  this  equation 
represents  a  circle,  when  6  and  60  are  conjugate  imaginaries  and  7  is  real ; 
its  centre  is  at  the  point  —  ^{0+  60),  i*(^—  ^oX  ^i^d  its  radius  is  (^^0  -  7)"- 

When  the  circle  is  subjected  to  the  homographic  transformation 

az  +  h 

cz  -\-d 

,  —dw  +  h.,,n  -doWo  +  bo 

we  have  z  = and  therefore  z„  = . 

cw  —  a  CqWq  —  tto 


258.]  SUBSTITUTIONS  "^  627 

Substituting  these  values,  the  relation  between  tu  and  Wq  is 

S'wwo  +  d'w  +  Oq'wq  +  7'  =  0, 

where  h'  =     dd^  — 6dco  — 6ocdo  + ycCo, 

6'  =  —  bod  +  Oaod+  6ocl>o  —  ycao, 

*      do  =  —  bdo  +  Ocoh  +  6oada  —  jCoa, 

7'  =     bbo  —  Ottob  —  dottbo  +  yaao. 

Here  8'  and  7'  are  real,  and  6'  and  Oq   are  conjugate  imaginaries ;  therefore 
the  equation  between  lu  and  Wn  represent?  a  circle. 

Ex.  1.     A  circle,  of  radius  ?•  and  centre  at  the  point  (e,  /),  in  the  s-plane  is  transformed 
into  a  circle  in  the  w-^\a.ne,  by  the  homographic  substitution 

az  +  b 

%o= • 

cz  +  d' 

shew  that  the  radius  of  the  new  circle  is 


ad— he 


where  A  =  (cr  cos  /3  +  e)^  +  (o-  sin  /3  +ff  -  r% 

d 
and  a,  (3  are  the  modulus  and  the  argument  respectively  o'f  - .     Find  the  coordinates  of 

the  centre  of  the  i(7-circle. 

Ux.  2.  The  inverse  of  a  point  P,  with  regard  to  a  circle,  is  Q ;  and  the  inverse  of  Q, 
with  regard  to  any  other  circle  is  R.  Prove  that  the  complex  variables  of  P  and  R  are 
connected  by  a  homographic  relation 

yz  +  8 

Moreover,  since  there  are  three  independent  constants  in  the  general 
homographic  transformation,  they  may  be  chosen  so  as  to  transform  any  three 
assigned  ^^-points  into  any  three  assigned  w-points.  And  three  points  on  a 
circle  uniquely  determine  a  circle :  hence  any  circle  can  be  transformed  into 
any  other  circle  {or  into  itself)  by  a  properly  chosen  homographic  transforma- 
tion. The  choice  of  transformation  can  be  made  in  an  infinite  number  of  ways  : 
for  three  points  on  the  circle  can  be  chosen  in  an  infinite  number  of  ways. 

A  relation  which  changes  the  three  points  z^,  z^,  z^  into  the  three  points 
Wi,  w^,  Ws  is  evidently 

{W  —  Wj)  (Wa  -  W3)  _  (Z  —  Zi)  {z.2  —  Z3) 
(w  -  W2)  (wi  -  W3)  ~  {z-  Z2)  {Zi  -Z3)' 

Hence  this  equation,  or  any  one  of  the  other  five  forms  of  changing  the  three 
points  z-^,  Z2,  Zo,  into  the  three  points  Wj,  Wg,  w^  in  any  order  of  correspondence, 
is  a  homographic  transformation  changing  the  circle  through  ^^j,  ^2,  ^3  into  the 
circle  through  w-^,  w^.  w^. 

.       40—2 


628        ■  "  HOMOGEAPHIC  [258. 

It  has  been  seen  that  a  transformation  of  the  form  w  =f{z)  does  not 
change  angles :  so  that  two  circles  cutting  at  any  angle  are  transformed  by 

w  = ,  into  two  others  cutting  at  the  same  angle.     Hence*  a  plane  crescent, 

of  any  angle,  can  be  transformed  into  any  other  crescent,  of  the  so.me  angle. 

The  expression  of  homographic  transformations  can  be  modified,  so  as  to 
exhibit  a  form  which  is  important  for  such  transformations  as  are  periodic. 

If  we  assume  that  w  and  z  are  two  points  in  the  same  plane,  then  there 
will  in  general  be  two  different  points  which  are  unaltered  by  the  transforma- 
tion ;  they  are  called  the'  fixed  (or  double)  points  of  the  transformation. 
These  fixed  points  are  evidently  given  by  the  quadratic  equation 

au  +  b 

u  — -J, 

cu  -h  a 

that  is,  cu^  —  (a  —  d)  u  —  b  =  0. 

Let  the  points  be  a  and  ^,  and  let  M  denote  (d  —  ay  +  46c ;  then 

2ca  =  a-d  +  M^,     2c/3  =  a-d-  M^. 

If,  then,  the  points  be  distinct,  we  have 

w.—  OL  _  {z  —  a){a  —  ca)  z  —  a 

w-^~  {z  -  /3)  (a  -  c/3)  "       7^ ' 

a  —  ca      a  +  d  —  M^ 


where  K  = 


a-cl3     a  +  c?  +  ilf  2 ' 


/  1  Y      (a  +  df 


and  therefore  (  JR  +  -?=  I  = 


ad  —  bo' 

The  quantity  K  is  called  the  midtiplier  of  the  substitution. 

If  there  be  a  2^-curve  in  the  plane  passing  through  a,  the  w-curve  which 

arises  from  it  through  the  linear  substitution  also  passes  through  a.     To  find 

the  angle  at  which  the  ^-curve  and  bhe  w-curve  intersect,  we  have  w  =  a+  Bw, 

z=a+  Sz:  and  then 

Sw  =  K8z, 

so  that  the  angle  between  the  tangents  to  the  w-cnrve  and  the  ^^-curve  is  the 
argument  of  K.  Similarly,  if  a  ^-curve  pass  through  ^,  the  angle  between 
the  tangents  to  the  ^^-curve  and  the  w-curve  is  the  argument  of  ^. 

The  form  of  the  substitution  now  obtained  evidently  admits  of  reapplica- 
tion ;  if  z^  be  the  variable  after  the  substitution  has  been  applied  n  times,  (so 
that  Zq  =  z,  z^  =  lu),  we  have 


Zn-J3  z-^' 

Kirchhoff,  Vorlesungen  ilber  matheviatische  Phifsik,  i,  p.  286. 


258.]  SUBSTITUTIONS  629 

The  condition  ttat  the  transformation  should  be  periodic  of  the  7zth  order 
is  that  Zn  =  z  and  therefore  that  K'^  =  1 ;  hence 

(a  -\-  dy  =  ^  {ad  —  he)  cos-  —  , 

where  s  is  any  integer  different  from  zero  and  prime  to  ?i;  K  cannot  be 
purely  real,  and,  in  general,  M  is  not  a  real  positive  quantity.  The 
various  substitutions  that  arise  through  different  values  of  s  are  so  related 
that,  if  points  z-^^,  z^,  ...,  Zn  be  given  by  the  continued  application  of  one 
substitution  through  its  period,  the  same  points  are  given  in  a  different 
cyclical  order  by  the  continued  application  of  the  other  substitution,  through 
its  period. 

JVote  The  formula  in  the  text  may  be  regarded  as  giving  the  ?ith  power  of  a  substi- 
tution. The  form  of  the  substitution  obtained  is  equally  effective  for  giving  the  ?ith  root 
of  a  substitution  :  all  that  is  necessary  is  to  express  K  in  the  form  pe  ^,  and  the  nth. 
root  is  then 

^1-°        1     1 

n 

Ex.  1.     The  value  of  Zn  has  been  given  by  Cayley  in  the  form 

{K»  +  i -l){az  +  b)  +  {E'' - K) {- dz  +  b)  _ 
{K''+'^-l){cz  +  d)  +  {K''-K){cz-a)    " 
obtain  this  expression.  , 

Ex.  2.  Periodic  substitutions  can  be  applied,  in  connection  with  Kirchhoff's  result 
that  a  plane  crescent  can  be  transformed  into  another  plane  crescent  of  the  same  angle  ; 
the  plane  can  be  divided  into  a  limited  number  of  regions  when  the  angle  of  the  crescent 
is  commensurable  with  tt. 

Let  ACBDA  be  a  circle  of  radius  unity,  having  its  centre  at  the  origin  :  draw  the 
diameter  AB  along  the  axis  of  y.  Then  the  semi-circle  ACE  can  be  regarded  as  a  plane 
crescent,  of  angle  \it  ;  and  the  semi-circle  ABD  as  another,  of  the  same  angle.  Hence 
they  can  be  transformed  into  one  another. 

We  can  effect  the  transformation  most  simply  by  taking  A  (  =  i)  and  5(=  —  ^)  as  the 
fixed  points  of  the  substitution,  which  then  has  the  form 

w  —  i_  jT-z  —  i 
w  +  i         z+i' 

The  line  AB  for  the  w-curve  is  transformed  from  the  z-circular  arc  AGB:  these  curves 
cut  at  an  angle  ^tt,  which  is  therefore  the  argument  of  K.  Considerations  of  symmetry 
shew  that  the  2- point  C  on  the  axis  of  x  can  be  transformed  into  the  w-origin,  so  that 

-1+?.' 

whence  K=i,  so  that  the  substitution  is 

iu  —  i_.z—i 
iv  +  i        z+i' 

It  is  periodic  of  order  4,  as  might  be  expected :  when  simplified  it  takes  the  form 

1-1-2 
^  =  1—.- 


630 


EXAMPLES 


[258. 


The  figure  (fig.  91)  shews  the  effect  of  repeated  application  of  the  substitution  through 
a  period.  The  first  application  changes  the  interior  of  ACBA  into  the  interior  of  ABDA  : 
by  a  second  application,  the  latter  area  is  transformed  into  the  area  on  the  positive  side  of 
the  axis  of  3/  lying  without  the  semi-circle  ADB ;  by  a  third  application,  the  latter  area  is 


e,<- 


C^3\ 


transformed  into  the  area  on  the  negative  side  of  the  axis  of  3/  lying  without  the  semi- 
circle ACB ;  and  by  a  fourth  application,  completing  the  period,  the  latter  area  is 
transformed  into  the  interior  of  ACBA,  the  initial  area. 

The  other  lines  in  the  figure  correspond  in  the  respective  areas. 


Ex.  3.     Prove  that  the  substitution 


is  periodic  of  order  six  ;  express  it  in  canonical  form  ;  and  trace  the  geometrical  eflfect  of 
the  application  of  the  successive  powers  of  the  substitution  upon 

(i)     the  straight  line  joining  ?■  and  -i; 

(ii)     the  semi-circle  on  the  last  line  lying  to  the  right  of  the  axis  of  y. 


Ux.  4.     Shew  that,  if  the  plane  crescent  of  Ex.   2  have  an  angle  of  -tt  instead  of 

^TT  but  still  have  +i  and  -  i  for  its  angular  points,  then  the  substitution 

z+t 

where  t  denotes  tan  ;^ ,  is  a  periodic  substitution  of  order  2n  which,  by  repeated  appli- 
2n 

cation  through  a  period  to.  the  area  of  the  crescent,  divides  the  plane  into  2n  regions,  all 

but  two  of  which  must  be  crescent  in  form.     Under  what  circumstances  will  all  the  2ii 

regions  be  crescent  in  form  ? 


258.]  HOMOGRAPHIC   SUBSTITUTIONS  631 

Ex.  5.  If  in  the  plane  of  the  complex  variable  z,.  two  circles  be  drawn  entirely  exterior 
to  one  another,  sketch  the  proof  of  the  theorem,  that  a  function  of  z  exists  which  is  real 
at  the  circumferences  of  these  circles  and,  exterior  to  the  circles,  is  everywhere  finite  and 
continuous  save  at  5=oo,  where  its  infinite  part  is  ^s™,  A  being  a  real  assigned  constant 
and  n  an  assigned  positive  integer. 

If  o,  /3  be  the  complex  arguments  of  the  limiting  points  of  these  circles,  and  a,  h  the 

modulus  of  - — -   upon  the  circles  surrounding  a,  /3  respectively,  express  this  function 

Z  —  a 

2  h 

in  terms  of  ^u  —  ^c;  where  <pu  is  Weierstrass's  elliptic  function  formed  with  2  and  — ;  log  - 

as  periods,  ii  =  — ;  log  (  -  ''- ) ,  c=  — .  log  (  -  ) . 

^  TTi     °\az  —  aj         71-1°  \a/ 

Determine  the  geometric  meaning  of  the  transformation  ~ —  =  ( - )  —  '•  ^^^  ^^^ 
effect  upon  the  function 

2     (-)M-^  ^        ""^ 

=  -  00  \C 


)i=  — 0 


and  state  the  relation  connecting  this  function  with  'Qu  above. 

(Math.  Trip.,  Part  II.,  1893.) 

259.  Homographic  substitutions  are  divided  into  various  classes,  according 
to  the  fixed  points  and  the  value  of  the  multiplier.  As  the  quantities  a,  h, 
c,  d  can  be  modified,  by  the  association  of  an  arbitrary  factor  with  each  of 
them  without  altering  the  substitution,  we  may  assume  that  ad  —  be  =  1; 
we  shall  suppose  that  all  substitutions  are  taken  in  such  a  form  that  their 
coefficients  satisfy  this  relation.  Figures  which,  by  them,  are  transformed 
into  one  another  are  called  congruent  figures. 

If  the  fixed  points  of  the  substitution  coincide,  it  is  called*  a  parabolic 
substitution. 

There  are  three  classes  of  substitutions,  which  have  distinct  fixed  points. 

If  the  multiplier  be  a  real  positive  quantity,  the  substitution  is  called 
hyperbolic. 

If  the  multiplier  have  its  modulus  equal  to  unity  and  its  argument 
different  from  zero,  it  is  called  elliptic. 

If  the  multiplier  have  its  modulus  different  from  unity  and  its  argument 
different  from  zero,  it  is  called  loxodromic. 

These  definitions  apply  to  all  substitutions,  whether  their  coefficients  be 
real  or  be  complex  constants;  when  we  consider  only  those  substitutions, 
which  have  real  coefficients,  only  the  first  three  classes  occur.  Such  sub- 
stitutions are  often  called  real. 

The  quadratic  equation,  which  determines  the  common  points  of  a  real 
substitution,  has  its  coefficients  real ;  according  as  the  roots  of  the  quadratic 

*  All  these  names  are  due  to  Klein :  I.e.,  p.  605,  note. 


632  CLASSES   OF   HOMOGRAPHIC  [259. 

are  imaginary,  equal,  or  real,  the  real   substitution  will  be  proved  to  be 
elliptic,  parabolic,  or  hyperbolic  respectively.     For  all  of  these,  we  take 

z  -\-  -  =  X  +  iy,       w =  X+  iY, 

c  c 

which  imply  a  transference  of  the  respective  origins  along  the  respective  axes 
of  real  quantity ;  and  then 

^       .„         ad  —  be      1 

X  +  iY= -^ 

c^       X  +  ly 

X  —  iy 


so  that 


c^  {x'^  +  y'^  ' 
Y  1 


y      c"  {x"  +  2/2)  ■ 

The  axes  of  x  and  of  X  have  been  unaltered  by  any  of  the  changes  made  in 
the  substitution ;  and  Y,  y  have  the  same  sign  and  vanish  together ;  hence 
the  effect  of  a  real  transformation  is  to  conserve  the  axis  of  real  quantities,  by 
transforming  the  half  of  the  ^-plane  above  the  axis  of  x  into  the  half  of  the 
w-plane  above  the  axis  of  X. 

A  real  transformation,  which  changes  z  into  %v,  also  changes  z^  into  Wo 
(these  being  conjugate  complexes).  A  circle,  having  its  centre  on  the 
axis  of  X  and  passing  through  a,  /3,  passes  through  a^,  /Sq  also :  hence  a 
transformation,  which  changes  a  circle  through  a,  /S  with  its  centre  on 
the  axis  of  x  into  one  through  7,  S  with  its  centre  on  the  axis  of  X,  is 

z  —  a     yS  —  tto  _  '^^  —  7     ^  —  7o 
z  —  a^^'  ^  —  OL      w  —  Jo'  S  —  y  ' 

Ex.  1.     Shew  that,  if  this  circle,  through  a,  /3,  ao,  /So,  cut  the  axis  of  x  in  h  and  Jc, 

a  —  h   B  —  h 
where  h  lies  in  /S^o  a'ld  k  in  aao ,  and  if  [a/3]  denote  7 .  - — j  ,  a  real  quantity  greater 

than  1,  then 

(a-^o)(/3-ao)      {l+[a^Jf 

Ex.  2.  ,  Prove  that  the  magnification  at  any  point,  by  a  real  substitution,  is  Yjy. 

(Poincare.) 

Ex.  3.  Any  2-circle,  having  its  centre  on  the  axis  of  .v,  is  transformed  by  a  real 
substitution  into  a  zy-circle,  having  its  centre  on  the  axis  of  X. 

Ex.  4.  Obtain  the  real  transformation,  which  makes  three  points  2  =  a,  /3,  y  on  the 
real  axis  correspond  to  the  three  points  w=0,  1,  00,  in  the  form 

_z—a ^—y 
2  — 7/3-a' 

Ex.  5.  The  points  w=0,  1,  00  can  be  grouped  in  six  ways;  find  the  sgt  of  transforma- 
tiong  which  change  any  one  grouping  into  the  rest  of  the  groupings. 


259.]  SUBSTITUTIONS  633 

Ex.  6.  A  circle,  of  radius  unity  and  having  its  centre  at  the  point  0,  1,  is  drawn  in 
the  ^<;-plane.  A  very  small  slit  is  made  in  the  circumference  at  the  point  0,  2  ;  and  two 
straight  lines  are  drawn  towards  +oo  from  this  point,  just  above  and  just  below  the  real 
axis,  and  practically  coinciding  with  that  axis. 

Draw  the  figures  in  the  ^-plane  which  arise  from  this  figure  by  the  respective  substitu- 
tions 

1       ,  1         z-\  z 

Let  the  classes  of  real  substitutions  be  considered  in  order. 

(i)  For  real  parabolic  substitutions,  the  quadratic  has  equal  roots :  let 
their  common  value  be  a,  necessarily  a  real  quantity,  so  that  the  fixed  points 
of  the  substitution  coalesce  into  one  on  the  axis  of  os.  The  quantity  M  is 
then  zero,  so  that  (d  +  a)-  =  4.  We  may,  without  loss  of  generality,  take 
d  +  a  =  2.  If  both  origins  be  removed  to  the  point  a,  then,  in  the  new 
form,  zero  is  a  repeated  root  of  the  quadratic,  so  that  b  =  0,  and  a  —  d  =  0. 
Hence  a  =  d  =  1,  and  the  real  substitution  is 

z 

ID  = 


cz  +  1 


that  is*,  —  =-  +  c. 

tu     z 

The  equations  of  transformation  of  real  coordinates  are 

X  y       x^  +  y"- 

Z  -  c  (X^  +  Y'^^Y^X^  +  F^ 

1 


{\  -  cxy ^- c^Y^' 

Ex.  1.  A  2-circle  passing  through  the  origin  is  transformed,  by  a  real  parabolic 
substitution  having  the  origin  for  its  common  point,  into  a  w-circle,  passing  through  the 
origin  and  touching  the  s-circle  :  and  a  ^-circle,  touching  the  axis  of  x  at  the  origin,  is 
transformed  into  itself 

Ex.  2,  Let  J  be  a  circle  touching  the  axis  of  x  at  the  origin  :  and  let  Cq  be  the 
extremity  of  its  diameter  through  the  origin.  Let  a  real  parabolic  substitution,  having 
the  origin  for  its  common  point,  transform  Cq  into  Cj,  c^  into  C2,  c-i  into  Cg,  and  so  on  :  all 
these  points  being  on  the  circumference  oi  A. 

Prove  that  the  radii  of  the  successive  circles,  which  have  their  centres  on  the  axis  of  x 
and  pass  through  the  origin  and  c^,  the  origin  and  Cj,  ...  respectively,  are  in  harmonic 
progression,  and  that,  if  these  circles  be  denoted  by  ^j,  C2,  ■••,  then  C^  is  the  locus  of  all 
points  Cfc  arising  through  different  initial  circumferences  A. 

Ex.  3.     What  is  the  effect  of  the  inverse  substitution,  applied  as  in  Ex.  2  ? 

Ex.  4.  Shew  that,  if  a  curve  of  finite  length  be  drawn  so  as  to  be  nowhere  infini- 
tesimally  near  the  axis  of  ^,  it  can  cut  only  a  finite  number  of  the  circles  G  in  Ex.  2. 

{Note.     All  these  results  are  due  to  Poincare.) 

*  If  the  origins  be  not  removed  to  the  point  a,  the  form  is = he 

°  jr>  10  -a      z-a 


634  ELLIPTIC   SUBSTITUTIONS  [259. 

(ii)  For  real  elliptic  substitutions,  a  and  /3  are  conjugate  complexes; 
hence  M  is  negative,  so  that 

(d  -  of  +  46c  <  0, 
or  {d  +  a)'  <  4  {ad  -  he)  <  4. 

The  value  of  K,  by  using  the  relation  ad  —  hc  =  \,  is 

K  =  ^  [{a  +  dy-2-i  (a  +  d)  [4,  -  {a  +  dffy 

It  is  easy  to  see  that  \K\  =  1  and  that  its  argument  is  cos"^  {^  (a  +  df -1],  so 
that,  if  this  angle  be  denoted  by  cr,  we  have 

K  =  e<^^ 
shewing  that  the  substitution  is  elliptic. 

It  is  evident  that,  if  z  describe  a  circle  through  a  and  /3,  its  centre  being 
therefore  on  the  axis  of  x,  then  w  also  describes  a  circle  through  a  and  0 
cutting  the  ^■-circle  at  an  angle  cr.  The  two  curves  together  make  a  plane 
crescent  of  angle  <t  having  a,  /3  for  its  angular  points. 

Ex.  Shew  tliat  a  real  elliptic  substitution  transforms  into  itself  any  circumference, 
which  has  its  centre  on  a^  produced  and  cuts  the  line  a/3  harmonically.  (Poincare.) 

(iii)  For  real  hyperbolic  substitutions,  the  roots  of  the  quadratic  are  real 
and  different ;  hence  the  fixed  points  of  the  substitution  are  two  (different) 
points  on  the  axis  of  oc.     The  quantity  M  is  positive,  so  that 

(a  +  dy  >4!: 

we  may  evidently  take  a  +  d  >  2.     Moreover  K  is  real  and  positive,  shewing 
that  the  substitution  is  hyperbolic. 

Taking  one  of  the  fixed  points  for  origin  and  denoting  by  /  the  distance 
of  the  other,  we  have  0  and  /  as  the  roots  of 

au  +  b 

u  = T, 

cu  +  d 

with  the  conditions  ad  —  be  =  1,  a  +  d  >  2.     Hence  h  =  0,  a  —  d  =  cf,  ad  =  1, 

K  =  -j',  then  K  is  greater  or  is  less  than  1  according  as  cf  is  positive  or  is 

negative.     We  shall  take  K  >1  as  the  normal  case ;   and   then  the    sub- 
stitution is 

a2 


w  = T 

cz  +  d 


with  a>l>  d,  a  +  d>2,  ad  =  l. 


Ex.  1.  A  2-eurve  is  drawn  through  either  of  the  fixed  points  of  a  real  hyperbolic 
substitution  :  shew  that  the  w-curve,  into  which  it  is  changed  by  the  substitution,  touches 
the  z-curve.  Hence  shew  that  any  2-circle  through  the  two  fixed  points  of  the  substi- 
tution is  transformed  into  itself. 


259.]  ■  ELLIPTIC   SUBSTITUTIONS  635 

Ex.  2.  Let  A  be  a  circle  through  the  origin  and  the  point  / ;  and  let  Cq  be  the  other 
extremity  of  its  diameter  through/.  Let  a  real  hyperbolic  substitution,  having  the  origin 
and  /  for  its  fixed  points,  transform  Cq  into  Ci,  Cj  into  C2,  C2  into  C3,  and  so  on  :  all  these 
points  being  on  the  circumference  of  A. 

Shew  that  the  radius  of  a  circle  C„,  having  its  centre  on  the  axis  of  x  and  passing 
through  c„  and  the  origin,  is 

so  that  C,i  is  the  locus  of  all  the  points  c„  arising  through  different  initial  circumferences 
A.     What  is  the  limit  towards  which  C„  tends  as  n  becomes  infinitely  great  1 

Ex.  3.  Apply  the  inverse  substitution,  as  in  Ex.  2,  to  obtain  the  corresponding  result 
and  the  corresponding  limit. 

Ex.  4.  Prove  that  a  curve  of  finite  length  will  meet  an  infinite  number,  or  only  a 
finite  number,  of  the  circles  (7„,  according  as  it  meets  or  does  not  meet  the  circle  having 
the  line  joining  the  common  points  of  the  substitution  for  diameter. 

{Note.     All  these  results  are  due  to  Poincare.) 

It  follows,  from  what  precedes,  that  no  real  substitution  can  be  loxodromic ; 
for,  when  the  multiplier  of  a  real  substitution  is  not  real,  its  modulus  is 
unity. 

It  is  not  difficult  to  prove  that  when  a  substitution,  with  complex 
coefficients  a,  h,  c,  d,  is  parabolic,  elliptic,  or  hyperbolic,  then  a  +  d  is 
either  purely  real  or  purely  imaginary.  In  all  other  cases,  the  substitution 
is  loxodromic 

Any  loxodromic  substitution  can  be  expressed  in  the  form 

VI  —  a_j^z  —  a 

the  coefficients  of  the  quadratic  determining  a  and  ^  are  generally  not  real, 
and  the  multiplier  K,  defined  by 

2K  =  {a  +  dy-2-  (a  +  d)  {{a  +  df  -  4}^, 
is  a  complex  quantity  such  that,  if 

K  =  pe''", 
where  p  and  w  are  real,  then  p  is  not  equal  to  unity  and  w  is  not  zero. 

Ex.  Shew  that,  if  a,  b,  c,  d  are  real  or  complex  integers  and  ad-  he  is  equal  to  1  or  j, 
the  only  possible  elliptic  substitutions  are  periodic  of  order  2,  3,  or  6 :  and  construct  an 
example  of  each.  (Math.  Trip.,  Part  II.,  1898.) 

260.  Further,  it  is  important  to  notice  one  property,  possessed  by  elliptic 
substitutions  and  not  by  those  of  the  other  classes  :  viz.  an  elliptic  substitution 
is  either  periodic  or  infinitesimal. 

Any  elliptic  substitution  of  which  a  and  /3  are  the  distinct  fixed  points, 
(they  are  conjugate  imaginaries),  can  be  put  into  the  form 

«;  —  «_„  z  —  a 


636  INFINITESIMAL   AND   PERIODIC   SUBSTITUTIONS  [260. 

where  \K\  =  1:  let  K  =  e^\     Then  the  mth  power  of  the  substitution  is 


Wr».  —  CL      z  —  a 


Now  if  6  be  commensurable  with  27r,  so  that 

ej^TT  =  X/fM, 

then,  taking  m  =  /u,,  we  have 


z  —  a 


that  is,  w^  =  z, 

or  the  substitution  is  jDeriodic. 

But  if  Q  be  not  commensurable  with  Itt,  then,  by  proper  choice  of  m,  the 
argument  inQ  can  be  made  to  differ'from  an  integral  multiple  of  Itt  by  a  very 
small  quantity.  For  we  expand  ^/27r  as  an  infinite  continued  fraction :  let 
jj/g,  'p'\(i  be  two  consecutive  convergents,  so  that  p'q  —  pq'  =  ±  1-     We  have 

/=^  +  xr<-a,whereO<X<l, 
lir      q  \q       q) 

=  1^1 
q      f 

where  |  ?;  |  <  1,  that  is,  qQ  —  2p7r  =  27r7;  -  , 

where  77,  being  real,  is  numerically  less  than  1.     Hence,  taking  m  =  q,  we 
have 


Wa-a      z-a  /^-^  _z-a 


i  +  ^»-+ 


Wq  —  0        Z  —  13  Z  —  ^ 

where,  by  making  q  large,  we  can  neglect  all  terms  of  the  expansion  after  the 
second.     Then 

_{z-a){z-  /3)  27777  • 
a— /8  q 

that  is,  by  taking  a  series  of  values  of  q  sufficiently  large,  we  can,  for  every 
value  of  z  find  a  value  of  w  differing  only  by  an  infinitesimal  amount  from 
the  value  of  z.  Such  a  substitution  is  called  infinitesimal ;  and  thus  the 
proposition  is  established. 

But  no  parabolic  and  no  hyperbolic  substitution  is  infinitesimal  in  the 
sense  of  the  definition.     For  in  the  case  of  a  parabolic. substitution  we  have 

1  1 

= +  qc, 

lUq  —  a      z  —  a 

which  does  not,  by  a  proper  choice  of  q,  give  Wg  nearly  equal  to  z  for  every 
value  of  z :  and  a  parabolic  substitution  is  not  substitutionally  periodic,  that 
is,  it  does  not  reproduce  the  variable  after  a  certain  number  of  applications. 
But  it   may  lead  to  periodic   functions  of  variables :   thus  {z,  z  +  co)  is   a 


260.]  INVERSION    CONNECTED   WITH   SUBSTITUTIONS  637 

parabolic  substitution.  And  in  the  case  of  a  hyperbolic  substitution,  we 
have 

Wq—  ^  Z  —  ^' 

where  X  is  a  real  quantity  which  differs  from  1.  No  value  of  q  gives  Wq 
nearly  equal  to  z  for  every  value  of  z :  hence  the  substitution  is  not  infini- 
tesimal.    And  it  is  not  substitutionally  periodic. 

Similarly,  a  loxodroraic  substitution  is  not  periodic,  and  is  not  infini- 
tesimal. 

Hence  it  follows  that,  in  dealing  Avith  groups  of  substitutions  of  the  kind 
above  indicated,  viz.  discontinuous,  all  the  elliptic  transformations  which  occur 
must  he  substitutionally  periodic  :  for  all  other  elliptic  transformations  are 
infinitesimal.  It  is  easy  to  see,  fi^om  the  above  equations,  that  the  effect  of 
an  unlimited  repetition  of  a  parabolic  substitution  is  to  make  the  variable 
ultimately  coincide  with  the  fixed  point  of  the  substitution ;  and  that  the 
effect  of  an  unlimited  repetition  of  a  hyperbolic  substitution  is  to  make  the 
variable  ultimately  coincide  with  one  of  the  fixed  points  of  the  substi- 
tution. These  common  points  are  called  the  essential  singularities  of  the 
respective  substitutions. 

261.  It  has  been  proved  (§  258)  that  a  linear  relation  between  two 
variables  can  be  geometrically  represented  as  an  inversion  with  regard  to  a 
circle,  followed  by  a  reflexion  at  a  straight  line.  The  linear  relation  can  be 
associated  with  a  double  inversion  by  the  following  proposition*,  due  to 
Poincare : — 

When  the  inverse  of  a  point  P  ivith  regard  to  a  circle  is  inverted  with 
regard  to  another  circle  into  a  point  Q,  the  coviplex  variables  of  P  and  Q  are 
connected  by  a  lineo-linear  relation. 

Let  z  be  the  variable  of  P,  u  that  of  its  inverse  with  regard  to  the  first 
circle  of  centre  /  and  radius  r ;  let  lu  be  the  variable  of  Q,  and  let  the  second 
circle  have  its  centre  at  g  and  its  radius  s.  Then,  since  inversion  leaves  the 
vectorial  angles  unaltered,  we  have 

{u-f){z,-f,)^r- 
for  the  first  inversion,  and 

{w-g){ua-go)  =  s' 

for  the  second.     From  the  former,  it  follows  that 


^-/ 


^•.2  o2 

and  therefore  ■ ,H =/o  —go, 

z-J     w-g     ' 

1     A-       ^  az  +  ^ 

leading  to  w  = §, , 

°  yz  +  o 

*  Acta  Math.,  t.  in,  (1883)\  p.  51. 


638  SUBSTITUTIONS   AS   INVEESIONS  [261. 

where,  when  olB  —  /By  =  1,  we  have 

rsa.  =g(fo-  go)  +  s^ 
rs^  =  gr-"  -fs'-fgifo-go), 
rsy=fo-go, 
rsB  =-f{fo-go)  +  r'. 
This  proves  the  proposition. 

Moreover,  as  the  quantities  f,  g,  r,  s  are  limited  by  no  relations,  and  as, 
on  account  of  the  relation  aS  -  yS7  =  1,  there  are  substantially  only  three 
equations  to  determine  them  in  terms  of  a,  /3,  y,  B,  it  follows  at  once  that  the 
lineo-linear  relation  can  he  obtained  in  an  infinite  number  of  ways  by  a  pair  of 
inversions,  and  therefore  in  an  infinite  number  of  ways  by  an  even  number  of 
inversions. 

Again,  taking  the  two  circles  used  in  the  above  proof,  we  have  ^ 

rs{a+B±V)  =  {r±sr-{f-g){f-g,) 

=  (r  +  sf  -  c^^ 

where  d  is  the  distance  between  the  centres  of  the  circles.  Hence  a  +  B 
is  real,  and  the  substitution  cannot  be  loxodromic.  Moreover,  if  the  circles 
touch,  the  substitution  is  parabolic ;  if  they  intersect,  it  is  elliptic ;  if  they 
do  not  intersect,  it  is  hyperbolic. 

Eliminating  r  and  s  between  the  equations  which  determine  a,  /3,  y,  B,  we 
find 

SO  that,  when  one  centre  is  chosen  arbitrarily,  the  other  centre  is  connected 
with  it  by  the  linear  substitution*. 

Ex.  1.  Shew  that,  if /and  g  lie  on  the  axis  of  real  quantities,  so  that  the  substitution 
is  real,  then 

^■'=(/-X)(/-mX       s^={.g-\){g-,j.\ 
where  X  and  ju,  are  the  fixed  points  of  the  substitution. 

Hence  prove  that,  if  two  real  substitutions  be  given,  it  is  generally  possible  to 
determine  three  circles  1,  2,  3,  such  that  the  substitutions  are  equivalent  to  successive 
inversions  at  1  and  2  and  at  1  and  3  respectively.     Discuss  the  reality  of  these  circles. 

(Burnside.) 

Ex.  2.  Shew  that,  if  a  loxodromic  substitution  be  represented  in  the  preceding 
geometrical  manner,  at  least  four  inversions  are  necessary.  (Bui'nside.) 

This  geometrical  aspect  of  the  lineo-linear  relation  as  a  double  inversion 
will  be  found  convenient,  when  the  relation  is  generalised  from  a  connection 
between  the  variables  of  two  points  in  a  plane  into  a  connection  between  the 
variables  of  two  points  in  space. 

*  Burnside,  Mess,  of  Math.,  vol.  xx,  (1891),  pp.  163—166. 


261.]  SOME   APPLICATIONS   OF    CONFORMAL   REPRESENTATION  639 


NOTE. 

Some  applications  of  conformal  representation  to 
mathematical  physics. 

It  may  be  useful  to  give  instances  of  the  manner  in  which  the  analysis 
arising  in  the  theory  of  conformal  representation  is  used  in  some  branches 
of  applied  mathematics.  The  branches  selected  are  merely  typical;  they 
are  not  exhaustive. .  In  each  instance,  the  account  is  at  once  elementary  and 
introductory  ;  for  the  fuller  development,  recourse  may  be  had  to  the  respec- 
tive authorities  mentioned. 

I.     Hydrodynamics, 

A.  The  simplest  applications  arise  in  connection  with  the  irrotational 
steady  motion  of  a  uniform  incompressible  fluid  in  two  dimensions.  It  is 
thereby  implied  that  the  motion  is  the  same  in  all  planes  parallel  to  one 
particular  plane ;  and  this  plane  is  taken  to  be  the  plane  of  x,  y,  that  is,  the 
^-plane. 

Let  p  and  q  denote  the  component  velocities  at  any  point  x,  y  of  the 
fluid,  parallel  to  the  axes  of  x  and  y  respectively.  When  there  is  a  velocity 
potential,  let  it  be  denoted  by  u;  then 


du  du 

^^"dx'     ^^~dy' 


The  equation  of  continuity  is 


dx     dy        ' 


that  is,  the  velocity  potential  u  satisfies  the  equation 

The  analysis  in  |  10  shews  that  there  exists  a  function  v,  associated  with 
the  function  u,  and  uniquely  determinate  (save  as  to  an  additive  constant) 

by  the  relations 

dv  ^     du        dv  _du  ■ 
dx         dy '      dy     dx' 

These  functions  are  such  that,  when  tu  denotes  u  +  iv  and  z  denotes  x  +  iy, 
then  a  functional  relation  exists  between  w  and  z,  the  only  other  quantities 
occurring  in  the  relation  being  constants. 


640   "  APPLICATIONS   OF    CONFORMAL  REPRESENTATION  [261. 

The  differential  equation  of  the  direction  of  motion  at  the  point,  that  is, 

of  the  stream  line,  is 

dx  _dy 

p  ~~  q' 
or,  what  is  its  equivalent, 

qdx  —  pdy  =  0. 

In  terms  of  the  function  v,  this  equation  becomes 

consequently,  the  stream  lines  are  given  by  the  relation 

V  =  constant. 
As  the  stream  line  is  the  direction  of  the  velocity  at  every  point  along  its 
course,  it  follows  that  there  is  no  flux  of  the  liquid  across,  a  stream  line. 
Thus  it  is  possible  to  consider 

V  =  some  particular  constant 
as  a  rigid  wall  along  which  the  frictionless  fluid  flows ;  and  then  the  streani 
lines  of  the  fluid  in  such  a  motion  are  given  hjv  =  constant. 
Moreover,  we  know  that  v  satisfies  the  differential  equation 

d^v      d^v  _ 

When  expressed  in  terms  of  p  and  q,  this  equation  is 

^  _  ^  =  0, 
dx      dy 

an  equation  which   is   the   analytical   expression   of  the  property  that  the 

motion  is  irrotational. 

It  thus  appears  that,  when  we  have  a  relation  between  complex  variables 

w  and  z  in  the  form 

u  +  iv  =  w  =f(2:)=f{x  +  iy), 

we  can  take  u  as  the  velocity  potential  and  v  as  the  stream  line  function  for 
the  type  of  fluid  motion  considered.  The  relations  between  the  functions 
u  and  V  are  an  analytical  statement  of  the  property  that  the  equipotential 
lines  and  the  stream  lines  are  orthogonal  to  one  another. 

Let  V  denote  the  resultant  velocity  at  the  point  x,  y,  so  that 

Tro       o       o     /duV      /Buy 

Now  (§  8) 

dw      dw     d.u      .  dv      du      .  da 
dz      dx      dx        dx      dx        dy 
consequently, 

-     y^     I  dw 
\  dz    ' 
so  that  the  modulus  of  dwjdz  gives  the  resultant  velocity  at  any  point. 


261.]  TO   HYDEODYNAMICS  641 

It  is  found  convenient  to  consider  the  quantity  dzjdw  in  some  hydro- 
dynamical  investigations.     We  write 


?= 

dz 
dw' 

^i= 

1 

^= 

1 

-p  + 

iq 

p  +  iq 

Thus 
and 


p^  +  q^  ' 

where  cf)  is  the  angle  between  the  direction  of  the  velocity  and  the  axis  of  x. 
(The  introduction  of  the  use  of  the  symbol  ^  is  due  to  Kirchhoff.) 

Further,  if  P  denote  the  pressure  at  any  point  and  if  p  denote  the 
(constant)  density,  we  have 

P 

— h  P"  +  o''^  =  constant 
P 

everywhere.  Thus  the  surfaces  of  constant  pressure  are  surfaces  of  constant 
velocity ;  in  particular,  a  free  surface  (such  as  a  surface  between  air  and 
water)  is  a  surface  of  constant  velocity  in  the  kind  of  motion  under  con- 
sideration. 

Some  examples  will  now  be  taken,  very  briefly,  to  illustrate  the  foregoing 
statements. 

Ux.  1.    Let 

to  =  az, 

where  a  is  a  real  constant.     Then 

u  =  ax,         V  =  ay. 

In  the  fluid  motion  the  velocity  is  constant,  and  equal  to  a  ;  the  stream  lines  are  parallel 
to  the  axis  of  x  ;  and  so  the  frictionless  fluid  can  be  regarded  as  flowing  with  constant 
velocity  between  two  parallel  walls. 

Ex.  2.     Let 

W  =  G  (  2  + 

where  a  and  c  are  real  constants.     Then 

u  =  c\r-{ —  1  cos  Q,         v  =  c\r 

when  polar  coordinates  are  used. 

The  stream  lines  are  given  by  v  =  constant.  In  particular,  the  stream  line  i'  =  0.  when 
traced,  is  the  axis  of  x  from  .r=QO  to  x  =  a,  corresponding  to  ^  =  0  ;  then  the  semi-circle, 
centre  the  origin  and  radius  a  ;  and  then  the  axis  of  x  from  x=  —a  to  .r=  —  oo,  corre- 
sponding to  6  =  7r.     So  the  fluid  can  be  regarded  as  flowing  along  a  rigid  wall  past  a 

F.   F.  41 


642  APPLICATIONS    OF    CONFORMAL   REPRESENTATION  [261. 

cylindrical  boss.     It  may  also  be  regarded  as  flowing  pas't  a  fixed  cylinder  in  a  direction 
perpendicular  to  the  axis. 

As  regards  the  velocity,  we  have 

dw 
dz 


so  that 


At  an  infinite  distance 


p=  —  c  f  1  — -ij-cos  2^  j ,  g- =  c -;2  sin  2^. 


so  that  the  velocity  of  the  fluid  towards  infinity  is  constant  and  is  parallel  to  the  straight 
wall. 

Ex.  3.     Let 

an  example  constructed  by  Helmholtz*,  who  was  the  pioneer  in  this  line  of  investigation. 

Then 

x=u+e^'' cosv,        ^  =  v  +  e'"-smv. 

The  stream  line  «7=7r  gives  ^  —  tt  for  all  values  of  u.  We  then  have  .r  =  tt  — e"  ;  as  ii 
varies  from  —  oo  to  0,  the  variable  x  varies  from  -  oo  to  - 1  ;  and  as  u  continues  its 
variation  from  0  to  +  oo ,  the  variable  x  varies  from  —  1  to  —  oo  :  that  is,  the  complete 
variation  of  u  from  —  oo  to  +  oo  requires  a  duplicated  variation  of  the  part  of  the  axis 
of  X,  from  —  00  to  —  1  in  the  first  place,  and  then  from  -  1  to  —  oo  in  the  second  place. 

Similarly  for  the  stream  line  »=  —  tt.  We  have  the  line  ?/=  —  tt,  for  all  values  of  u. 
The  range  of  x  is  the  same  as  for  the  stream  line  v  =  '7t. 

The  stream  line  v  =  0,  symmetrically  situated  between  these  two  stream  lines  ?;  =  7r 
and  v=  —  TT,  allows  the  full  variation  of  x  from  +  oo  to  —  qo  . 

As  regards  the  velocity,  we  have 

dz 

that  is, 

-P  +  ^^-lh" 

1  +  e"  cos  V  —  ie'"  sin  v 
~,    1+26^*008  v  +  e^"     ' 
so  that 

-P       _     -?     _  1 


1  +  e"  cos  V     e"  sin  v      1  +  2e"  cos  v  +  e^" " 

Between  v  =  'n-  and  v—0,  obviously  q  is  negative;  so  that,  in  that  range,  the  flow  is 
towards  the  axis  of  3/.  Between  v=  -tt  and  v  =  0,  obviously  q  is  positive;  so  that,  in 
that  range,  the  flow  is  again 'towards  the  axis  of  y. 

We  have  the  case  of  a  fluid  flowing  in  all  directions  along  the  proper  stream  lines  into 
a  canal  bounded  by  the  walls  y  =  7r  and  j/=  —  tt,  in  the  direction  of  the  range  from  x=  —  I 
to  .r=  -  00 ,  symmetrically  with  respect  to  the  plane  y=0. 

Ex.  4.     Discuss  similarly  the  hydrodynamical  interpretation  of  the  relations  : — 

(1)     tu  =  az'i  :  (n)     z  =aw2  ; 

(iii)    IV  =  a  cosh  z;  (iv)     z  =acosh.w  ; 

(v)     iv  =  alogz  ;  (vi)     w  =  ae^. 

*  Berl.  Monatsb.,  (1868),  pp.  215—228. 


.261.]  TO   HYDKODYNAMICS  643 

B.  The  preceding  illustrations  shew  how  interpretations  can  be  given 
for  assigned  functional  relations  between  w  and  z.  We  shall  now  sketch, 
very  briefly,  a  constructive  process  which  leads  to  the  functional  relations 
appropriate  to  many  propounded  problems  in  the  two-dimensional  irrotational 
motion  of  a  perfect  fluid. 

Reverting  to  the  symbol  ^,  we  have 

t=_—  * 

dw 

1     *• 

where  V  is  the  velocity  at  any  place  and  ^  is  the  inclination  of  the  stream 
line  to  the  axis  of  x.  Now  ^  itself  is  a  complex  variable ;  so,  in  association 
with  the  ^-plane  and  the  -it^-plane,  we  shall  find  it  convenient  to  consider  a 
f-plane. 

We  have  seen  that,  when  we  have  a  free  surface  in  the  w-plane,  it  is  a 
surface  along  which  the  velocity  is  constant.  Without  loss  of  generality, 
we  can  take  unity  as  this  velocity — it  is  only  an  assumption  as  to  the  unit 
of  time.  Thus  a  free  surface  in  the  w-plane  gives  a  circle  of  radius  unity  in 
the  ^-plane. 

The  moving  fluid,  where  it  has  not  a  free  surface,  is  bounded  (in  the 
problems  to  be  considered)  by  plane  walls.  All  of  these  are  represented  in 
the  w-plane  by  the  equation  v  =  constant ;  that  is,  by  a  succession  of  lines 
parallel  to  the  real  axis  in  that  plane.  Along  these  plane  walls  the  direction 
of  flow  is  straight ;  arid  so  </>  is  constant,  that  is,  the  corresponding  lines  in 
the  ^-plane  are  the  radii  of  the  circle  of  radius  unity  which  represents  the 
free  surface  in  the  w-plane.     We  thus  have  a  figure  in  the  ^-plane. 

Next,  introduce  another  complex  variable  ^',  defined  by  the  equation 

r  =  iog^; 

then  ^'  =  log  y  +  ^(/), 

so  that  we  have  another  figure  in  another  plane,  the  ^'-plane.  The  circle 
F  =  1  in  the  ^-plane  becomes  part  of  the  ^'-axis  of  imaginary  quantities, 
which  accordingly  corresponds  to  the  free  surface  in  the  motion,  given  in  the 
w-plane  by  a  particular  line  v  =  constant.  To  the  radial  lines  0  =  constant 
in  the  ^-plane,  correspond  straight  lines  in  the  f'-plane  parallel  to  the  real 
axis  in  that  plane,  that  is,  by  general  lines  v  =  constant. 

It  thus  is  necessary  to  construct  a  relation  which  shall  secure  that  a 
figure-  in  the  ■?^-plane  bounded  by  straight  lines  parallel  to  its  axis  of  real 
quantities  shall  correspond  to  a  figure  in  the  ^'-plane  bounded  by  straight 

41—2 


644  APPLICATIONS   OF    CONFOKMAL   REPRESENTATION  [261. 

lines,  neither  necessarily  nor  usually  parallel  to  the  axis  of  real  quantities  in 
the  ^'-plane. 

Finally,  this  result  is  achieved  by  making  the  figures  in  the  w-plane  and 
in  the  ^'-plane  respectively  to  be  represented  upon  one  and  the  same  half  of 
another  plane  of  a  final  complex  variable  t,  the  boundary  in  each  representation 
being  transformed  into  the  real  axis  in  the  i-plane. 

We  thus  have  a  succession  of  planes  of  complex  variables,  conformally 
represented  each  upon  the  next  and  therefore  all  upon  one  another.  There 
is  the  5-plane,  the  original  plane  of  motion  of  the  fluid.  There  is  the 
w-plane,  giving  the  velocity  potential  and  the  stream  line  function  of  the 
motion.  There  are  the  ^-plane,  relatively  unimportant,  and  the  ^'-plane 
(where  ^'  =  log  ^),  giving  the  magnitude  and  the  direction  of  the  velocity  of 
the  fluid.  And  there  is  the  i-plane,  on  the  same  positive  half  of  which 
both  the  variables  ^'  and  w  are  represented.  In  these  final  representations 
we  have  some  relations  of  the  types 

and  so,  eliminating  t  we  have  a  relation 
that  is,  a  relation 

iogf-;7i)=/(^)' 


which,  on  integration,  gives  the  connection  between  the  variables  2  and  tv. 

The  hydrodynamical  problem  can  thus  be  made  to  depend  upon  the 
solution  of  the  associated  problem  in  conformal  representation.  The  examples 
of  the  latter,  which  have  been  given  already,  can  be  used  to  solve  some  of 
the  hydrodynamical  problems  which  arise :  one  further  illustration  must 
suffice  here. 

&.  Imagine  a  fluid  moving  symmetrically  between  two  parallel  walls  inserted  into  a 
relatively  infinite  quantity  of  the  fluid,  so  as  to  form  a  sort  of  jet  coming  out  between 
the  walls.  We  shall  assume  these  to  have  their  ends  in  a  line  perpendicular  to  their 
direction. 

Then  in  the  s-plane,  we  have  between  the  walls  a  fluid  moving  along  the  canal  sym- 
metrically with  respect  to  the  middle  line  and  outside  the  canal  so  as  to  supply  the  canal. 
The  boundary  of  the  fluid  is  made  up  of  the  sides  of  the  two  walls  in  the  fluid  and  the 
double  free  surface  within  the  canal,  the  two  free  surfaces  being  symmetrical  with  respect 
to  the  middle  line. 

In  the  ^-plane,  we  have 

so  that  the  boundary  formed  by  one  wall  up  to  its  extremity  is  given  by  ^  =  0,  that  is,  by 
the  axis  of  real  quantities  from  00  to  +  1  ;  then  by  the  circumference  of  a  circle  of  radius 
V=  1  and  centre  ihe  origin,  representing  the  free  surface  ;  and  then  by  0  =  27r,  that  is, 
by  the  axis  of  real  quantities  from  + 1  to  oo  ,  representing  the  other  wall. 


261.]  TO   HYDRODYNAMICS  .645 

In  the  {"'-plane,  we  have 

so  the  boundary  of  the  configuration  in  the  ^'-plane  is 

^'  +  irj' =  ]og  l+i(f) 

viz.  it  is  two  lines  parallel  to  the  axis  of  real  quantities  given  by  (^  =  0  and  0  =  27r, 
together  with  the  part  of  the  axis  of  imaginary  quantities  intercepted  between  these 
lines. 

The  representation  of  this  bounded  area  upon  the  half  of  the  ^-plane  is  (§  269,  p.  673) 
given  by 

dC'^       P       . 
dt      (^2_i)4-' 

or,  adapting  the  scale  and  settling  the  origin,  we  can  take 

f'=2cosh-ii;. 

When  {■'  ranges  along  the  line  (^  =  0  from  the  origin  to  oo ,  <  ranges  from  1  to  oo  along  its 
real  axis.  When  f  ranges  along  the  line  (f)  =  2ir  from  the  axis  of  imaginary  quantities  to 
infinity,  t  ranges  from  —  1  to  —  oo  along  its  real  axis.  When  ^'  ranges  along  the  axis  of 
imaginary  quantities  from  7;'  =  0  to  r]'  =  2Tr,  t  ranges  from  1  to  —  1  along  its  real  axis. 

Now  for  the  ■?y-plane,  let  the  breadth  of  the  jet  towards  its  issue  (that  is,  at  a  great 

distance  from  the  extremities  of  the  walls  where  the  jet  enters  the  canal)  be  26  ;  so  that 

one  free  surface  is  given  by  v  =  6,  and  the  other  by  v  =  —  6.     Let  the  velocity  potential 

curve  through  the  finite  extremities  of  the  walls  be  u  =  ().     We  have  in  general  (§  269, 

p.  673) 

w  =  M\ogt-\-N. 

At  the  place  corresponding  to  the  extremity  of  the  lower  wall,  we  have 

w  =  (d  —  ib  ; 

and  at  the  place  corresponding  to  the  extremity  of  the  upper  wall,  we  have 

tt;  =  0  +  lb. 

Hence 

26, 
?«  =  —  log  t  —  %b. 

TV 

Consequently,  the  relation  between  z  and  vj  is  given  by  the  elihiination  of  t  between  the 

two  equations 

dz 
log  -r^  =  t'  =  2  cosh"  1 1 
^  dw     ^ 

26.      ^     ., 
IV  =  —  log  t  —  ib 

TT 

As  a  matter  of  fact,  it  is  more  convenient  to  keep  the  two  equations  and  not  eliminate  t. 

Along  the  free  stream  line,  we  have  V=\,  and  so 

dz  __       ^i 
dw~     ^    ' 
or  writing  6  =  (f)  +  iT,  we  have 

dz  _  ei  ^ 

dvj         ' 

consequently  along  that  stream  line  we  have 

ie  =  2Gosh-'^t, 
and  therefore 

t  =  GOSw0, 


646  APPLICATIONS   OF   CONFORMAL  REPRESENTATION  [261. 

where  t  ranges  from  0  to  1.     Also 

11=  —  log  cos  T!  6 

■n 

along  the  line.     But  u  is  the  velocity  potential ;  and  along  the  line,  we  have 
so  that  we  can  take 

llz=  —s. 

Thus  the  equation  of  the  free  stream  line  is 
Further,  we  have 


s  =  2  —  log  sec  \  6. 


1     _      dz  _       ie 
p  —  iq         div 

along  the  stream  line,  because p^  +  q^=\;  so 

p=-cos^,         y— —  sin^, 

and  therefore  6  is  the  inclination  to  the  axis  of  x  of  the  tangent  to  the  stream  line. 

The  foregoing  equation  is  therefore  the  intrinsic  equation  of  the  free  stream  line.     It 
is  easy  to  prove  that  the  equivalent  Cartesian  equations  of  the  line  are 

x  =  2-  (sin2 1(9  -  log  sec  h6)  ] 


y=    -(^  — sin^)  j 

TV  ' 

taking  the  origin  at  the  finite  extremity  of  the  wall.  As  6  ranges  from  0  to  tt,  ^  ranges 
from  0  to  -  00 ,  and  y  ranges  from  0  to  6,  that  is,  ?/  =  &  gives  the  asymptotic  distance 
between  the  free  stream  line  and  the  nearer  wall. 

Similarly  for  the  other  free  stream  line  and  the  other  wall.  Thus,  as  the  breadth  of 
the  issuing  jet  is  26,  the  distance  between  the  walls  is  6  +  26  +  &,  that  is,  4&.  Thus  the 
breadth  of  the  issuing  jet  is  one-half  the  distance  between  the  parallel  walls,  a  result  first 
stated  approximately  by  Borda  in  1766  and  first  proved  by  Helmholtz  in  1868. 

For  a  full  discussion  of  many  similar  applications  of  the  theory  of  conformal  repre- 
sentation to  the  irrotational  motion  of  a  liquid  in  two  dimensions,  reference  may  be 
made  to  Lamb's  Hydrodynamics  (4th  ed.),  ch.  iv.  "Some  later  investigations  are  due 
to  J.  G.  Leathern,  Phil.  Trans.  (1915),  pp.  439 — 487,  and  Proc.  Lond.  Math.  Soc,  Ser.  2, 
vol.  xvi,  (1917),  pp.  140 — 149;  he  gives  some  further  references. 

II.     Electrostatics. 

C.     In  all  the  space  free  from  electric  charges  in  a  planar  electrostatic 
field,  the  electric  potential  v  satisfies  the  equation 

d^v      d-v  _ 
dx^     dy- 
The  equipotential  lines  are  given  by 

V  =  constant. 
The  component  of  electric  force  in  any  direction  is 

dv 


/^ 


261.]  TO   ELECTROSTATICS  647 

where  dt  is  the  increment  in  the  positive  direction ;  thus  the  components  of 
electric  force  parallel  to  the  positive  directions  of  the  axes  are 

dv        dv 
dx'      dy' 

The  direction  of  the  line  of  electric  force  at  any  point  is  given  by  the 
equation 

'^/(-s)=''2'/(-|)' 

that  is,  by  . 

^r-  dec  —  ^  dy  =  0. 

dy  ox    ^ 

The  left-hand  side  is  a  perfect  differential,  because 

d_  /dv\  _^(_dv\. 

^y  \^y)    9^  V  9^/ ' 

denoting  it  by  du,  we  have 

du  _dv      du         dv 

dx     dy'    dy         dx' 
Eliminating  v,  we  have 

9%      dhi  _     _ 
dx"      dy"^         ' 

and  the  lines  of  electric  force  are  given  by 

u  =  constant. 

Alike  from  the  physical  properties,  and  from  these  mathematical  expressions 
for  equipotential  lines  and  from  the  lines  of  electric  force,  it  follows  that  the 
two  sets  of  lines  are  orthogonal  to  one  another  at  every  common  point. 

It  thus  follows  from  former  investigations  (§  8)  that,  if  w  =  u  +  iv  and 
z  =  X  +  iy, 

tu  =  some  function  of  z  =f{z). 

The  electric  intensitv  at  any  point  is 

that  is,  it  is  equal  to 

dw 
dz 

corresponding  to  the  velocity  in  the  hydrodynamical  problem  and  to  the 
magnification  in  the  conformation  problem.  Further,  v  =  constant  along 
any  conductor  ;  thus,  by  Coulomb's  law,  if  a  is  the  surface  density  of  electrifi- 
cation at  any  place  on  the  plane  conductor, 

1     dw  j 
^  ~  4!7r    dzV 


648  APPLICATIONS   OF   CONFORM AL  REPRESENTATION  [261. 

the  electric  intensity  being  measured   outwards   from  the  surface   of  the 
conductor. 

Thus  the  analysis  is  the  same  throughout  for  the  various  classes  of 
problems.  We  have  interpretations  in  the  various  vocabularies  of  the 
different  subjects  of  applied  mathematics.  We  shall  therefore  deal  very 
briefly  with  a  few  quite  simple  examples. 

Ex'.  1.     Let 

then  we  have  the  equipotential  lines  given  by 

1)  =  r^  sin  -J  6 
in  "polar  coordinates,  with  the  equivalent  equation 

in  Cartesian  coordinates.     They  are  a  family  of  coaxial  and  confocal  parabolas. 

In  particular,  when  v=0,  we  have  0  =  0  from  x  =  cc  to  .r=0  ;  and  then  ^  =  27r  from 
d;  =  0  to  ^  =  00 .  Thus  we  have  an  infinitely  thin  conductor,  in  the  form  of  a  straight  line 
stretching  along  the  axis  of  real  quantities  from  the  origin  to  infinity.  The  surface 
density  of  electrification  on  this  conductor  at  a  distance  d  from  the  origin  is 

1       7-1 
~d       2. 

OTT 

The  lines  of  electric  force,  given  by 

u  =  r2  cos  5  ^  =  constant, 
are  represented  in  Cartesian  coordinates  by  the  equation 

that  is,  they  are  a  family  of  coaxial  and  confocal  parabolas,  orthogonal  to  the  former 
family. 

Ex.  2.     Let 

w=sin~^s. 

We  have  (Ex.  6,  §  257) 

A^  =  sin  u  cosh  v,        y=cos  u  sinh  i\ 

The  equipotential  lines  v= constant  are  given  by  the  equation 

cos^  hv      sin^  hv       ' 
a  family  of  confocal  ellipses. 

In  particular,  when  v  =  0,  x  ranges  from  +1  to  —1  on  the  positive  side  and  on  the 
negative  side  along  the  real  axis.  Thus  there  is  an  infinitely  thin  conductor  in  the  fonn 
of  the  straight  line  between  —1  and  +1  on  the  axis  of  real  quantities.  The  surface 
density  of  electrification  on  the  conductor  at  a  distance  c  from  the  middle  is 

The  lines  of  electric  force  are  given  by 

x"^  y^    _^ 

sin^  u     cos^  u 

that  is,  they  are  the  family  of  confocal  hyperbolas,  orthogonal  to  the  former  family. 


261.]  TO    CONDUCTION   OF   HEAT  649 


■.  3. 

Construct  electrostatic  interpretations  of  the  following  equations  : 

(i) 

■iv  =  ism~^  z  ; 

(ii) 

w=\ogz  ; 

(iii) 

3  =  log  ?(? ; 

(iv) 

z  =^\iw  ; 

(V) 

w  =  sn  ^  ; 

(vi) 

e^  =  en  w. 

For  a  full  discussion  of  the  application  of  the  theory  of  conformal  representation  to 
general  problems  in  electrostatics,  reference  should  be  made  to  Sir  J.  J.  Thomson's  Recent 
Researches  in  Electricity  and  Magnetism,  ch.  iii ;  and  to  the  treatise  by  J.  H.  Jeans,  The 
Mathematical  Theory  of  Electricity  and  Magnetism. 


III.     Conduction  of  Heat. 

D.  The  same  analysis  can  be  used  for  the  discussion  of  any  number  of 
two-dimensional  problems  in  conduction  of  heat  when  the  temperature  is 
steady :  that  is,  when  the  temperature  varies  from  place  to  place  in  the 
plane  of  the  two  dimensions,  while  at  every  place  it  is  independent  of  the 
time. 

For  purposes  of  illustration,  we  shall  assume  that  the  conductivity  K  is 
constant.  The  temperature  at  any  point  will  be  denoted  hy  v;  so  the  flux 
of  heat  at  a  point  perpendicular  to  the  axis  of  oc  is 

and  at  a  point  perpendicular  to  the  axis  of  y  is 


f'-^,y 


The  condition  that  there  is  neither  gain  nor  loss  of  heat  at  any  place  is 
that  is,  K  being  constant, 


dx      dy 


The  curves  across  which  there  is  no  flux  of  heat — they  may  be  called  lines  of 

flow — are  given  by 

dx  _dy 

that  is, 

f^dx-fjy  =  0. 

Because 

d_fdv\  _d_  fdv^ 
dy  \dy)      dx  V     dxj 


650 


APPLICATIONS   OF   CONFORM AL  REPRESENTATION 


[261. 


this  last  expression  is  a  perfect  differential,  say  du ;  then 

du  _dv       du  _      dv 
doc     dy'     dy         dx' 
and  therefore 

d^u     dhi  _ 
dx^     dy- 

Once  more,  we  have  the  same  equations  as  before.  We  know  that  u  +  iv  can 
be  expressed  as  a  function  oi  x-\-  iy ;  so,  writing  tu  =  u  +  iv  and  z  =  x  -\-  iy,  and 
taking 

w=f{z\ 

where  f  is  any  functional  form,  we  can  interpret  the  analytical  relation  as  a 
result  in  a  steady  temperature  problem  in  the  conduction  of  heat,  by  taking 
the  lines 

V  =  constant 

as  the  isothermal  line's,  and  the  lines 

u  =  constant 

as  the  lines  of  flow.     The  total  flux  of  heat  at  any  point  is 


that  is,  it  is 


^<iy-r* 


K 


Ex.  1.     Consider  the  relation 
where  a  is  a  real  constant.     We  have 


diu  I 

dz  I ' 

=  i  cot  2, 

g  — 12   r    giz 


=  log 
=  log 


Thus 


and 


av  =  tail  ~  1 
=  tan~^ 


1  +  e2i^ 

l_2e-22'cos2A'  +  e-42' 
2e~2y  sin  2^ 


sin  2x 


au=\ 


sinh  2y ' 
l_2e-2j/cos2.r  +  e-*2'' 


As  regards  the  isothermal  lines  given  by  v  =  constant,  we  have  y=0  when  .r=0  so  long 
as  y  is- not  zero,  and  v  =  0  when  x  =  hiT  so  long  as  ?/  is  not  zero.  When  y=0  and  x  ranges 
from  0  to  ^TT,  we  have 


2a 


261.]  TO    CONDUCTION   OF   HEAT  651 

We  thus  have  a  half -infinite  rectangiilar  plane,  kept  at  temperature  zero  along  the  half- 
infinite  plane  wall  :?-=0,  at  temperature  7r/2a  along  the  finite  plane  wall  ?/  =  0  between  x=0 
and  hrr,  and  at  temperature  zero  along  the  half-infinite  plane  wall  a:  =  ^ir. 
The  isothermal  lines  in  the  solid  are  given  by 

sinh  2^— a  sin  2x, 
for  parametric  values  of  a  ;  and  the  lines  of  flow  are  given  by 

cosh  2y  =  /3  cos  2x, 
for  parametric  values  of  /3. 

The  actual  flux  of  heat  at  any  place  is 

dw 
~di 


K 

that  is, 


2/1        1 

a    I  sin  22  | 

AK  1 


a    cosh  2y  —  cos  'ix ' 
This  result,  in  substance,  was  first  given  by  Fourier*. 

Ex.  2.     We  shall  use  the  property  of  conformal  representation  to  change  this  solution 
of  a  given  problem  into  the  solution  of  another  problem.     The  relation  was 

gaw  _  l  got  Z. 

The  relation  /=  -  cos  2s  gives 

x'=-  cos  'Hx  cosh  2?/,        y' — sin  'ix  sinh  2?/. 
When  the  range  of  x  is  given  by  0^.r^^7r  and  of  y  is  ^iven  by  0=^.y,  y'  is  always 
positive  or  zero,  and  x'  ranges  from  -  oo  to  +  oo  .     Thus  the  open  half-infinite  rectangular 
plane  bounded  by  0  ^  ^  ^  Jtt,  0^  y,  is  represented  upon  the  positive  half  of  the  s' -plane. 

The  relation 


gives  (p.  619,  Ex.  13)  a  representation  of  the  positive  half  of  the  s'-plane  upou  the  interior 
of  a  circle,  of  radius  unity  and  centre  the  origin,  in  the  s"-plane. 

Accordingly,  we  seek  the  relation  between  the  original  w  and  this  final  s"  ;  it  is  easily 

found  to  be  ' 

i  —  2" 
1  —  ^V ' 

With  this  relation,  we  have  [a  being  real) 

2a(ti-^-^l.)  =  log (i+y,)2  +  y'2  ' 

and  therefore 

2ay=tan-i -^, . 

2x 

The  isothermal  lines  are  given  by  y  =  constant.  We  have  y  =  0  when  ^"2-f3/"2-l  =  0  so 
long  as  x"  is  not  zero.  When  ^-"  =  0  and  ^'Hy"^-  1  is  not  zero,  so  that  y"  can  range  from 
-  00  to  - 1  and  from  1  to  -I-  oo  ,  then  v=  irl4a.  And  when  x"  is  not  zero  within  the  boundary 
limits,  we  take  x"  positive.  We  thus  have  an  infinite  half-plane  with  a  semi-circular 
boss,  of  radius  unity  on  the  positive  side  of  the  half-plane.  The  boss  is  maintained  at 
temperatiu-e  zero  :  the  rest  of  the  boundary  of  the  infinite  plane  is  maintained  at  tem- 
perature 7r/4a. 

*  Theorie  de  la  Chaleur,  cb.  iii,  §  5. 


652  APPLICATIONS    OF   CONFORMAL   REPRESENTATION  [261. 

Ex.  3.     Obtain  the  lines  of  flow  for  the  transformation  in  the  last  example. 

Ex.  4.  Construct  the  relation  between  w  and  z  for  a  solid,  the  outer  surface  of  which 
is  outwardly  bounded  by  a  circular  cylinder  kept  at  temperature  zero  and  the  inner 
surface  of  which  is  another  circular  cylinder  kept  at  temperature  unity,  the  two  axes  being 
parallel, 

Ex.  5.     Apply  the  transformation 

to  obtain  the  steady  temperatui'e  within  a  sector  of  a  circle,  bounded  by  radii  ^  =  0  and 
6  =  a  which  are  kept  at  temperature  zero,  and  a  circular  are  given  by  ?'=!  between  these 
radii  kept  at  temperature  unity. 

Note.  Very  special  cases  as  regards  the  boundary  temperatures  have 
been  taken.  When  any  problem  has  been  solved  for  a  more  general  assign- 
ment of  temperature  along  a  boundary,  then  any  number  of  problems  can 
be  solved  for  steady  temperatures  in  a  plane  limited  by  other  boundaries 
conformally  representable  upon  the  first.  For  a  number  of  applications, 
reference  ma}^  be  made  to  Christoffel  *,  Mathieuf,  and  Carslawij:. 

*  See  reference  in  foot-note,  p.  666. 

t  Cours  de  plnjsique  mathematique,  ch.  iii. 

+  Fourier's  series  and  integrals,  cli.  xiii. 


CHAPTER  XX. 

CONFORMAL  REPRESENTATION  :  GENERAL  THEORY. 

262.  In  Gauss's  solution  of  the  problem  of  the  conformal  representation 
of  surfaces  there  is  a  want  of  determinateness.  On  the  one  hand,  there  is  an 
element  arbitrary  in  character,  viz.,  the  form  of  the  function ;  on  the  other 
hand,  no  limitation  to  any  part  of  either  surface,  as  an  area  to  be  represented, 
has  been  assigned.  And  when,  in  particular,  the  solution  is  applied  to  two 
planes,  then,  corresponding  to  any  curve  given  in  one  of  the  planes,  a  curve 
or  curves  in  the  other  can  be  obtained,  partially  dependent  on  the  form 
of  functional  relation  assumed,  different  curves  being  obtained  for  different 
forms  of  functional  relation. 

But  now  a  converse  question  suggests  itself  Suppose  a  curve  given  also  in 
the  second  plane  :  can  a  function  be  determined,  so  that  this  curve  corresponds 
to  the  given  curve  in  the  first  plane  and  at  the  same  time  the  conformal 
similarity  of  the  bounded  areas  is  preserved,  with  unique  correspondence 
of  points  within  the  respective  areas  ?  in  fact,  does  the  conformal  corre- 
spondence of  two  arbitrarily  assigned  areas  lead  to  conditions  which  can 
be  satisfied  by  the  possibilities  contained  in  the  arbitrariness  of  a  functional 
relation  ?     And,  if  the  solution  be  possible,  how  far  is  it  determinate  ? 

An  initial  simplification  can  be  made.  If  the  areas  in  the  planes, 
conformally  similar,  be  T  and  R,  and  if  there  be  an  area  >Sf  in  a  third  plane 
conformally  similar  to  T,  then  S  and  R  are  also  conformally  similar  to  one 
another,  whatever  S  may  be.  Hence,  choosing  some  form  for  8,  it  will 
be  sufficient  to  investigate  the  question  for  T  and  that  chosen  form.  The 
simplest  of  closed  curves  is  the  circle,  which  will  therefore  be  taken  as  S : 
and  the  natural  point  within  a  circle  to  be  taken  as  a  point  of  reference  is  its 
centre. 

Two  further  limitations  will  be  made.  It  will  be  assumed  that  the  plane 
surfaces  are  simply  connected*  and  one-sheeted.     And  it  will  be  assumed 

*  The  conformal  representation  of  multiply  connected  plane  surfaces  is  considered  by 
Schottky,  Crelle,  t.  Ixxxiii,  (1877),  pp.  300 — 351.  Some  special  cases  are  considered  by  Burnside, 
Lond.  Math.  Soc.  Proc,  vol.  xxiv,  (1893),  pp.  187—206. 


654  riemann's  theorem  [262. 

that  the  boundary  of  the  area  T  is  either  an  analytical  curve*  or  is  made  up 
of  portions  of  a  finite  number  of  analytical  curves — a  limitation  that  arises  in 
connection  with  the  proof  of  the  existence-theorem.  This  limitation,  initially 
assumed  by  Schwarz  in  his  early  investigations f  on  conformal  representation 
of  plane  surfaces,  is  not  necessary :  and  Schwarz  himself  has  shewn  J  that  the 
problem  can  be  solved  when  the  boundary  of  the  area  T  is  any  closed  convex 
curve  in  one  sheet.  The  question  is,  however,  sufficiently  general  for  our 
purpose  in  the  form  adopted. 

Then,  with  these  limitations  and  assumptions,  Riemann's  theorem  §  on 
the  conformation  of  a  given  curve  with  some  other  curve  is  effectively  as 
follows  : — 

Any  simply  connected  part  of  a  plane  hounded  by  a  curve  T  can  always  he 
conformally  represented  on  the  area  of  a  circle,  the  two  areas  having  their 
elements  similar  to  one  another;  the  centre  of  the  circle  can  be  made  the 
homologue  of  any  point  Oo  within  T,  and  any  point  on  the  circumference  of  the 
circle  can  he  made  the  homologue  of  any  point  0'  on  the  boundary  of  T ;  the 
conformal  representation  is  then  uniquely  and  completely  determinate. 

263.  We  may  evidently  take  the  radius  of  the  circle  to  be  unity,  for  a 
circle  of  any  other  radius  can  be  obtained  with  similar  properties  merely  by 
constant  magnification.  Let  w  be  the  variable  for  the  plane  of  the  circle,  z 
the  variable  for  the  plane  of  the  curve  T ;  and  let 

log  w  =  t  —  m  +  ni. 

Evidently  n  will  be  determined  by  m  (save  as  to  an  additive  constant),  for 
m  +  ni  is  a  function  of  z  :  and  therefore  we  need  only  to  consider  m. 

At  the  centre  of  the  circle  the  modulus  of  w  is  zero,  that  is,  e'^  is  zero : 
hence  m  must  he  —  oo  for  the  centre  of  the  circle,  that  is,  for  (say)  z  =  Zo  in  T. 

At  the  boundary  of  the  circle  the  modulus  of  iv  is  unity,  that  is,  e^  is 
unity :  hence  m  must  be  0  along  the  circumference  of  the  circle,  that  is,  along 
the  boundary  of  T. 

Moreover,  the  correspondence  of  points  is,  by  hypothesis,  unique  for  th'e 
areas  considered :  and  therefore,  as  e'""  and  n  are  the  polar  coordinates  of  the 
point  in  the  copy  arid  as  m  is  entirely  real,  m  is  a  one-valued  function, 
which  within  T  is  to  be  everywhere  finite  and  continuous  except  only  at 
the  point  z^.  Hence,  so  far  as  concerns  m,  the  conditions  are : — 
(i)  m  must  be  the  real  part  of  some  function  of  z : 
(ii)    m  must  be  —  oo  at  some  arbitrary  point  z^ : 

*  A  curve  is  said  to  be  an  analytical  curve  (§  265)  when  the  coordinates  of  any  point  on  it 
can  be  expressed  as  an  analytical  function  (§  34)  of  a  real  parameter, 
t  Crdle,  t.  Ixx,  (1869),  pp.  105—120. 
t  Ges.  Werke,  t.  ii,  pp.  108—132. 
§  Ges.  Werke,  p.  40. 


263.]  ON   CONFORMAL   REPRESENTATION  655 

(iii)    m  must  be  0  along  the  boundary  of  T : 

(iv)  for  all  points,  except  Zo,  within  T,  m  must  be  one-valued,  finite  and 
continuous. 

Now  since  m  +  ni  =  log  w  =  log  R-\-i%,  the  negatively  infinite  value  of  m 
at  2^0  arises  through  the  logarithm  of  a  vanishing  quantity ;  and  therefore,  in 
the  vicinity  of  z^^,  the  condition  (ii)  will  be  satisfied  by  having  some  constant 
multiple  of  log  {z  —  z^)  as  the  most  important  term  in  m  +  ni ;  and  the  rest  of 
the  expansion  in  the  vicinity  of  z^  can  be  expressed  in  the  form  p{z  —  z^),  an 
integral  series  of  positive  powers  of  z  —  z^,  because  m  is  to  be  finite  and 
continuous.     Hence,  in  the  vicinity  of  ^'o,  we  have 

log  lu  =  m  +  ni  =  -  log  (z  -Zf)+p{z-  z^), 

where  \  is  some  constant.  This  includes  the  most  general  form :  for  the 
form  of  any  other  function  for  m  +  ni  is 

-\og[{z-z,)g{z-z,)\+P{z-z,), 

where  g  is  any  function  not  vanishing  when  z  =  Zq:  and  this  form  is  easily 
expressed  in  the  form  adopted.     Hence 

1 

Since  w  is  one-valued,  we  must  have  X.  the  reciprocal  of  an  integer ;  and 
since  the  area  bounded  by  T  is  simply  connected  and  one-sheeted  we  must 
have  z  —  z^  H.  one-valued  function  of  w.  Hence  \  =  1 ;  and  therefore,  in  the 
vicinity  of  ^o; 

w  =  {z  —  Zq)  e^^^~^'>\ 

a  form  which  is  not  necessarily  valid  beyond  the  immediate  vicinity  of  Zf^, 
for  ]}  (z  —  Zq)  might  be  a  diverging  series  at  the  boundary.  Thus,  assuming 
that  p  (z  —  Zo)  is  0  when  z  =  Z(,,  we  have,  in  the  immediate  vicinity  of  Zq, 

m  +  ni  =  log  (z  —  Zq), 

a  form  which  satisfies  the  second  of  the  above  conditions. 

It  now  appears  that  the  quantity  m  must  be  determined  by  the  con- 
ditions : — 

(i)  it  must  be  the  real  part  of  a  function  of  z,  that  is,  it  must  satisfy 
the  equation  V^wi  =  0  : 

(ii)     along  the  boundary  of  the  curve  T,  it  must  have  the  value  zero  : 

(iii)  at  all  points,  except  Zq,  in  the  area  bounded  by  T,  m  must  be 
uniform,  finite  and  continuous :  and,  for  points  z  in  the 
immediate  vicinity  of  z^,  it  must  be  of  the  form  logr,  where 
r  is  the  distance  from  z  to  Zq. 


656  riem-ann's  theorem  [263. 

When  m  is  obtained,  subject  to  these  conditions,  the  variable  w  is  thence 
determinate,  being  dependent  on  z  in  such  a  way  as  to  make  the  area 
bounded  by  T  conformally  represented  on  the  circle  in  the  w-plane. 

264.  The  investigations,  connected  with  the  proof  of  the  existence- 
theorem,  shewed  that  a  function  exists  for  any  simply  connected  bounded 
area,  if  it  satisfy  the  conditions,  (1)  of  acquiring  assigned  values  along  the 
boundary,  (2)  of  acquiring  assigned  infinities  at  specified  points  within  the 
area,  (3)  of  being  everywhere,  except  at  these  specified  points,  uniform,  finite, 
and  continuous,  together  with  its  differential  coefficients  of  the  first  and  the 
second  order,  (4)  of  satisfying  V^w  =  0  everywhere  in  the  interior,  except  at 
the  infinities.     Such  a  function  is  uniquely  determinate. 

But  the  preceding  conditions  assigned  to  m  are  precisely  the  conditions 
which  determine  uniquely  the  existence  of  the  function :  hence  the  function 
m  exists  and  is  uniquely  determinate.  And  thence  the  function  w  is 
determinate. 

It  thus  appears  that  any  simply  connected  hounded  area  can  he  conformally 
represented  on  the  area  of  a  circle,  with  a  unique  correspondence  of  points  in 
the  areas,  so  that  the  centre  of  the  circle  can  he  made  the  homologue  of  an 
internal  point  of  the  hounded  area. 

An  assumption  was  made,  in  passing  from  the  equation 

w  =  {z  —  Zq)  e'P  '^"^o' 

to  the  equation  which  determines  the  infinity  of  m,,  viz.  that,  when  z  —  Zq, 
the  value  of  ^  (^  —  z^)  is  0.  If  the  value  of  ^  (^;  —  Zq)  when  z  =  Zo  be  some 
constant  other  than  zero,  then  there  is  no  substantial  change  in  the  conditions: 
instead  of  having  the  infinity  of  m  actually  equal  to  \og\z  —  Zo\,  the  new 
condition  is  that  m  is  infinite  in  the  same  way  as  logl^^  — ^o|,  and  then  a 
constant  factor  must  be  associated  with  w.  A  constant  factor  may  also  arise 
through  the  circumstance  that  n  is  determined  by  m,  save  as  to  an  additive 
constant,  say  7 :  hence  the  form  of  w  =  e''^'^^'^  will  be 

w  =  A'ey^u  =  All. 

Since  displacement  in  the  plane  makes  no  essential  change,  we  may  take 
a  form  tu  =  Au  +  B,  where  now  the  conformal  transformation  given  by  w  is 
over  any  circle  in  its  plane,  the  one  given  by  u  being  over  a  particular  circle, 
centre  the  origin  and  radius  unity. 

The  conformation  for  w  is  derived  from  that  for  u  by  three  operations : — 

(i)     displacement  of  the  origin  to  the  point  —  B/A  : 

(ii)    magnification  equal  to  A' : 

(iii)    rotation  of  the  circle  round  its  centre  through  an  angle  7 : 


264]  DERIVATIVE   FUNCTIONS   REQUIRED  65T 

these  operations  evidently  make  no  essential  change  in  the  conformation. 
If  the  limitation  to  the  particular  circle,  centre  the  origin  and  radius  1, 
be  made,  evidently  J5  =  0,  A'  =1,  but  7  is  left  arbitrary.  This  constant 
can  be  determined  by  assigning  a  condition  that,  as  the  curve  C  has  its 
homologue  in  the  circle,  one  particular  point  of  C  has  one  particular  point 
of  the  circumference  for  its  homologue  :  the  equation  of  transformation  is 
then  completely  determined. 

This  determination  of  A',  B,  7  is  a  determination  by  very  special  con- 
ditions, which  are  not  of  the  essence  of  the  conformal  representation :  and 
therefore  the  apparent  generality  for  the  present  case  should  arise  in  the 
analysis.     Now,  if  w  =  J.  ?<  +  5,  we  have 


d2  \    ^  \dz)]      dz  \    ^  \dz) 


which  is  the  same  for  the  two  forms ;  and  therefore  the  function  to  he 
sought  is 

when  the  area  included  by  C  is  to  be  represented  on  a  circle  so  that  a  given 
point  internal  to  C  shall  have  the  centime  of  the  circle  as  its  homologue. 
The  arbitrary  constants,  that  arise  when  w  is  thence  determined,  are  given 
by  special  conditions  as  above. 

Again,  if  the  conformation  be  merely  desired  as  a  representation  of  the 
2^-area  bounded  by  the  analytical  curve  C  on  the  area  of  a  circle  in  the 
w-plane  (without  the  specification  of  an  internal  point  being  the  homologue 
of  the  centre),  there  will  be  a  further  apparent  generality  in  the  form  of  the 
function.  From  what  was  proved  in  §  258,  a  circle  in  the  zt-plane  is  trans- 
formed into  a  circle  in  the  vj-plane  by  a  substitution  of  the  form 

_Au+  B 
'^~Cu  +  B' 

so  that,  if  u  be  a  special  function,  w  will  be  the  more  general  function  giving 
a  desired  conformal  representation ;  and,  without  loss  of  this  generality,  we 
may  assume  AB  —  BC  —  1.     Using  [w,  z]  to  denote 


^(\.J-^\-i 


dz" 


{^"^thi  s('» 


d  /■,      dw 
dz 


that  IS,  — -. 4 


w        ■"  \w 
called  the  Schwarzian  derivative  by  Cayley*,  we  have 

{w,  z]  =  [u,  z], 

*  Camb.  Phil.  Trans.,  vol.  xiii,  (1879),  p.  5,    Coll.  Math.  Papers,  vol.  xi,  p.  148  ;    for  its 
properties,  see  the  memoii'  just  quoted,  and  my  Treatise  on  Differential  Equations,  pp.  106 — 108. 

F.  F.  42 


658  SOLUTION   BY   BELTRAMI   AND   CAYLEY  [264. 

which   is   the  same  for  the  two  forms:    and  therefore  the  function  to   he 

sought  is 

{w,  z], 

luhen  the  area  included  by  the  analytical  curve  C  is  to  be  conformally  repre- 
sented on  a  circle.  The  (three)  arbitrary  constants,  that  arise  when  w  is 
thence ,  determined,  are  obtained  by  special  conditions. 

These  two  remarks  will  be  useful  when  the  transforming  equation  is 
being  derived  for  particular  cases,  because  they  indicate  the  character  of  the 
initial  equation  to  be  obtained :  but  the  importance  of  the  investigation  is 
the  general  inference  that  the  conformal  representation  of  an  area  bounded 
by  an  analytical  curve  on  the  area  of  a  circle  is  possible,  though,  as  the  proof 
depends  on  the  existence-theorem,  no  indication  is  given  of  the  form  of 
the  function  that  secures  the  representation. 

Further,  it  may  be  remarked  that  it  is  often  convenient  to  represent  a 
^-area  on  a  w-half-plane  instead  of  on  a  «^-circle  as  the  space  of  reference. 
This  is,  of  course,  justifiable,  because  there  is  an  equation  of  unique  trans- 
formation between  the  circular  area  and  the  half-plane ;  it  has  been  given 

(Ex.  13,  §  257).     Moreover,  a  further  change,  given  by  u'  = -, ,  is  still 

possible :  for,  when  a,  b,  c,  d  are  real,  this  transformation  changes  the  half- 
plane  into  itself,  and  these  real  constants  can  be  obtained  by  making  points 
p,  q,  r  on  the  axis  change  into  three  points,  say  0,  1,  oo ,  respectively — the 

transformation  then  being 

,     u—pq—r 

u  = — . 

u  —  r q  —  p 

265.  Before  discussing  the  particular  forms  just  indicated,  we  shall 
indicate  a  method  for  the  derivation  of  a  relation  that  secures  conformal 
representation  of  an  area  bounded  by  a  given  curve  C. 

Let  *  the  curve  C  be  an  analytical  curve,  in  the  sense  that  the  coordinates 
w  and  y  can  be  expressed  as  functions  of  a  real  parameter,  say  of  u,  so  that 
we  have  x=p  (u),  y  =  q  (u) ;  then 

z  =  x  +  iy  =p +  iq  =  <f){u). 
If  for  u  we  substitute  w=u  +  iv,  we  have 

z  =  4>  (w) ; 
and  the  curve  G  is  described  by  z,  when  lu  moves  along  the  axis  of  real 
quantities  in  its  plane. 

When  the  equation  x  -\-iy  =  (^  {ii  +  iv)  is  resolved  into  two  equations 
involving  real  quantities  only,  of  the  form  x  —  \  {u,  v),  y  =  fx  {u,  v),  then  the 
eliminations  of  v  and  of  u  respectively  lead  to  curves  of  the  form 
yjr  (x,  y,  u)  =  0,  X  (^.  2/.  ^)  =  0, 
*  Beltrami,  Ann.  di  Mat.,  2""  Ser.,  t.  i,  (1867),  pp.  329—366 ;  Cayley,  Quart.  Journ.  Math., 
vol.  XXV,  (1891),  pp.  203—226,  Ooll.  Math.  Papers,  vol.  xiii,  pp.  170—190;  Schwarz,  Ges.  Werke, 
t.  ii,  p.  150. 


265.]  FOR  ANALYTICAL  CURVES  659 

which  are  orthogonal  trajectories  of  one  another  when  u  and  v  are  treated  as 
parameters.     Evidently  x  {^>  y>  0)  =  0  is  the  equation  of  C :  also 

So  far  as  the  representation  of  the  area  bounded  by  (7  on  a  half-plane  is 
concerned,  we  can  replace  w  by  an  arbitrary  function  of  Z{=X  +  iY)  with 
real  coefficients  :  for  then,  when  Y=  0,  we  have  w=f(X)  and 

^=p{f(X)],     y  =  q[f{X)], 
which  lead  to  the  equation  of  G  as  before,  for  all  values  of  /.     This  arbi- 
trariness in  character  is  merely  a  repetition   of  the   arbitrariness   left   in 
Gauss's  solution  of  the  original  problem. 

Now  let  the  w-plane  be  divided  into  infinitesimal  squares  with  sides 
parallel  and  perpendicular  to  the  axis  of  real  quantities.  Then  the  area 
bounded  by  Cis  similarly  divided,  though,  as  the  magnification  is  not  every- 
where the  same,  the  squares  into  which  the  area  is  divided  are  not  equal  to 
one  another.  The  successive  lines  parallel  to  the  axis  of  u  are  homologous 
with  successive  curves  in  the  area,  the  one  nearest  to  that  axis  being  the 
curve  consecutive  to  G.     Similarly,  if  the  ^-plane  be  divided. 

Conversely,  if  a  curve  consecutive  to  G,  say  G',  be  arbitrarily  chosen,  then 
the  space  of  infinitesimal  breadth  between  G  and  G'  can  be  divided  up  into 
infinitesimal  squares.  Suppose  the  normal  to  C  at  a  point  L  meet  G'  in  L' : 
along  G  take  LM=  LL',  and  let  the  normal  to  G  at  M  meet  G'  in  M'\  along 
G  take  MN=  MM',  and  let  the  normal  to  C  at  iV  meet  G'  in  N':  and  so  on. 
Proceeding  from  G'  with  L'M',  M'N', ...  as  sides  of  infinitesimal  squares,  we 
can  obtain  the  next  consecutive  curve  G",  and  so  on;  the  whole  area  bounded 
by  G  may  then  be  divided  up  into  an  infinitude  of  squares.  It  thus  appears 
that  the  arbitrary  choice  of  a  curve  consecutive  to  G  completely  determines 
the  division  of  the  whole  area  into  infinitesimal  squares,  that  is,  it  is  a 
geometrical  equivalent  of  the  analytical  assumption  of  a  functional  form 
which,  once  made,  determines  the  whole  division. 

Next,  we  shall  shew  how  the  form  /  of  the  function  can  be  determined 
so  as  to  make  the  curve  consecutive  to  C  a  given  curve.  As  above,  the 
curve  G  is  given  by  the  elimination  of  a  (real)  parameter  between 

x  =  p  (u),      y=^q{u); 

and  the  representation  is  obtained  by  taking 

x  +  iy  =  z=p{w)  +  iq  (w)=p  [f{Z)}  +  iq  [f{Z)]. 

Let  the   arbitrarily  assumed  curve  G',  consecutive   to  G,  be  given   by  the 
elimination  of  a  (real)  parameter  6  between 

x=p  +  eP,       y  =  q  +  eQ, 

where  p,  P,  q,  Q  are  functions  with  real  coefficients,  and  e  is  an  infinitesimal 
constant :  the  form  of/  has  to  be  determined  so  that  the  curve  corresponding 

42—2 


660  CONFORMATION    OF   AREA  [265. 

to  an  infinitesimal  value  of  Y  is  the  curve  C  Taking  u=f{X),  where  u 
and  X  are  real,  we  have,  for  the  infinitesimal  value  of  T, 

^  +  ii/=p{fi^)]+n{fm 

so  that  ^  =  p  — y-y^^,  y  =  9+^'yYP' 

dashes  denoting  difi'erentiation  with  regard  to  u.  This  is  to  be  the  same 
as  the  curve  C,  given  by  the  equations 

x=p  +  eP,     y  =  q  +  eQ. 

Hence  the  (real)  parameter  6  in  the  latter  differs  from  u  only  by  an  infini- 
tesimal quantity :  let  it  be  ti  —  fi,  so  that  we  have 

x=p  —  jxp'  +  eP,     y  =  o  —  fJ^(l'  +  eQ, 
the  terms  involving  products  of  e  and  yu.  being  neglected,  because  they  are 
of  at  least  the  second  order.     Hence 

-^p'  +  eP  =  -Y^q',  -^q^+eQ=Y^p'; 

whence  /^  (p''  +  q'^)  =  e  (Pp'  +  Qq'), 

and  e{p'Q-q'P)=Y^{f^  +  (^^r. 

Now  e  is  a  real  infinitesimal  constant,  as  is  also  F  for  the  present  purpose : 
so  that  we  may  take  6  =  AY,  where  J.  is  a  finite  real  constant :  and  A  may 
have  any  value  assigned  to  it,  because  variations  in  the  assumed  value 
merely  correspond  to  constant  magnification  of  the  ^-plane,  which  makes 
no  difference  to  the  division  of  the  area  bounded  by  C     Thus 

A{p'Q-q'P)=^{p'-^  +  q% 

and  therefore  AX  =  I  —rp^ 77:;  du, 

J  pQ-qP 

the  inversion  of  which  gives  u=f{X)  and  therefore  w  =f{Z),  the  form 
required. 

Also  we  have  fi  =  A  Y?K^^^ , 

p^  +  q^ 

shewing  that,  if  the  point  oc=p  +  eP,  y  =  q  +  eQ  on  C  lie  on  the  normal  to  C 
a,t  x=p,  y  =  q,  the  parameters  in  the  two  pairs  of  equations  are  the  same; 
the  more  general  case  is,  of  course,  that  in  which  the  typical  point  on  C  is  in 

*  Beltrami  obtains  this  result  more  directly  from  the  geometry  by  assigning  as  a  condition 
that  the  normal  distance  between  the  curves  is  equal  to  the  arc  given  by  du :  I.e.,  (p.  658,  note), 
p.  343. 


265.]  BOUNDED  BY  ANALYTICAL  CURVE  661 

the  vicinity  of  C.     And  it  is  easy  to  prove  that  the  normal  distance  between 
the  curves  at  the  point  in  consideration  is 

„  ds 
dX' 
where  ds  is  an  arc  measured  along  the  curve  C. 

Ex.  1.  As  an  illustration*  let  Cbe  an  ellipse  x^  I  a?' + y'^  jl)^  =  \  and  let  C  be  an  interior 
confocal  ellipse  of  semi-axes  a  —  n,  6-/3,  where  a  and  /3  are  infinitesimally  small ;  so  that, 
since 

(a-a)2-(6-^)2  =  a2-62  =  c2, 

we  have  aa=bj3—C€  say;  then  the  semi-axes  of  C  are  a  —  e,  b-^e.     We  have 

a  b 


p  =  a  cos  u,  q=bsinu, 

c 
b' 


P= — COS.U,        ^=-^sini«, 


so  that  AX=\  —  du=~u, 

J  c  c     ' 

or,  taking  A= — ,  we  have  X=u  and  therefore  Z=w.  Hence  the  equation  of  transform- 
ation is 

z=x  +  iy  =  a  cos  Z+ib  sinZ; 

or,  if  a  =  c  cosh  Yq,  6  =  csinh  Fq,  and  if  Y'  denote  Fq-  Y,  the  equation  is 

z  =  c  cos  {X+iY')  =  c  cos  Z'. 

The  curves,  corresponding  to  parallels  to  the  axes,  are  the  double  system  of  confocal 
conies. 

Ex.  2.  When  the  curve  C  is  a  parabola,  with  the  origin  as  focus  and  the  axis  of  real 
quantities  as  its  axis,  and  C  is  an  external  confocal  coaxial  parabola,  the  relation  is 

z=a{Z+if; 

substantially  the  same  relation  as  in  Ex.  9,  §  257. 

Ex.  3.  When  C  is  a  circle  with  its  centre  on  the  axis  of  real  quantities  and  C  is 
an  interior  circle,  having  its  centre  also  on  the  axis  but  not  coinciding  with  that  of  C,  the 
circles  being  such  that  the  axis  of  imaginary  quantities  is  their  radical  axis,  the  relation 
can  be  taken  in  the  form 

z=c  tan  Z.  (Beltrami ;  Cayley.) 

.  Note.  Although,  in  the  examples  just  considered,  the  successive  curves 
G  ultimately  converge  to  a  curve  of  zero  area  (either  a  point  or  a  line),  so 
that  the  whole  of  the  included  area  is  transformed,  yet  this  convergence 
is  not  always  a  possibility,  when  a  consecutive  to  G  is  assigned  arbitrarily. 
There  will  then  be  a  limit  to  the  ultimate  curve  of  the  series,  so  that  the 
representation   ceases   to   be    effective  beyond  that  limit.      The  limitation 

may  arise,  either  through  the  occurrence  of  zero  or  of  infinite  values  of  -^^ 

for  areas  and  not  merely  for  isolated  points,  or  through  the  occurrence  of 
branch-points  for  the  transforming  function.  In  either  case,  the  uniqueness 
of  the  representation  ceases. 

*  Beltrami,  I.e.,  (p.  658,  note),  p.  344  ;  Cayley,  (ib.),  p.  206. 


662  EXAMPLES  •  [265. 

Ex.  4.     Consider  the  area,  bounded  by  the  cardioid 

r=2a  (1+cos^); 
then  we  can  take 

x=p  =  'ia  (l  +  cos«)  cos  u,        y  =  q='2a  {\  +  cos  u)  sin  u, 

where  evidently  u  =  d  along  the  curve.     Let  the  consecutive  curve  be  given  by 

^=  -a6  +  2a(l  +  e)  i\  +  GOsu')cosu',        y  =  2a(l  +  e)  (1  +  cos  M')sin  u', 
so  that,  to  determine  X,  we  assume  P=  — a+2a  (1  +  cosm)  cos^f,   ^  =  2a  (1+cosm)  sinw, 
for  m'  —  2«=  -/a  a  small  quantity. 

We  have  p"^  +  q''^  =  16a^  cos^  ^i, 

q'P—p'Q=12a?  cos2  ■!«., 
p'P+q'Q=  —  2a^  sin  u ; 
and  then,  proceeding  as  before  and  choosing  A  of  the  text  as  equal  to  —  |^,  (which  implies 
that  e  is  negative  and  therefore  that  the  interior  area  is  taken),  we  find 

X=u, 

therefore  Z=  w.     Thus  the  cardioid  itself  and  the  consecutive  curves  are  given  by 

iZ 

zz=p  +  iq  =  2a{l+cos  Z)e    . 
To  trace  the  curves,  corresponding  to  lines  parallel  to  the  axes  of  X  and  Y,  we  have 

'^^''  =  2cos^Zei'^, 


Hence,  multiplying,  we  have 


and,  dividing,  we  have 


(f)*  =  2c„.K.«-i^ 


r  =  4:ae       {cos  ^  Z  cos  ^  Zq) 
=  2ae       (cosh  T+  cos  X) ; 


iX  cos^Z 
cos  l^o' 


,•,    ,  .  i{X-e)_cos^X cosh^T+isin  ^Xsmh^Y 

^   ^^'  ^  ~ cos  i X  cosh  I r-  i  sin  ^X sinh  ^Y' 


and  therefore 

Moreover,  we  have 


tan  ^{X-0)  =  tan  |  X  tan  h  i  Y. 

dz     ^   .  iz,,  ,    iZ. 
-^=2aie     (1  +  e    ), 


which  vanishes  when  Z=Tr  (2%  +  l),  that  is,  at  the  point  X={'2,n+\)Tt,  F=0;  whence  the 
cusp  of  the  cardioid  is  a  singularity  in  the  rej)resentation. 

When   F=0,  then  X=^d  and  r=2a  (1 +  cos(9),  which  is  the  cardioid;  when  F  is  very 
small  and  is  expressed  in  circular  measure,  then 

tan|(Z-^)=irtaniX, 
or  .r=(9+rtani^, 

so  that  r =2a  {I  +  cos  6) -AaY. 

It  is  easy  to  verify  that 

6=u'  +  \Yisji\u', 
agreeing  with  the  former  result. 

The  relation  may  be  taken  in  the  form 

(«/a)*  =  2  cos  lZe^^^  =  e'^+l, 


265.] 


OF   ANALYTICAL   CURVES 


663 


which  shews  that  z  =  a  is  a  branch -point  for  Z.  Two  different  paths  from  any  point  to  a 
point  P,  which  together  enclose  a,  give  different  values  of  Z  at  P.  Hence  the  representa- 
tion ceases  to  be  effective  for  any  area  that  includes  the  point  a. 

Consider  a  strip  of  the  Z-plane  between  the  lines  F=0,  Y=+<x^,  X=  —  ^tt,  X=  +|7r. 

First,  when  Z=^Tr-\-iY,  we  have  X=\n,  so  that 


and  therefore 
whence 


tan  ( jTT  -  \6)  =  tanh  ^  Y ; 
tan  \6  =  e      , 
2a 


H-cos^' 

a  part  of  a  parabola.     And  when  Y  varies  from  oo  to  0,  ^  varies  from  0  to  Jtt. 

Secondly,  when  Z=X,  so  that  Y=0,  we  have  X=6,  and  then 

r  =  2a(l-l-cos(9): 

and,  when  X  varies  from  \-n-  to  -  ^tt,  6  varies  from  \tt  to  —\it. 

Thirdly,  when  Z=  —hTr  +  iY,  we  have  X=  —  ^jt,  so  that 

tan  (  Jtt  -I-  \e)  =  tanh  J  F, 

whence  tan  \d=  —e      , 

so  that,  as  Y  varies  from  0  to  oo  ,  ^  varies  from  —  \ir  to  0.     And  then 

._      2a 
'*~  1  +  cos^ ' 

another  part  of  the  same  parabola  as  before. 
Lastly,  when  Y  is  infinite  and  X  varies  from 

tani(Z-(9)  =  tan|J', 

so  that  ^  =  0 ;  and  then  r=a,  in  effect  the  point  of  the  s-plane  corresponding  to  the  point 
at  infinity  in  the  Z-plane. 

We  thus  obtain  a  figure  in  the  s-plane  ABCDA  corresponding  to  the  strip  in  the 
Z-plane :  the  boundary  is  partly  a  parabola  DAB,  of  focus  0  and  axis  OA,  and  partly 
a  cardioid  with  0  for  cusp — the  inverse  of  the  parabola  with  regard  to  a  circle  on  the 
latus  rectum  BD  as  diameter:  the  angles  at  B  and  D  are  right. 


—  to  +  -  ,  we  have 


Fig.  92. 

To  trace  the  division  of  the  space  between  the  axes  of  the  cardioid  and  of  the  parabola 
corresponding  to  the  division  of  the  plane  strip  into  small  squares,  we  can  proceed  as 
follows. 


664  EXAMPLES  [265. 


Let  e       =  c :  theu  we  have 


—  =c(-  +  c|  +  2c  cos  X. 
a        \c       ] 


or,  if  i2=ap,  then  p  =  l+c2+2ccos  Z; 


1  -c 
and  tan  ^  (Z  -  ^)  =  t——  tan  \  JT, 


so  that 


and  therefore 


"■~sin(X--|^)' 
cos  \6  sin  \6 


1  +  c  cos  X     c  sin  X      „! ' 


so  that  coos  A'=/3i  cos^^  — 1,         csin  A'=p2  sin  i^, 

from  which  the  curves,  corresponding  to  c  =  constant  and  to  A'=  constant,  are  at  once 
obtained.  They  are  exhibited  in  the  figure,  the  whole  of  the  internal  space  being 
divisible. 

By  combination  with  the  transformation,  which  (Ex.  18,  §  257)  represents  a  strip  of 
the  foregoing  kind  on  a  circle,  the  relation  can  be  obtained,  leading  to  the  representation 
of  the  figure  on  a  circle. 

Ex.  5.  Shew  that,  if  a  straight  line  be  drawn  from  the  cusp  to  the  point  r=a,  ^  =  0, 
so  as  to  prevent  z  from  passing  round  2=0  or  2  =  a,  then  the  area  bounded  by  the  cardioid 
and  this  line  can  be  rej)resented,  on  a  strip  of  the  w-plane  given  by  F=0,  F=co, 
X—  —  77,  X=  +  TT,  by  the  equation 

iw=\og  {{zja)^  -  1}.  (Burnside.) 

Ex.  6.  In  the  same  way,  treating  the  curve  (the  Cissoid  of  Diodes)  {2r  —  x)  y'^  =  x^ , 
and  taking  the  equations 

_     .   ,                   _    sin^M 
x=zr  sm-  %,         y='zr , 

as  defining  the  points  on  the  curve,  we  may  assume  the  consecutive  curve  defined  by  the 
equations 

X  =  e  +  (2r  -  e)  sm2  «,  y  =  (2r  -  e)  ——i  , 

another  cissoid  with  the  same  asymptote.  Proceeding  as  before,  we  find  the  value  of  X 
to  be  tanw  +  j^tan^w,  on  taking  A=—^r. 

The  relation,  which  changes  the  cissoidal  arc  into  the  axis  of  X  and  a  consecutive 
cissoidal  arc  into  a  line  parallel  to  the  axis  of  X  at  an  infinitesimal  distance  from  it, 

is  then 

^  siu^  tv 

z=2r e™^ 

cos  iV 

where  the  relation  between  w  and  Z  is 

Z=  tan  iv + Y^o-  tan^  w. 

Npte.  The  method  is  applicable  to  any  curve,  whose  equation  can  be  expressed  in  the 
form  r—f{6) :  a  first  transfoi-mation  is 

The  determination  of  lo  in  terms  of  Z  depends  upon  the  character  of  the  consecutive 
curve  chosen ;  this  curve  also  determines  the  details  of  the  conformation. 


266.]  CONTRACTION  OF  AREAS  665 

266.  It  has  been  pointed  out  (§  265,  Note)  that,  though  a  curve  and  its 
consecutive  in  the  ^-plane  correspond  with  a  curve  and  its  consecutive  in  the 
w-plane,  the  conformation  is  only  effective  for  parts  of  the  included  areas,  in 
which  the  magnification,  if  it  is  not  uniform,  becomes  zero  or  infinite  only  at 
isolated  points,  and  in  which  no  branch-points  of  the  transforming  relation 
occur.  The  immediate  vicinity  of  a  curve  G  is  conformable  with  the 
immediate  vicinity  of  a  corresponding  curve  S,  arbitrarily  chosen  limits 
being  assigned  for  the  vicinity. 

But,  as  remarked  by  Cayley*,  when  a  curve  is  given,  then  the  con- 
secutive curve  can  be  so  chosen  that  the  whole  included  area  is  conformable 
with  the  whole  corresponding  area  in  the  .^-plaue.  For  a  circle  can  be  thus 
represented,  the  ultimate  limit  of  the  squares  when  consecutive  curves  are 
constructed  being  then  a  point :  this  can  be  expressed  by  saying  that  the 
area  can  be  contracted  into  a  point.     For  instance,  the  relation 

z{iv-ir  1)  -F  i{iv  -  1)  =  0 

transforms  the  5-half-plane  into  the  area  included  by  a  w-circle  of  radius 
unity.  The  lines  parallel  to  the  axis  of  x  are  internal  circles  all  touching 
one  another  at  the  point  (—1,  0) :  and  the  lines  parallel  to  the  axis  of  y  are 
circles  orthogonal  to  these,  having  their  centres  on  a  line  parallel  to  the  axis 
of  Y  and  all  touching  at  the  point  (—  1,  0).  Similarly  for  the  contraction  of 
any  circle,  by  making  it  one  of  two  systems  of  orthogonal  circles :  the  form  of 
the  necessary  equation  is  obtained  as  above  by  taking  the  next  circle  of  the 
same  system  as  the  consecutive  curve :  and  a  circle  can  thus  be  contracted  to 
its  centre  (the  infinitesimal  squares  being  bounded  by  concentric  circles  and 
by  radii)  when  the  w-circle  is  derived  from  a  strip  of  the  ^-half-plane  by  the 
relation  lu  =  e'^.     Such  a  contraction  of  a  circle  is  unique. 

But,  by  Riemann's  theorem,  it  is  known  that  the  area  of  a  given 
analytical  curve  can  be  conformally  represented  on  the  area  of  a  given 
circle,  so  that  a  given  internal  point  is  the  homologue  of  the  centre  and 
a  given  point  on  the  curve  is  the  homologue  of  a  given  point  on  the 
circumference  of  the  circle :  and  that  the  representation  is  unique.  Hence 
it  follows  that,  when  an  analytical  curve  C  is  given,  a  consecutive  curve 
C"  can  be  chosen  in  such  a  manner  as  to  secure  that  the  construction  of 
the  whole  series  of  consecutive  curves  by  infinitesimal  squares  will  make 
the  curve  G  contract  into  an  assigned  point  f. 

267.  The  areas,  already  considered  in  special  examples,  have  been 
bounded  by  one  or  by  two  analytical  curves :  we  shall  now  consider  two 
special  forms  of  areas  bounded  by  a  number  of  portions  of  analytical  curves. 
These  areas  are  (i)  the  area  included  within  a  convex  rectilinear  polygon, 
(ii)  the  area  bounded  by  any  number  of  circular  arcs,  and  especially  the  area 

*  I.e.  (p.  658,  note),  pp.  213,  214. 

t  For  further  developments,  see  Gayley's  memoir  cited  p.  658,  note. 


666  RECTILINEAE   POLYGON  [267. 

bounded  by  three  circular  arcs.  For  the  sake  of  analytical  simplicity,  the 
former  will  be  conformally  represented  on  the  half-plane,  the  trfinsformation 
to  the  circle  being  immediate  by  means  of  the  results  of  §  257. 

In  regard  to  the  representation*  of  the  rectilinear  polygon,  convex  in 
the  sense  that  its  sides  do  not  cross,  we  shall  take  the  case  corresponding 
to  the  first  of  the  two  forms  of  §  264 ;  it  will  be  assumed  that  the  origin  in 
the  w-plane  is  left  unspecified  and  that  the  magnification  is  subject  to  an 
unspecified  increase,  constant  over  the  plane.  Our  purpose,  therefore,  is  to 
represent  the  w-area  included  by  a  polygon  on  the  half  of  the  ^-plane ;  the 
boundary  of  the  polygonal  area  in  the  tv-T^lane  is  to  be  transformed  into  the 
axis  of  real  quantities  in  the  ^-plane. 

It  follows,  from  Schwarz's  continuation-theorem  (§  36),  that  a  function 
defined  for  a  region  in  the  positive  half  of  a  plane  and  acquiring  continuous 
real  values  for  continuous  real  values  of  the  argument  can  be  continued 
across  the  axis  of  real  quantities :  and  the  continuation  is  such  that  conju- 
gate values  of  the  function  correspond  to  conjugate  values  of  the  variable. 
Moreover,  the  function,  for  real  values  of  the  variable,  can  be  expanded  in 
a  converging  series  of  powers,  so  that 

w  —  Wo  =  (oc  —  c)  P  (x  —  c), 

where  P  is  a  series  of  positive,  integral  powers  with  real  coefficients  that 
does  not  vanish  when  c  is  the  value  of  the  real  variable  x. 

Suppose  a  convex  polygon  given  in  the  w-plane,  the  area  included  by 
which  is  to  be  represented  on  the  ^-plane,  and  the  contour  of  which  is  to  be 
represented  along  the  axis  of  x  by  means  of  a  relation  between  tu  and  2. 

First,  consider  a  point  say  /3  on  the  side  Ar-iA,.  which  is  not  an  angular 
point.     Then,  if  6  denote  the  inclination  of  A,.-iAr  to 
the  axis  of  u,  the  function 

(w  -  /3)  e-^'-+e) 

is  real  when  w  lies  on  the  side  Ar-iA,. :  it  changes  sign 
when  w  passes  through  /3 :  and  for  all  other  points  tu, 
lying  either  in  the  interior  or  on  the  other  sides  of  the 
polygon,  it  has  the  same  properties  as  w.  Hence,  if  b  be 
a  (purely  real)  value  of  ^  corresponding  to  lu  =  /3,  we  have 

{w  -  ^)  e-^(-+e)  =  (2^b)P{2-  h), 

*  lu  connection  with  the  succeeding  investigations  the  following  authorities  may  be 
consulted : 

Schwarz,  Ges.  Werke,  t.  ii,  pp.  65—83;  Christoffel,  Ann.  di  Mat.,  2''='  Ser.,  t.  i,  (1867), 
pp.  95—103,  ib.,  t.  iv,  (1871)^  pp.  1—9;  Schlafli,  Crelle,  t.  Ixxviii,  (1873),  pp.  63—80; 
Darboux,  Theorie  generale  des  surfaces,  t.  i,  (2nd  ed.),  pp.  507 — 512 ;  Phragmen,  Acta 
Math.,  t.  xiv,  (1890),  pp.  229—231. 


267.]  REPRESENTED   ON   A   HALF-PLANE  667 

for  points  in  the  vicinity  of  /S  :  the  series  P {z  —  h)  does  not  vanish  for  z=h, 

because  the  magnification  I  -^    is  not  to  vanish  at  any  current  point ;  and, 

when  w  lies  on  the  side  AyAr-i,  then  z  =  x. 

Next,  consider  the  vicinity  of  an  angular  point  of  the  polygon.  Let  7  be 
the  coordinate  of  Ay,  let  imtt  be  the  internal  angle  of  the  polygon,  and  let  i/r  be 
the  inclination  oi  A^Ar+i  to  the. axis  oiu:  and  consider  the  function 

When  tu  lies  on  the  side  J.,.  J.,._i  at  a  distance  d  from  Ar,  then 

so  that  the  function  is  then  real  and  positive. 

When  IV  lies  in  the  interior  of  the  polygon,  the  function  has  the  same 
properties  as  lu,  and  its  argument  is  negative. 

When    lu  lies   on   the   side   A.yAr+i    at   a    distance   d'   from   Ar,   then 

w-y  =  d'ei'^,  so  that  the  function  is  d'e-'^^-^+^-'^\  that  is,  d'e"''^''.     Hence 

1 

is  real  and  positive  along  the  side  A^-iAr,  and  is  real  and  negative  along 
the  side  AyAr+i.  If  then  2^  =  c  be  the  value  corresponding  to  w  =  y,  we 
can  expand  this  function  in  the  form  (z  —  c)  Q' (z  -  c),  where  Q'{z  —  c)  does 
not  vanish  for  c;  and  therefore 

(w  -  7)  e-*('^+«)  =  {z-  cfR  (z  -  c), 
where  R  (=  Q''^)  does  not  vanish  for  z  =  c. 

These  forms  assume  that  neither  b  nor  c  is  infinite.  The  point  on  the 
boundary  of  the  polygon  (if  there  be  one),  corresponding  to  a;  =  o) ,  can  be 
obtained  as  follows.     We  form  a  new  representation  of  the  ^-plane  given  by 

which  conformally  represents  the  upper  half  of  the  ^-plane  on  itself:  and 
then,  on  the  assumption  that  such  point  at  infinity  does  not  correspond 
to  an  angular  point  of  the  polygon,  we  have  ^  =  0  corresponding  to  an 
ordinary  point  of  the  boundary,  so  that 

where  Q  does  not  vanish  when  z  =  ao . 

If  an  angular  point,  at  which  /u-tt  is  the  internal  angle  of  the  polygon, 
has  its  homologue  at  infinity,  a  similar  argument  shews  that,  in  its  immediate 
vicinity,  the  transformation  is 


zi^      \zj 


where  R  does  not  vanish  when  z  =  <x> 


668  RECTILINEAR   POLYGON  [267. 

All  kinds  of  points  on  the  boundary  of  the  ty-polygon  have  been  con- 
sidered, corresponding  to  points  on  the  axis  of  x. 

We  now  consider  points  in  the  interior.  If  lu'  be  such  an  interior  point 
and  z  be  the  corresponding  ^--point,  then 

w  —  iv'  =  {z  —  z')  S  {z  —  z'), 

where  S  does  not  vanish  for  z  =  z' ,  because  at  every  point  -7  -  must  be  different 

from  zero :  for  otherwise  the  magnification  from  a  part  of  the  ^•-plane  to  a 
part  in  the  interior  of  the  polygon  would  be  zero  and  the  representation 
would  be  ineffective. 

Now  in  the  present  case,  just  as  in  the  first  case  suggested  in  §  264,  it  is 
manifest  that,  if  a  particular  function  w'  give  a  required  representation,  then 
Aw'^-B,  where  |^|  =  1,  will  give  the  same  w-polygon  displaced  to  a  new 
origin  and  turned  through  an  angle  a,  =  arg.  A,  that  is,  no  change  will  be  made 
in  the  size  or  in  the  shape  of  the  polygon,  its  position  and  orientation  in  the 
w-plane  not  being  essential.  Hence  the  function  to  be  obtained  may  be 
expected  to  occur  in  the  form  lu  =  Aiu'  +  B ;  hence 

dw  _    .  dw' 
dz  dz  ' 

and  therefore  log  -y-  =  log  e^"  +  log  -j— , 

,,    ,  d  L      dw\      d  /,      dw'\ 

BO  that  £rs&j=sr»3rj- 

Thus,  in  representing  a  figure  hounded  hy  straight  lines,  the  function  to  he 
ohtained  is 

Now  in  the  vicinity  of  a  boundary-point  /3,  not  being  an  angular  point 
and  corresponding  to  a  finite  value  of  z,  we  have 

w-l3  =  e^(-^+^\z-b)P(z-h), 

and  therefore  Z=  Pi{z  —  b), 

having  z  =  b  for  an  ordinary  point. 

For  a  boundary-point  /3',  not  being  an  angular  point  and  corresponding 
to  an  infinite  value  of  z  on  the  real  axis,  we  have 

w-/3'  =  e*'(-+^'>  IqI]), 

2      ]        /1\ 
and  therefore  Z  — h  —  Qi  ( -  1 , 

z      z'-        \z ) 

where  Qi  is  not  zero  for  ^  =  00  .     Thus  Z  vanishes  for  such  a  point. 


267.]  REPRESENTED   ON   A   HALF-PLANE  669 

In  the  vicinity  of  an  angular  point  7,  corresponding  to  a  finite  point  c  on 
the  real  axis,  we  have 

and  therefore  Z  = \-  B.,(z  —  c), 

Z  —  G 

where  i^^  has  z-=g  for  an  ordinary  point. 

In  the  vicinity  of  an   angular   point  7',   corresponding  to    a   place   at 
infinity  on  the  real  axis,  we  have 

z         z-       \zj 

where  R^  is  not  zero  when  z  =  ao  .     Thus  Z  vanishes  at  such  a  point. 

Lastly,  for  a  point  lu'  in  the  interior  of  the  polygon,  we  have 

w-iu'={z-z')S{z-z'), 

and  therefore  Z  =  Si(z  —  /), 

having  z  =  z'  for  an  ordinary?-  point. 

Hence  Z,  considered  as  a  function  of  z,  has  the- following  properties: — 

It  is  an  analytical  function  of  z,  real  for  all  real  values  of  its  argument, 
and  zero  when  a;  is  infinite : 

It  has  a  finite  number  of  accidental  singularities  each  of  the  first  order 
and  all  of  them  isolated  points  on  the  axis  of-^:  and  at  all  other 
points  on  one  side  of  the  plane  it  is  uniform,  finite  and  continuous, 
having  (except  at  the  singularities)  real  continuous  values  for  real 
continuous  values  of  its  argument. 

The  function  Z  can  therefore  be  continued  across  the  axis  of  x,  conjugate 
values  of  the  function  corresponding  to  conjugate  values  of  the  variable:  and 
its  properties  make  it,  by  §  48,  a  rational  meromorphic  function  of  z. 

Let  a,h,c,...,l  be  the  points  (all  in  the  finite  part  of  the  plane)  on  the 
axis  of  X  corresponding  to  the  angular  points  of  the  polygon,  and  let 

OTT,   /StT,  ryir,  ...,  XlT 

be  the  internal  angles  of  the  polygon  at  the  respective  points. 

The  residues  at  these  points  are  a— 1,  ;8-l,7  —  1,...,A,—  1  respectively ; 
so,  after  §  48,  we  consider  the  function  W  given  by 

[z  —  a      z  —  0       z  -  c  z  —  L 

It  is  finite  everywhere  in  the  finite  part  of  the  plane  z. 


670  RECTILINEAR   POLYGON  [267. 

If  no  angular  point  of  the  polygon  corresponds  to  ^^=00,  then  for  large 
values  of  z  we  have 

where  T\-\  is  finite  for  large  values  of  z.     But 


S  (tt  —  om)  =  sum  of  the  external  angles  of  the  polygon 
=  27r, 
so  that  2  (a  -  1)  =  -  2. 

Thus  W  is  zero  of  the  second  order  when  z  =  cc  . 

If  an  angular  point  of  the  polygon  corresponds  to  z  =  00 ,  then  for  large 
values  of  z  we  have 

n.=_'i±.i+\ij.(i)_l2(«-i)-lr(i), 

z         z^       \zj      z  z^      \zj 

where  T{-\  is  finite  for  large  values  of  z.     But  now 

(tt  —  /i7r)  +  S  (tt  -  air)  =  sum  of  the  external  angles  of  the  polygon 
•        =  -  27r, 
so  that  1  -  /i  +  2  (1  -  a)  =  -  2, 

that  is,  /A  +  l  +  2(a-l)  =  0. 

Thus  W  is.  zero  of  the  second  order  when  z  —  ^  . 

Consequently  W  has  no  pole  anywhere  in  the  ^r-plane ;  so  it  is  a  constant. 
At  infinity,  W  is  zero,  so  that  this  constant  is  zero.     Hence  we  have  , 


d  L      fdw\]  _  «  —  1  A.  —  1 

dz\    °  \dz )\      z  —  a      '"      z  —  i 


dz 

and  each  of  the  quantities  a,  /3,  ...,  X-  is  less  than  2.     This  equation*,  when 
integrated,  gives 

w  =  GJ(z  -  ay-'  (z  -  by-'  ...(z-  ly-'  dz  +  C, 
where  G  and  C  are  arbitrary  constants,  determinable  from  the  position  of 
the  polygon •(-. 

•     It  is  to  be  noted  that,  when  no  angular  point  of  the  w-polygon  has  its 
homologue  at  infinity,  the  relation 

2  («-!)  = -2 

is  satisfied ;  and  that  this  relation  does  not  hold  when  an  angular  point  of 
the  w-polygon  has  its  homologue  at  infinity. 

The  final  expression  for  w  in  terms  of  z  is  commonly  called  the  Schwarz- 
Christoffel  transformation. 

*  This  relation  is  made  the  basis  of  some  interesting  applications  in  hydrodynamics,  by  Michell, 
Phil.  Trans.,  (1890),  pp.  389 — 43 L  See  also  the  authorities  quoted  in  the  "  Note  on  some  appli- 
cations of  couformal  representation  to  mathematical  physics,"  pp.  639 — 652,  ante. 

t  This  result  was  obtained  independently  by  Christoffel  and  by  Schwarz:  I.e.,  p.  666,  note. 


268.]  TRIANGLE   ON   A   HALF-PLANE  671 

268.  It  may  be  remarked,  first,  that  any  three  of  the  real  quantities 
a,  h,  c,  . . . ,  I  can  be  chosen  arbitrarily,  subject  to  the  restrictions  that  the 
points  a,  h,  c,  ...,  I  follow  in  the  same  order  along  the  axis  of  w  as  the  angular 
points  of  the  polygon  and  that  no  one  of  the  remaining  points  passes  to 
infinity.     For  if  three  definite  points,  say  a,  b,  c,  have  been  chosen,  they  can, 

.  by  a  real  substitution 

where  p,  q,  r,  s  are  real  quantities  satisfying  ps  —  qr  =  1,  be  changed  into 
other  three,  say  a,  b',  c' :  and  then,  substituting 

and  using  the  relation  S(a  —  1)=  —  2, 

we  have  tu  =  TJ(^-ay-'(^-by-'...{^-iy-'d!;+C\ 

where  F  is  a  new  constant.  By  the  real  substitution,  the  axis  of  real 
quantities  is  preserved :  and  thus  the  new  form  equally  effects  the  con- 
formal  representation  of  the  polygon. 

But,  secondly,  it  is  to  be  remarked  that  when  three  of  the  points  on  the 
axis  of  a;  are  thus  chosen,  the  remainder  are  then  determinate  in  terms  of 
them  and  of  the  constants  of  the  polygon. 

269.  The  simplest  example  is  that  of  a  triangle  of  angles  utt,  /Sir,  j-jr,  so 

that 

a  +  /3  +  y  =  l. 

Then  a  particular  function  determining  the  conformal  representation  of  this 

w-triangle  on  the  half  2^-plane  is 

_  r dz 

^^  ~  j  (^  -  a)i-  {z  -  by-^  (z  -  cf-y ' 

dz 

so  that  T- = (^  -  «y~"  (^  -  by~^  (^  -  oy-^  > 

aw 
a  differential  equation  of  the  first  order*. 

For  general  values  of  a,  yS,  7,  which  are  rational  fractions,  the  integral- 
function  w  is  an  Abelian  transcendent  of  some  class  which  is  greater  than  1 : 
and  then,  after  §§  110,  239,  z  is  no  longer  a  definite  function  of  w,  and  the 
path  of  integration  must  be  specified  for  complete  definition  of  the  function. 

If  a  =  0,  the  only  instance  when  the  integral  is  a  uniform  function  of  w 
is  when  yS  =  i,  7  =  i  :  and  then  the  function  is  simply-periodic.  In  such 
a  case,  the  w-figure  is  a  strip  of  the  plane  of  finite  breadth,  extending  in  one 

*  Equations  of  this  type  are  discussed  in  my  Theory  of  Differential  Equations,  Part  11,  vol.  ii ; 
in  particular,  see  Chapters  ix,  x.  The  cases  when  the  integrals  are  uniform  functions,  either 
algebraic,  simply-periodic,  or  doubly-pei  iodic,  are  discussed  in  §  137  of  the  volume  quoted. 


672  SQUARE   ON    A   CIRCLE  [269. 

direction  to  infinity  and  terminated  in  the  finite  part  of  the  plane  by  a  straight 
line  perpendicular  to  the  direction  of  infinite  extension. 

Ex.  Discuss  the  case  when  a  =  0,  i3=0,  7=1,  pointing  out  the  relation  between  the 
half  of  one  plane  and  a  strip  in  the  other. 

If  no  one  of  the  quantities  a,  ^,  7  be  zero,  then  on  account  of  the  condition 

aJ^  ^  -\.  ^  =  1,  the  only  cases  when  the  integral  gives  ^^  as  a  uniform  function 
of  IV  are  as  follows  : — 

(I)  a  =  i,   /3  =  J,  7=3;  an  equilateral  triangle  : 

(II)  0L  =  \,  /3  =  \,   7  =  i;  an  isosceles  right-angled  triangle  : 

(III)     a  =  i,  ^  =  i,  7  =  i;  ^  right-angled  triangle,  with  one  angle  equal 

to  ^TT. 

In  each  of  these  cases  z  is  a,  uniform  doubly-periodic  function  of  w  ;  and 
after  arranging  the  constants  of  integration,  the  respective  relations  are 

(I)  -j  =  ^'(^o), 

where  the  invariants  g2,  gz  are  =  0,  4  ;  . 

(H)  ^=r(^). 

where  the  invariants  g^,  gz  are  =  —  4,  0 ; 

(HI)  1^.  =  ^'W' 

where  the  invariants  g^,,  gz  are  0,  4. 

The  integral  expressions  for  these  cases  were  given  by  Love*,  who  also 
discussed  a  further  case,  (due  to  Schwarz),  in  which  z  occurs  as  a  two-valued 
doubly-periodic  function  of  w.  The  triangle  is  then  isosceles  with  an  angle 
of  f TT,  the  values  of  a,  /3,  7  being  a  =  f,  ;8  =  ^,7  =  |-;  the  relation  is 

where  the  invariants  g^  and  g.^  are  0,  4. 

The  example  next  in  point  of  simplicity  is  furnished  by  a  quadrilateral, 
in  particular  by  a  rectangle  :  then 

a  =  ;S  =  7  =  8  =  i. 

When  no  one  of  the  homologues  of  the  angular  points  is  at  infinity  on 
the  real  axis  in  the  ^-plane,  the  general  form  is 

'w=j[{z—a){z  —  h){z  —  c){z  —  d)\~^dz, 

so  that  2^  is  a  doubly-periodic  function  oi  z. 

*  Amer.  Journ.  of  Math.,  vol.  xi,  (1889),  pp.  158—171. 


269.]  SQUARE   ON   A    CIRCLE  673 

When  one  (but  only  one)  of  the  homologues  of  the  angular  points  is  at 
infinity,  the  form  is 

w=J[(2-a)(z-  b)  {z  -  c)]~^dz, 

so  that  again  ^  is  a  doubly-periodic  function  of  z. 

When  two  of  the  homologues  of  the  angular  points  are  at  infinity — and 
there  cannot  be  more  than  two — then  after  the  explanations  of  §  267,  the 
form  of  relation  is 


dz\       dz  J      z  —  b     z  —  c' 

We  shall  consider  the  last  form  at  once.  There  are  two  alternatives 
according  as  b  and  c  are  distinct  or  as  they  coincide.  When  they  are  distinct, 
we  take  6  =  +  l,  c  =  —  1;  when  they  coincide,  we  take  6  =  c  =  0. 

For  the  first  case,  we  easily  find 

w  =  A  \ — :  +  B 

-'    {Z'  -  1)2 

=  A  cosh~^  z  +  B. 

Dropping  the  constant  B  (which  only  affects  the  origin  in  the  w-plane), 
we  have 

w 


z  =  cosh    ,  , 
A 


and  we  shall  take  A  real.     Thus 


u         V  .   ,    u    .     V 

^  cos  -^  ,     2/  =  smh  J  sm  ^ ; 


the  upper  half  of  the  ^-plane  is  conformally  represented  upon  the  interior  of 
a  strip  in  the  w-plane  bounded  by  the  lines  v  =  0,  v  =  7rA,  and  the  part  of 
u  =  0  between  these  two  lines. 


For  the  second  case, 

we  have 

d  A      dw\ 
dz\  ^^  dz)  ~ 

1 

z  ' 

and  so 

we  easily  find 

w  =  A  log  z  4- 

■B. 

Again 

dropping  the  constant  B  and  taking 

A  real, 

we 

have 

u 

X  =  e^  cos  -T  ,     y  = 

=  e^  sin 

V 

the  upper  half  of  the  ^;-plane  is  conformally  represented  upon  the  interior  of 
the  strip  in  the  w-plane  bounded  by  the  lines  v  =  0,  v  =  irA. 

Passing  now  to  the  general  form,  we  shall  first  suppose  that  the  rectangle 
is  a  square.     We  choose  oo ,  1,  0  as  points  on  the  axis  of  x  corresponding  to 
three   of  the   angular  points   in  order.     The  symmetry  of  the  figure   then 
F.  F.  43 


674 


RECTANGLE   ON 


[269. 


enables  us  to  choose  —  1  as  the  point  on  the  axis  of  x  corresponding  to  the 
remaining  angular  point.  As  the  homologue  of  one  angular  point  is  at 
infinity,  and  as  —  1,  0,  1  are  the  homologues  of  the  other  three,  we  have  the 
relation  between  w  and  z  in  the  form 


w  =  CJ{z  {z  -  1)  (^  +  V)\-^dz  +  G' 
dz 


=  0 


-T  +  C, 


{z(z^-l)}^ 

C  and  C   being  dependent  upon  the  position  and  the  magnitude  of  the 
w-square. 

Again,  the  half  ^-plane  is  transformed  (§  257,  Ex.  13)  into  the  interior  of 
a  ^-circle,  of  radius  1  and  centre  the  origin,  by  the  relation 

I  +  Z 

Then  except  as  to  a  constant  factor,  which  can  be  absorbed  in  C,  the  integral 

in  IV  changes  to 

dZ 


so  that,  by  the  relation 


W  = 


(1  -  Z'f 
dZ 


0  (1  _  z^y 

the  interior  of  a  ^-circle,  centre  the  origin  and  radius  1,  is  the  conformal 
representation  of  the  interior  of  some 
square  in  the  TT-plane.    Denoting  by 

L  the  integral      1 ,  so  that  2L 

is  the  length  of  adiagonal,  the  angular 
points  of  the  square  are  D,  A,  B,  G 
on  the  axes  of  reference :  and  these 
become  d,  a,  h,  c  on  the  circumference 
of  the  circle.  They  correspond  to 
representation  on  the  half-plane; 

Ex.  Shew  that  the  area  outside  a  square  in  the  w-plane  can  be  contbrmally  repre- 
sented on  the  interior  of  a  circle  in  the  s-plane,  centre  the  origin  and  radius  unity,  by  the 
equation 


Fig.  94. 
1,  0,  1,  GO    on  the  axis  of  x  in  the 


J  1  2" 
the  2-origin  corresponding  to  the  infinitely  distant  part  of  the  ?6'-plane.  (Schwarz.) 

Secondly,  let  the  rectangle  have  unequal  sides.     Then  the  symmetry  of 
the  figure  justifies  the  choice  of  j,  1,—  1,  — ^  as  four  points  on  the  axis  of  x 


269.]  A   HALF-PLANE  675 

corresponding  to  the  angular  points  of  the  rectangle  when  it  is  represented 
on  the  half-plane.     We  thus  have 

=  g!"  {(1  -  z')  (1  -  A;V^)|-  idz+  G\ 


w 

J  0 


If  the  rectangle  be  taken  so  that  its  angular  points  are  a,  a  +  2bi,  -a  +  2bi, 
-  a  in  order,  these  corresponding  to  1,  ^  ,  -  r ,  -  1  respectively,  then  we  have 


so  that  the  relation  is 


and  then 


0: 

=  C', 

a- 

=  CK, 

a 

+  2bi  : 

=  C(K  +  iK'); 

w 
a 

K  = 

'Jo 

{(1-. 

K' 
K~ 

q  =  e 

_2b 

a  ' 

27r6 
a 

whence 

where  q  is  the  usual  Jacobian  constant :  this  equation  determines  the  relation 
between  the  shape  of  the  rectangle  and  the  magnitude  of  k. 

In  the  particular  case  when  the  rectangle  is  a  square,  we  have  b  =  a  and 

K'  -        1  - 

so  ^  =  e-2^  or  -^  =  2  :  and  therefore  *  ^^  -  3  -  VS  or  y  =  3  -f-  V8.  The  differ- 
ence from  the  preceding  representation  of  the  square  is  that,  there,  the  point 
z  =  i  was  the  homologue  of  the  centre  of  the  square,  whereas  now,  as  may 
easily  be  proved,  the  point  z  =  i{\l2  -[- 1)  is  the  homologue  of  the  centre. 

Ex.     Discuss  the  transformation 


/"2  1  1 

tO=\      -,(1-202  COS  a -I- 3*)  "2(^0^ 


shewing  that  the  perimeter  of  a  rectangle  in  the  w-plane  is  transformed  into  the  circum- 
ference of  a  circle  in  the  z-plane.  Discuss  also  the  correspondence  of  the  areas  bounded 
by  these  perimeters. 

But  in  the  case  of  a  quadrilateral,  in  which  such  symmetrical  forms  are 
obviously  not  possible,  and,  in  the  case  of  any  convex  polygon,  only  three  points 
can  be  taken  arbitrarily  on  the  axis  of  x  :  the  most  natural  three  points  to  take 
are  0,  1,  x  for  three  successive  points.  The  values  for  the  remaining  points 
must  be  determined  before  the  representation  can  be  considered  definite. 

*  This  is  derived  at  once  by  means  of  the  quadric  transformation  in  elliptic  functions. 

43—2 


676  QUADRILATERAL  [269. 

Thus  in  the  case  of  a  quadrilateral,  taking  oo ,  0,  1  as  the  homologues  of 
D,  A,  B  respectively  and  -  as  the  homologue  of  C,  c 

(where  yu.<l),  the  equation  for  conformal  representation 

is 

w  =  Cu  +  C, 
where  * 

ti  =      z'^-'^  (1  -  zf-^  (1  -  fxzy'-'^  dz  =      Zdz,  say.  pig.  95. 

Jo  '  .'0 

If  the  w-origin  be  taken  at  A,  and  the  real  axis  along  AB,  we  have 

0  =  C", 


a=c\^  Xdx+G', 

.'o 


de^'"'  =  G\     Xdx+C, 

0 


6e»-(i-^)  =  (7r  Xdx+C, 


being  the  equations  for  the  four  angular  points.  They  determine  only  three 
quantities  G,  G',  /u.,  so  that  they  coexist  in  virtue  of  a  relation,  which  is  in 
effect  the  relation  between  the  sides  and  the  angles  of  a  quadrilateral. 

An  equation  to  determine  fi  is 


a       Xdx  =  de^"""      Xdx ; 


the  second  equation  serves  to  determine  G,  because  G'  =  0. 

The  equation  determining  /i  can  be  modified  as  follows*,  so  as  to  be  expressed  in 
terms  of  the  hypergeometric  series. 

d    ■ 
Let  -  e'^"'  =  \,  so  that  the  equation  is 
a 


I     Xdx  =  X  I     Xdx. 
Jo  Jo 


Now  to  compare  these  integrals  with  the  definite  integrals  which  are  solutions  of  the 
diflFerential  equation  of  the  hypergeometric  sei'ies,  we  take 

a'  =  l--y,  /3'  =  a,  'Y'  =  a  +  fi, 

so  that  .Y=x^'~^(l-.r)^'"^'"\l-;x.x-)""'. 

And  a'>0<l,  y'-^'>0,  a'  + 1 -y' =  2 -y- a- /3  =  S  >  0, 

SO  that,  as  jLt  <  1,  the  definite  integral  is  finite  at  all  the  critical  points. 

*  For  the  analytical  relations  in  reference  to  the  definite  integrals,  see  Goursat,  "  Sur 
r^quation  differentielle  lineaire  qui  admet  pour  integrale  la  s^rie  hypergeometrique,"  Ann. 
de  VEc.  Norm.  Sup.,  2°^=  Ser.,  t.  x,  (1881),  Suppl.,  pp.  3—142;  and  for  the  relations  between  the 
hypergeometric  series,  see  my  Treatise  on  Differential  Equations,  4th  ed.,  pp.  211 — 230,  290 — 293, 
the  notation  of  which  is  here  adopted. 


269.] 

We  have 


DETERMINATION    OF    CONSTANT 


677 


/>^=-j«ji&;=«^K  ,■„.,,) 


r(a)r(/3) 


Ti; 


,r.(ff-i)r-(g)r(y)      _ 
r(/3+y)  '^*' 


xi^U'-/3',   l-,3',  y'-a'-/3'  +  l, 


_  -..■(i3+v)r(y)r(S) 


Xi^(^^'-y'  +  l,   1-a',  2-y',   -^ 


r(y+8) 


^T-'I^2• 


Hence 

Now,  if 


r(a)r(^)     ^  ^.•(/3-i)rO)r(y)  x.  ,   -«(0+y)r^yOr^) .. 
^^    '^  r(a+/3)  -^^-'  r(y+S)  ^*+'  TFPsy    ■ 


J/l=- 


n(y-i)n(-a')n(-/y) 


r(a+^)r(y)r(i-a) 


n(l-y')n(y'-a'-l)n(y'-/3'-l)      T  (y  +  S)  T  (1 -8)  T  (/3)  ' 

^,^  ^     n(-aOn(-^')     _      r(y)r(i-«) 


n(y'-a'-i3')n(-y')     r(/3  +  y)r(y-i-S-i)'      ■ 
then  ri  =  Af^Yo+N^Yi. 

Substituting,  we  have 

*L^       ^  r(a+^)  r(^+y)  _ 

^L  r(y  +  S)       ^         ^   r(a  +  i3)       i_ 

By  using  properties  of  the  Gamma  functions,  the  coefficient  of  Y^  can  be  proved  equal  to 

e"'"      r((3)r(y),,    .     ,        ^,  -     ^    -,  e""siny7r     r  (^)  r  (y) 

asmoTT    r(/3  +  y)    ^  ^         '  '  asmoTr        r(/3  +  y)  ' 

and  the  coefficient  of  Y.2  can  be  proved*  equal  to 

e"°     r(y)r(g),^   .     .   _^^^    ,  w«_L.   ^     )         e"Siny7r,r(y)r(8) 

— ^ '/     .  ^N  {asm  (a  +  j3  +  y)  7!--a  sm  (/S+y)  7r}= ^ ^—  h  —rj-^ — ^  . 

asmaiv  r(y  +  8)   ^  ^    'MT-//  \t-^  r/    j  a  sm  ott       r(y  +  S) 

Moreover 

l-y 


SO  that 


72=^20=^4  =  /    ^(l-/^f  ""     ^  F{l-a',   1-0',   2-y',  m}, 

"^>{l-a',   1-/3',  y'-a'-/3'  +  l,   1-/.}: 
Fa  i^{y,    1-a,   y  +  8,  /x}      . 


^4=y24=J/8  =  /^^     ^   (1-M 


Fi      /'{y,    1-a,  y+ft   1-;^}' 
and  therefore  an  equation  to  determine  /i  is 

F{y,    l-g,   y  +  d,   ^}      ^C  V{^)V{y  +  b) 
F{y,    1-a,   y  +  /3,    l-^a}       6  T  (S)  T  (y+/3)  ' 


*  In  reducing  the  coeflBcients  to  these  forms,  limiting  cases  (such  as  ;8  +  y=l)  of  the  quadri- 
lateral are  excluded. 


678  LIMITING   CASE   OF   POLYGON  [269. 

Ex.  1.  A  convex  quadrilateral  in  the  w-plane  has  its  sides  AB,  BC  equal  to  unity,  its 
angles  A  and  C  each  equal  to  air,  and  its  angle  B  equal  to  /Stt  ;  prove  that  it  can  be 
conformally  represented  upon  the  positive  half  of  the  s-plane  by  the  relation 


-1/1      ^.•iNa-i  j„_  /    -^-1/1      „2n'^~i 


"l 

2 

Ts 

_2 

(1- 

-^")  »  dx= 

/      ^1- 

-s")  ^dz. 

0 

^ 

/o 

x^-\l-x'f    'dx=       z^    \\-z^T     dz. 

(Math.  Trip.,  Part  II.,  1895.) 

Ex.  2.  A  regular  polygon  of  n  sides,  in  the  w-plane,  has  its  centre  at  the  origin 
and  one  angular  point  on  the  axis  of  real  quantities  at  a  distance  unity  from  the  origin. 
Shew  that  its  interior  is  conformally  represented  on  the  interior  of  a  circle,  of  radius 
unity  and  centre  the  origin,  in  the  2-plane  by  means  of  the  relation 

(Schwarz.) 

Ex.  3.  A  plane  non-reentrant  hexagon  ABCDEF  is  symmetrical  about  its  diagonal 
AD:  prove  that  the  enclosed  area  can  be  conformally  represented  upon  a  half -plane, 
so  that  the  angular  points  correspond  to  points  co ,  —  1,  — /x,  0,  /x,  1  on  the  real  axis, 
provided  /x  is  determined  by  the  equation 

]_ 

where  the  variable  6  is  real  throughout  the  two  ranges  of  integration,  b  and  c  are  the 
sides  AB  and  BC  respectively,  and  air,  /Stt,  yn  are  the  internal  angles  A,  B,  G  respectively, 

(Trinity  Fellowship,  1898.) 

270.  It  is  natural  to  consider  the  form  which  the  relation  assumes  when 
we  pass  from  the  convex  polygon  to  a  convex  curve,  by  making  the  number 
of  sides  of  the  polygon  increase  without  limit.  The  external  angle  between 
two  consecutive  tangents  being  denoted  by  d\lr,  and  the  internal  angle  of  the 
polygon  at  the  point  of  intersection  of  the  tangents  being  ^tt,  we  have 

TT  —  ^TT  =  dyjr, 

dyl/' 
so  that  ^—1  — . 

TT 

Let  X  be  the  point  on  the  axis  of  real  quantities,  which  corresponds  to  this 
angular  point  of  the  polygon ;  then  the  limiting  form  of  the  relation 

d  /,      dw\  _  n;>  a  —  1 


dz\°dzj         2  —  a 
d   (.      dw\  1    i    dylr 


dz  \         dz]  TT  J  z  —  x^ 

where  x  is  the  point  on  the  real  axis  in  the  ^-plane  corresponding  to  the 
point  on  the  tu-cwxve  at  which  the  tangent  makes  an  angle  i/r  with  some 
fixed  line,  and  the  integral  extends  round  the  curve,  which  is  supposed  to  be 
simple  (that  is,  without  singular  points)  and  everywhere  convex. 


270.]  AS  A  CONVEX  CURVE  679 

The  disadvantage  of  the  form  is  that  x  is  not  known  as  a  function  of  -v/r ; 
and  its  chief  use  is  to  construct  curves  such  that  the  contour  is  conformally 
represented,  according  to  any  assigned  law,  along  the  axis  of  real  quantities 
in  the  ^-plane.  The  utility  of  the  form  is  thus  limited :  the  relation  is  not 
available  for  the  construction  of  a  function  by  which  a  given  convex  area  in 
the  w-plane  can  be  conformally  represented  on  the  half  of  the  2^-plane*. 

Ex.     Let  A'  =  tan  ■^■v//- :    then  taking  the  integral  from  —  tt  to  +7r,  we  have 


log 


dz\    ^  dz  J  TT  j  _^^-tan|■v//■ 


7r  7  _i„3  — tan 
The  integral  on  the  right-hand  side  is 


p'^       d(i>       _  fo 
j  0    2  -  tan  (^      j  1 


z  +  tan  (f) 
d(t> 


/  0    z^  —  tan^  (p 


and  therefore 


d^  /       ^\  _  _  _2_ 
dz\^dz)~     z-V 


which,  on  further  integration,  leads  to  the  ordinary  expression  for  a  circle  on  a  half- 
plane. 

271.  In  regard  to  the  conformal  representation  on  the  half  of  the 
^-plane  of  figures  in  the  'Zt'-plane  bounded  by  circular  arcs,  we  proceed-f- 
in  a  manner  similar  to  that  adopted  for  the  conformal  representation  of 
rectilinear  polygons. 

*  See  Christoffel,  Gott.  Naehr.,  (1870),  pp.  283—298. 

t  For  the  succeeding  investigations  the  following  authorities  may  be  consulted : — 

Schwarz,  Ges.  Werke,  t.  ii,  pp.  78—80,  221—259. 

Cayley,  Camb.  Phil.  Trans.,  vol.  xiii,  (1879),  pp.  5—68;  Coll.  Math.  Papers,  vol.  xi, 
pp.  148—216. 

Klein,  Vorlesungen  ilber  das  Ikosaeder,  Section  I.,  and  particularly  pp.  77,  78. 

Darboux,  Theorie  generale  des  surfaces,  t.  i,  (2nd  ed.),  pp.  512 — 526. 

Klein-Fricke,  Theorie  der  elliptischen  Modulfunctionen,  t.  i,  pp.  93 — 114. 

Goursat,  I.e.,  p.  676,  note. 

Schonflies,  3Iath.  Ann.,  t.  xlii,  (1893),  pp.  377—408,  ib.,  t.  xliv,  (1894),  pp.  105—124. 


680  SCHWARZIAN   DERIVATIVE  [271. 

It  is  manifest  that,  if  u=f{z)  determine  a  eonformal  representation  on 
the  2^-plane  of  a  w-polygon  bounded  by  circular  arcs  and  having  assigned 
angles,  then 

Au  +  B 

Cu  +  D' 

where  A,  B,  C,  D  may  be  taken  subject  to  the  condition  AD  —  BG  =  1,  will 
represent  on  the  half  ^-plane  another  such  polygon  with  the  same  assigned 
angles :  for  the  homographic  transformation,  preserving  angles  unchanged, 
changes  circles  into  circles  or  occasionally  into  straight  lines.  Hence,  as 
in  §  264,  when  the  transforming  function  is  being  obtained,  it  is  to  be  expected 
that  it  will  be  such  as  to  admit  of  this  apparent  generality :  and  therefore, 
since 

[w,  z]  =  [u,  z], 

where  [w,  z\  is  the  Schwarzian  derivative,  it  follows  that,  in  obtaining  the 
eonformal  representation  of  a  figure  bounded  by  circidar  arcs,  the  function  to 
be  constructed  is 

We  proceed,  as  in  the  case  of  the  rectilinear  polygon,  to  find  the  form  of 
the  appropriate  function  in  the  vicinity  of  points  of  various 
kinds.     But  one  immediate  simplification  is  possible,  which 
enables  us  to  use  some  of  the  earlier  results. 

Let  C  be  an  angular  point,  CA  and  CB  two  circular 
arcs,  one  of  which  may  be  a  straight  line  :  if  both  were 
straight  lines,  the  modification  would  be  unnecessary.  In- 
vert the  figure  with  regard  to  the  other  point  of  intersection 
of  CA  and  GB  :  the  two  circles  invert  into  straight  lines  cutting  at  the  same 
angle  /jltt.  Take  the  reflexion  of  the  inverted  figure  in  the  axis  of  imaginary 
quantities  :  and  make  any  displacement  parallel  to  the  axis  of  real  quantities  : 
if  W  be  the  new  variable,  the  relation  between  lo  and  W  is  of  the  form 


aW  +  b 

cW  +  d~'"' 

rhere  ad  - 

-bc  =  l;  and  therefore 

{ W,  z]  =  [w,  z 

Consider  the  function  for  the  T^-plane.  Let  T  be  the  point  corresponding 
to  G,  an  angular  point  of  the  polygon,  having  ^r  =  c  as  its  homologue  on  the 
axis  of  x,  account  being  taken  of  the  possibility  of  having  c  =  x  ;  let  /3  be  any 
point  on  either  of  the  straight  lines  corresponding  to  a  point  on  the  contour 
of  the  polygon  not  an  angular  point,  having  z  =  b  as  its  homologue  on  the 
axis  of  x.  If  a  contour  point  not  an  angular  point  have  ^  =  oo  as  its 
homologue  on  the  axis,  denote  it  by  /3'. 


271.]  FOR   EEPRESENTATION   ON   A    CIRCLE  681 

Then  for  the  vicinity  of  ^,  we  have  (as  in  §  267)  a  relation  of  the  form 
F-  /3  =  e*>+«) (z-b)P(z-b); 


then 


so  that 


dW 

log  -7—  =  const.  4-  log  Pi  (z  —  b), 

{W,z]=P,(z-b), 


where  Pg  is  an  integral  function  of  z  —  b,  converging  for  sufficiently  small 
values  of  \  z  —  b\.' 

For  the  vicinity  of  /3',  we  have  similarly 

z      \z 


then 


and  therefore 


f=^"  }=«.©. 


:f,^}=--I 


where  Q2  is  finite  for  ^^  =  co  . 

In  the  vicinity  of  the  angular  point  F,  having  a  finite  point  on  the  axis  of 
X  for  its  homologue,  we  have 

W-T  =  e''(-+«)  (z  -  cY  R{z-  c), 

and,  proceeding  as  before,  we  find  that 

r 

+  R^{z-  c), 


Tf,  ^}  =  *^^~^'^  ■     ^' 


+ 


{z  —  c)~       z  —  c 

where  G^  depends  on  the  coefficients  in  the  series  R{z  —  c). 

But  if  the  angular  point  F  have  the  point  at  infinity  on  the  axis  of  x  for 
its  homologue,  we  have 

F-F  =  e*"(-+^)-Tf- 
then,  proceeding  as  before,  we  find  that 


\W,z]=^ 


ia-/^^) ,  1 


-^^U' 


where  T^i-]  is  finite  when  z  ^  00 


682  CURVILINEAR   POLYGON  [271. 

Lastly,  for  a  point  W  in  the  interior  having  its  homologue  at  2^  =  z',  we 

have 

W-  W'  =  {z-z)S{z-z'), 

and  then  [W,z]=  S^ {z  -  z). 

Hence  [  W,  z],  considered  as  a  function  of  z,  has  the  following  properties  : — 

(i)  It  is  an  analytical  function  of  z,  real  for  all  real  vaTues  of  the 
argument  z ;  and  if  ic  =  00  do  not  correspond  to  an  angular 
point  of  the  polygon,  then  for  very  large  values  of  z 

where  Q^  is  finite  when  z  =  00  . 

(ii)  It  has  a  finite  number  of  accidental  singularities,  all  of  them 
isolated  points  on  the  axis  of  x :  and  at  all  other  points  on  one 
side  of  the  plane  it  is  uniform,  finite,  and  continuous,  having 
(except  at  the  accidental  singularities)  real  continuous  values 
for  real  continuous  values  of  its  argument.  Its  form  near  the 
singularities,  and  its  form  for  infinitely  large  values  of  z,  it 
^  =  00  be  the  homologue  of  an  angular  point,  are  given  above. 

Hence  {IT,  ^r|  can  be  continued  across  the  axis  of  x,  conjugate  values  of 
[W,  z]  corresponding  to  conjugate  values  of  z:  and  thus  its  properties  make 
it  a  rational  meromorphic  function  of  z. 

Two  cases  have  to  be  considered. 

First,  let  the  angular  points  of  the  polygon  have  their  homologues  at 
finite  distances  from  the  ^-origin,  say,  at  a,  6,  . . . ,  ^ :  and  let  air,  ^ir,  ...,  Xtt  be 
the  internal  angles  of  the  polygon  at  the  vertices.     Then 


w   ^|_V_^^_iV 


2  — 


z  —  a      "     (z  —  of 

has  no  infinity  in  the  plane ;  it  is  a  uniform  analytical  function  of  z,  and 
must  therefore  be  a  constant,  which,  by  the  value  at  z=  ao  ,  is  seen  to  be 
zero.     Hence 

[W,z]=X-^+^X^—^^  =  2J{z\ 
^         '         z  —  a  {z  -ay 

the  summation  being  for  the  homologues  of  all  the  angular  points  of  the 
polygon.     But  when  z  is  very  large,  we  have,  in  this  case 


F..)  =  ^ft(l-), 


271.]  REPRESENTED   ON   A    CIRCLE  683 

SO  that,  expanding  2J{z)  in  powers  of  -  and  comparing  with  the  latter  form, 
we  have,  on  equating  coefficients  of  z'~'^,  z~^,  z~^, 

0  =  2^0^'  +  2a  (1  -  a^), 

relations  among  the  constants  of  the  problem. 

Secondly,  let  one  angular  point,  say  a,  of  the  polygon  have  its  homologue 
on  the  axis  of  x  at  infinity,  and  let  oltt  be  the  internal  angle  at  a :  and  let  the 
homologues  of  the  others  be  h,  ...,  k,  I,  the  internal  angles  of  the  polygon 
being  /Stt,  . . . ,  kit,  Xtt.     Then  the  function 

has  no  infinity  in  the  plane :  it  is  a  uniform  analytical  function  of  z,  and 
must  therefore  be  a  constant,  say  M;  thus 

'         ^  z  —  b{z  —  by 

But,  when  z  is  very  large,  we  have 


tr,.)  =  ^<i-'"'> 


z     \z 


Z" 

'  because  a;  =  oo  is  the  homologue  of  the  vertex  a  of  the  polygon,  the  angle 
■  there  being  cctt  ;   also,   T  (-)   is  finite  when  z  =  ao .     Hence,  expanding  in 

powers  of  -  and  comparing  coefficients,  we  have 

25o  =  0, 

so  that  lW,z]  =  ^^  +  it^—^.=  2I{z), 

z  —  b  {z  —  bf 

where  the  summation  is  for  the  homologues  of  all  the  angular  points  other 
than  a,  and  the  constants  are  subject  to  the  two  conditions 

%B,b=^{i-o?)-\x{i-n 

The  form  of  the  function  { W,  z]  is  thus  obtained  for  the  two  cases,  the 
latter  being  somewhat  more  simple  than  the  former :  and  the  exact  expansion 
of  W  in  the  vicinity  of  a  singular  point  can  be  obtained  with  coefficients 
expressed  in  terms  of  the  constants. 


684  CEESCENT  [2T2. 

272.  In  either  case  the  equation  which  determines  W  is  of  the  third 
order;  but  the  determination  can  be  simplified  by  using  a  well-known 
property  of  linear  differential  equations*.  If  i/i  and  3/2  be  two  solutions  of 
the  equation 

the  quotient  of  which  is  equal  to  the  quotient  of  two  solutions  of 

dx- 

dP 
where  I=Q--j P-,  being  the  invariant  of  the  equation  for  linear  trans- 

formation  of  the  dependent  variable,  and  where  Y/y  =  e^^'^^,  then  the  equation 

satisfied  by  s,  =  y-ily-,,  is 

[s,  x]  =  21. 

Hence  for  the  present  case,  if  we  can  determine  two  independent  solutions 
Z^  and  Z2  of  the  equation 

drZ 

dz^ 


+  ZJ{z)  =  0  , 


for  the  first  case,  or  two  independent  solutions  of  the  equation 

for  the  second  case,  then 

AZ,  +  BZ, 
CZ,  +  DZ, 

is  the  general  solution  of  the  equation 

{W,z]  =  ^J{z)oy2I{z\ 

and  therefore  is  the  function  by  which  the  curvilinear  w-polygon  is  conform- 
ally  represented  on  the  ^-half-plane. 

273.  As  a  first  example,  consider  the  w-area  between  two  circular  arcs 
which  cut  at  an  angle  Xtt.  The  ^r-origin  can  be  conveniently  taken  as  the 
homologue  of  one  of  the  angular  points,  and  the  ^-point  at  infinity  along  the 
axis  of  X  as  the  homologue  of  the  other.     Then  we  have 

[W,z]=  —  +  ^^^—^ —  , 
'      z  z^ 

provided  ^  =  0,  vl .  0  =  |(1 -\-) -i(l ->^'X 

both  of  which  conditions  are  satisfied  by  J.  =  0 ;  and  so 

'■   See  my  Treatise  on  Differential  Equations,  §§  59 — 62. 


273.] 


CURVILINEAR   TRIANGLE 


685 


The  linear  differential  equation  is 

so  that  Z,  =  z^-^^^'^\     Z,  =  ^i(i-^>; 

and  therefore  the  general  solution  for  Tl^  is 

cz^  +  d' 
The  (three)  arbitrary  constants  can  be  determined  by  making  z  =  0  and 
z  =  QC  correspond  to  the  angular  points  of  the  crescent,  and  the  direction  of 
the  line  z  —  z^  (which  is  the  axis  of  x)  correspond  to  one  of  the  circles,  the 
other  of  the  circles  being  then  determinate. 

If  the  w-circles  intersect  in  —  i  (the  homologue  of  the  5-origin)  and  +  i 

(the  homologue  of  «  =  oo  ),  and  if  the  centre  of  one  of  the  circles  be  at  the 

point  (cot  a,  0),  then  the  relation  is 

.  z^  —  ce~"'' 

w  =  ^  — -, 

^^  +  ce~°-^ 

where  c  is  an  arbitrary  constant,  equivalent  to  the  possible  constant  magnifi- 
cation of  the  5^-plane  without  affecting  the  conformal  representation :  it  can 
be  determined  by  fixing  homologous  points  on  the  contour  of  the  crescent. 

More  generally,  if  the  w-circles  intersect  in  w-^  and  Wg)  respectively  homo- 
logous to  ^  =  0  and  z=  cc  ,  then 

is  the  form  of  the  relation. 

Evidently  a  segment  of  a  circle  is  a  special  case. 

274.  Next,  consider  a  triangle  in  the  w-plane  formed  by  three  circular 
arcs  and  let  the  internal  angles  be  Xtt,  fiir,  vtt.  The  homo- 
logue of  one  of  the  angular  points,  say  of  that  at  yu,7r,  can  be 
taken  at  z—co;  of  one,  say  of  that  at  Xtt,  at  the  2^-origin ; 
and  of  the  other,  say  of  that  at  vir,  at  a  point  z=l:  all  on 
the  axis  of  cc.     Then  we  have 


B        C 
z      z  —  \ 


.  1  -  \2 
+  h 7r-+^ 


''{z-iy 

where  the  constants  B  and  G  are  subject  to  the  relations 

5+0=0, 
5 . 0  +  (7 . 1  =  i  (1  -  /^^)  -  i  (1  -  X^)  -  i  (1  -  v% 


Fig.  97. 


SO  that 

and  therefore 


-B=C=^{X^-fi?  +  v''-l\ 


iz-^y 


+ 


^    V  -  yU,^  -I-  V^  -    1 

^       z{z-\) 


686 


CON  FORMAL   REPRESENTATION 


[274. 


But  I{z)  is  the  invariant  of  the  differential  equation  of  the  hypergeometric 

series  * 

<PZ     7-(a+;8+l)^  dZ  _      a/3 

~d^^        z(l-z)         dz      z(l-z)  ' 

provided  \^  =  (l-y)\     ^^  =  {pi-^y,     v^  =  (y- a- /3y ; 

so  that,  if  Zi  and  Z.2  be  two  particular  solutions  of  this  equation,  the  function 
which  gives  the  conformal  representation  of  the  w-triangle  on  the  ^^-half- 
plane  is 


w  = 


AZ,  +  BZ, 


CZ,  +  DZ,' 

The  transforming  function  thus  depends  upon  the  solution  of  the  differential 
equation  of  the  hypergeometric  series,  and  for  general  values  of  \,  /m,  v 
which  are  >  0  <  1  we  shall  obtain  merely  general  values  of  a,  /3,  7 ;  hence 
the  transforming  function  will  be  obtained  as  a  quotient  of  two  particular 
solutions  of  the  equation  of  the  series.  Now  according  to  the  magnitude  of 
\z\,  these  solutions,  which  are  in  the  form  of  infinite  series,  change:  and  thus 
we  have  w  equal  to  an  analytical  function  of  z,  which  has  different  branches 
in  different  parts  of  the  plane. 

The  distribution  of  the  values  ^^  =^  0,  1,  00  as  the  homologues  of  the  three 
angular  points  was  an  arbitrary  selection  of  one  among  six  possible  arrange- 
ments, which  change  into  one  another  by  the  following  scheme : — 


1-z 

1 

1 

1-z 

z 

2-1 

z-\ 

z 

0 

1 

00 

1 

0 

00 

1 

0 

1 

00 

00 

0 

00 

00 

0 

0 

1       1       1 

The  quantities  in  the  first  row  are  the  homographic  substitutions,  conserving 
the  positive  half-plane  and  interchanging  the  arrangements. 

These  substitutions  are  the  functions  of  z  subsidiary  to  the  derivation  of 
Kummer's  set  of  24  particular  solutions  of  the  equation  of  the  hypergeometric 
series. 

Ex.  Take  the  case  when  two  of  the  angles  of  the  triangle  are  right,  say  v  =  h,  ^■  =  ^. 
X  =  ~ .     Then,  when  n  is  finite t,  a  transforming  relation  is 


l  +  (i-2)*' 

and,  when  n  is  infinite,  a  transforming  relation  is 

1         l-(l-3)* 

«<^  =  log  i -- 

l+(l-2)^ 

*  Treatise  on  Differential  Equations,  §  116. 


t  ib.,  §  131. 


and 


274.]  OF    CURVILINEAR   TRIANGLE  687 

obtainable  either  as  a  limiting  form  of  the  above,  or  by  means  of  the  solutions  F{a,  ^,  y,  z) 
and  i^(a,  /3,  a  +  /3  -  7+ 1,  1  -«)  of  the  differential  equation  of  the  hypergeometric  series.  In 
the  respective  cases  the  general  relations,'  establishing  the  conformal  representation,  are 

aw  +  by  _!-{!- z)^ 
^«'  +  ^/    ~l+(l-2)4' 

aw  +  b     ,      l-n-z)i 
,  =  log ^ — . 

CW  +  d  ^l  +  (i._^)4 

The  three  circles,  arcs  of  which  form  the  triangle,  divide  the  whole  of  the 
w-plane  into  eight  triangles  which  can  be  arranged 
in  four  pairs,  each  pair  having  angles  of  the  same  . 

magnitude.     Thus  D' 

D,  D'  have  angles  Xir,  /xtt,  vtt.  ^^ 


A,  A'    Xtt,  (l-/x)7r,  (l-z/)7r,  /        \  /'"A     ^ 

B,  B'       i\  —  X)  TT ,  IXTT ,   {1  —  v)  IT , 


A 


M' 


and  C,  C    (1  —  X)7r,  (1  —  yu,)  vr,  z^tt  ;  \  \C/     B 

and  when  any   one   of  the   triangles  is  given,  it 

determines  the  remaining  seven.     It  is  convenient 

to  choose  the  particular  pair  which  has  the  sum  of  its  angles  not  greater 

than  the  sum  for  any  of  the  others.     A  triangle  of  the  selected  pair  is  called 

the  reduced  triangle*;   let  it  be  the  triangle  D.      Let  Sir  be  this  smallest 

sum,  so  that  8=\  +  fx-\-v\  as  the  sum  of  the  angles  in  each  of  the  other 

pairs  of  triangles  is  equal  to,  or  greater  than,  Sir,  we  have 

4*S^7r  <  sums  for  pairs  A,  B,  C,  J) 

<  Gtt, 

so  that  ^  <  I ,  that  is,  A,  +  //,  4- 1-  <  f .     (The  only  case  of  exception  arises  when 
all  the  four  sums  are  equal  to  one  another ;  and  then  A,  =  ^u,  =  z/  =  ^.) 

We  have  already,  in  part,  considered  the  case  in  which  X  +  fj,  +  v  =  1. 
For,  when  this  equation  holds,  inversion  with  the  other  point  having  Xir  for 
its  angle  as  centre  of  inversion,  changes  f  D  into  a  triangle  bounded  by 
straight  lines  and  having  Xir,  /jltt,  vtt  as  its  angles;  and  therefore,  in  that 
case,  the  problem  is  merely  a  special  instance  of  the  representation  of  a 
w-rectilinear  polygon  on  the  ^•-half-plane. 

But  there  is  a  very  important  difference  between  the  cases  for  which 
\  +  [x-\-v  <  1  and  those  for  which  \  +  iju  +  v  >1:  in  the  former,  the  ortho- 
gonal circle  (having  its  centre  at  the  radical  centre  of  the  three  circles)  is  real, 
and  in  the  latter  it  is  imaginary.     The  cases  must  be  treated  separately,      ,^ 

*  Schwarz,  Ges.  Werke,  t.  ii,  p.  236. 

t  The  figure  in  the  text  does  not  apply  to  this  case,  because,  as  may  easily  be  proved,  the 
three  circles  must  meet  in  a  point. 


688  FUNCTIONAL    RELATION  [275. 

275.  First,  we  take  \  +  //,  +  v  <  1.  Then  of  the  two  triangles,  which 
have  the  same  angles,  one  lies  entirely  within  the  orthogonal  circle  and  the 
other  entirely  without  it ;  and  each  is  the  inverse  of  the  other  with  regard  to 
the  orthogonal  circle  *.  Let  inversion  with  regard  to  the  angular  point  Xtt  in 
A  take  place  :  then  the  new  triangle  is  bounded  by  two  straight  lines  cutting 
at  an  angle  Xtt  and  by  a  circular  arc  cutting  them  at 
angles  /xtt  and  vir  respectively,  the  convex  side  of  the 
arc  being  turned  towards  the  straight  angle.  The 
new  orthogonal  circle  is  the  inverse  of  the  old  and  its 
centre  is  A,  the  angular  point  at  Xir ;  its  radius  is  the 
tangent  from  A  to  the  arc  GB,  and  therefore  it  com- 
pletely includes  the  triangle  ABC. 

The  homologue  of  A  is,  as  before,  taken  to  be  the  ^r-origin  0,  that  of  C  to 
be  the  point  z  =  1,  say  c,  and  that  of  5  to  be  ^  =  co  on  the  axis  of  x,  say  b  for 
+  00  and  6'  for  —  00  . 

Suppose  that  we  have  a  representation  of  the  triangle  on  the  positive 
half-plane  of  z.  The  function  [w,  z]  can  be  continued  across  the  axis  of  x 
into  a  negative  half-plane,  if  the  passage  be  over  a  part  of  that  axis,  where 
the  function  is  real  and  continuous,  that  is,  if  the  passage  be  over  Oc,  or  over 
c&,  or  over  h'O ;  and  therefore  w  is  defined  for  the  whole  plane  by  [w,  z]  =  2/  (z), 
its  branch -points  being  0,  a,  b.  Any  branch  on  the  other  side,  say  w^,  will 
give,  on  the  negative  half-plane,  a  representation  of  a  triangle  having  the 
same  angles,  bounded  by  circular  arcs  orthogonal  to  the  same  circle,  and 
having  0,  c,  b  for  the  homologues  of  its  angular  points.  Thus  if  the  con- 
tinuation be  over  cb,  the  new  w-triangle  has  CB  common  with  the  old,  and 
the  angular  point  J.'  lies  beyond  CB  from  A. 

To  obtain  the  new  triangle  A  'CB  geometrically,  it  is  sufficient  to  invert 
the  triangle  AGB,  with  regard  to  the  centre  of  the  circular  arc  CB.  This 
inversion  leaves  CB  unaltered;  it  gives  a  circular  arc  CJ.'  instead  of  CA 
and  a  circular  arc  BA'  instead  of  BA:  the  angles  of  A'CB  are  the  same  as 
those  of  A  CB.  Since  the  orthogonal  circle  of  A  CB  cuts  CB  at  right  angles 
and  CB  is  inverted  into  itself,  the  orthogonal  circle  is  inverted  into  itself; 
therefore  the  triangle  A'CB  has  the  same  orthogonal  circle  as  the  triangle 
ACB. 

The  branch  lu^ ,  by  passing  back  across  the  axis  round  a  branch-point  into 
the  positive  half-plane,  leads  to  a  new  branch  lUo,  which  gives  in  that  half-plane 
a  representation  of  a  triangle,  again  having  the  angles  Xtt,  /mtt,  vtt  and  having 
0,  c,  b  for  the  homologues  of  its  angular  points.  Thus  if  'the  passage  be 
over  Oc,  the  new  w-triangle  has  A'C  common  with  A'CB  and  the  angular 
point  B"  lies  on  the  side  of  CA'  remote  from  B :    but  if  the   passage   be 

*  For  the  general  properties  of  such  systems  of  circles,  see  Lachlan,  Quart.  Journ.  Math., 
vol.  xxi,  (1886),  pp.  1—59. 


275.]  FOR   CURVILINEAR   TRIANGLE  689 

over  cb,  then  we   merely  revert   to   the   original   triangle  CAB.      The   new 
triangle  has,  as  before,  the  same  orthogonal  circle  as  A'GB. 

Proceeding  in  this  way  by  alternate  passages  from  one  side  of  the 
axis  of  X  to  the  other,  we  obtain  each  time  a  new  w-triangle,  having  one  side 
common  with  the  preceding  triangle  and  obtained  by  inversion  with  respect 
to  the  centre  of  that  common  side :  and  for  each  triangle  we  obtain  a  new 
branch  of  the  function  w,  the  branch-points  being  0,  1,  oo .  If,  by  means  of 
sections  such  as  Hermite's  (§  103),  we  exclude  all  the  axis  of  x  except  the  part 
between  two  branch-points,  the  function  is  uniform  over  the  whole  plane  thus 
bounded. 

All  these  triangles  lie  within  the  orthogonal  circle,  and  they  gradually 
approach  its  circumference :  but  as  the  centres  of  inversion  always  turn  that 
circle  into  itself,  while  the  sides  of  the  triangle  are  orthogonal  to  it,  they  do 
not  actually  reach  the  circumference.  The  orthogonal  circle  forms  a  natural 
limit  (§81)  to  the  part  of  the  w-plane  thus  obtained. 

Ex.  Shew  that  all  the  inversions,  necessary  to  obtain  the  complete  system  of  triangles, 
can  be  obtained  by  combinations  of  inversions  in  the  three  circles  of  the  original  triangle. 

(Burnside.) 

Each  of  the  triangles,  thus  formed  in  successive  alternation,  gives  a 
w-region  conformally  represented  on  one  half  or  on  the  other  of  the  2^- plane. 
If,  then,  the  original  triangle  be  combined  with  the  first  triangle  that  is 
conformally  represented  on  the  negative  half-plane,  every  other  similar 
combination  may  be  regarded  as  a  symmetrical  repetition  of  that  initial 
combination :  each  of  them  can  be  conformally  represented*  upon  the  whole 
of  the  ^-plane,  with  appropriate  barriers  along  the  axis  of  x. 

The  number  of  the  triangles  is  infinite,  and  with  each  of  them  a  branch 
of  the  function  w  is  associated:  hence  the  integral  relation  -  between  w 
and  z  which  is  equivalent  to  the  differential  relation  [w,  z]  =  2I  (z),  when 
\  +  /J,  +  v  kI,  is  transcendental  in  w. 

In  the  construction  of  the  successive  triangles,  the  successive  sides  passing 
through  any  point,  such  as  C,  make  the  same  angle  each  with  its  predecessor : 
and  therefore  the  repetition  of  the  operation  will  give  rise  to  a  number  of 
triangles  at  G  eaqh  having  the  same  angle  A-tt. 

If  X,  be  incommensurable,  then  no  finite  number  of  operations  will  lead  to 
the  initial  triangle :  each  operation  gives  a  new  position  for  the  homologous 
side  and  ultimately  the  w-plane  in  this  vicinity  is  covered  an  infinite  number 
of  times,  that  is,  we  can  regard  the  w-surface  as  made  up  of  an  infinite 
number  of  connected  sheets. 

If  A,  be  commensurable,  let  it  be  equal  to  l/l',  where  I  and  I'  are  finite 
integers,  prime  to  each  other.  When  I  is  odd,  21'  triangles  will  fill  up  the 
w-space  immediately  round  C,  and  the  {21'+  l)th  triangle  is  the  same  as  the 

F.  F.  44 


690  SPECIAL  [275. 

first :  but  the  space  has  been  covered  I  times  since  ^I'Xir  =  2^7r,  that  is,  in  the 
vicinity  of  C  we  can  regard  the  w- surface  as  made  up  of  I  connected  sheets. 
When  I  is  even  (and  therefore  V  odd),  V  triangles  will  fill  up  the  space  round 
G  completely,  but  the  (Z'+ l)th  triangle  is  not  the  same  as  the  first:  it  is 
necessary  to  fill  up  the  space  round  G  again,  and  the  (2Z'+l)th  triangle  is 
the  same  as  at  first ;  the  space  has  then  been  covered  I  times,  so  that  again 
the  w-surface  can  be  regarded  as  made  up  of  I  connected  sheets.  The 
simplest  case  is  evidently  that  in  which  X  is  the  reciprocal  of  an  integer,  so 
that  l  =  \\  and  the  w-surface  must  be  regarded  as  single-sheeted. 

Similar  considerations  arise  according  to  the  values  of  /u,  and  of  v. 

If  then  either  A,,  fx,  or  v  be  incommensurable,  the  number  of  t«-sheets  is 
unlimited,  that  is,  ^  as  a  function  of  w  has  an  infinite  number  of  values,  or  the 
equation  between  z  and  w  is  transcendental  in  z.  Hence,  when  X.  +  yu,  +  y  <  1 
and  either  \  or  /j,  or  v  is  incommensurable,  the  integral  relation  between  w  and 
z,  luhich  is  equivalent  to  the  differential  relation  [w,  z]  =21  (z),  is  transcend- 
ental both  in  tv  and  in.z. 

If  all  the  quantities  X,  (x,  v  he  commensurable,  the  simplest  case  of  all 
arises  when  they  are  the  reciprocals  of  integers*.  Then  ^  is  a  uniform 
transcendental  function  of  w,  satisfying  the  equation 

{w,z]  =  2I{z); 

or,  making  z  the  dependent  and  w  the  independent  variable,  we  have  the 
result : — 


A  function  z  that  satisfies  the  equation 

1 

/V   1    "      n^       .  P'      m-     n 


d^z  dz      ^  fd^zy 


dw^  dw      ^  \dw^J 


1      1  1       1  111/ 


fi'-Y 


2      ^2     +h(^2_iy+l-        ^(^_i)        ]\dwj' 


111 

where  I,  m,  n  are  integers,  such  that  -j+~-\--<l,isa  uniform  transcendental 

I      m      n  -' 

function  of  w. 

Restricting  ourselves  to'  this  case,  merely  for  simplicity  of  explanation, 
it  is  easy  to  see  that  the  whole  of  the  space  within  the  orthogonal  circle  is 
divided  up  into  triangles,  with  angles  \ir,  iiir,  vir  bounded  by  circular  arcs 

*  The  cases,  when  X,  /x,  v  are  commensurable  with  one  another  but  are  not,  each  of  them 
the  reciprocals  of  integers,  have  not  yet  been  fully  investigated.  In  an  earlier  edition  of  this 
work  (2nd  edn.,  p.  654),  a  theorem  was  stated  which  by  a  special  example  was  proved  to  be 
inccrect ;  see  a  pamphlet  Ueher  die  Vervielfdltigung  von  Kreishogendreiecken  nach  dem  Syvi- 
metriegesetze,  by  Dr  Paul  Ziihlke  (1903);  the  special  example  (I.e.,  p.  20)  appears  to  be  due  to 
Schwarz.  Reference  may  also  be  made  to  a  paper  by  Hodgkinson,  Proc.  L.  M.  S.,  vol.  xv,  (1916), 
pp.  166—181. 

As  regards  the  whole  matter,  see  a  brief  note  at  the  end  of  this  chapter  (p.  712). 


275.]  CASES  .  691 

which  cut  that  circle  orthogonally :  and,  by  the  inversion  which  connects  the 
space  external  to  the  circle  with  the  internal  space,  the  whole  of  the  outside 
space  is  similarly  divided.     Moreover,  it  has  been  seen  that  every  triangle 

can  be  obtained  from  any  one  by  some  substitution  of  the  form  Wr  =  ^^'^  "^  ^^  . 

CrW  +  d^ 
therefore  the  division  of  the  interior  of  the  circle  into  triangles  is  that 
which  is  considered,  in  the  next  chapter,  for  the  more  general  case  of  division 
into  polygons,  the  orthogonal  circle  of  the  present  case  being  then  the 
'  fundamental '  circle.  The  uniform  transcendental  function  of  w  is  therefore 
automorphic :  the  infinite  group  of  substitutions  is  that  vrhich  serves  to 
transform  a  single  triangle  into  the  infinite  number  of  triangles  within  the 
circle  *. 

One  or  two  special  cases  need  merely  be  mentioned. 

If  any  one  of  the  three  quantities  X,  /x,  v  be  zero  and  if  X  +  fi  +  v  is 
not  equal  to  unity,  the  triangle  can  be  included  under  the  general  case 
just  treated.  For  let  \  =  0,  and  suppose  that  fi  +  v  is  not  greater  than  unity : 
if  /j,  +  v  were  greater  than  unity,  the  triangle  would  be  a  particular  instance 
of  the  class  about  to  be  discussed.  The  division  of  the  area  within  the 
(real)  orthogonal  circle  is  of  the  same  general  character  as  before :  a 
particular  illustration  is  provided  by  the  division  appropriate  to  the 
elliptic  modular- functions,  for  which  /tx  =  J ,  v  =  ^  (§  284).  When  two 
triangles,  one  of  which  is  obtained  from  the  other  by  continuation  in  the 
^^- plane  across  the  axis  of  real  variables,  are  combined,  they  give  a  w-space 
(corresponding  to  the  whole  of  the  2--plane)  for  which  A,  =  0,  /x' =  ^,  v  =  l. 
Since  the  orthogonal  circle  is  real,  it  forms  a  natural  limit  to  these  spaces ; 
when  it  is  transformed  into  the  axis  of  real  variables  in  the  w-plane  by 
a  homographic  substitution,  the  positive  half  of  the  w-plane  is  divided  as 
in  figure  108  (p.  723). 

The  extreme  case  of  the  present  class,  for  which  X  +  /x  +  v  is  less  than 
unity,  is  given  by  A,  =  0,  /x  =  0,  i'  =  0  :  the  triangle  is  then  the  area  between 
three  circles  which  touch  one  another.  Reverting  to  the  differential  equa- 
tion of  the  hypergeometric  series,  we  have  7  =  1,  a  =  /9  =  i;  the  equation  is 

d^Z  ^    l-2z   dZ  i       ^_Q 

dz^      z{\—z)dz      z(l—z) 

which  is  the  differential  equation  of  the  Jacobian  quarter-periods  in  elliptic 
functions  with  modulus  equal  to  z'^.     If 

K=^r   (1  - z  sin' (f>yi dcji,     K' =\^   [I -{I  -  z)sin^  (f)]'^  d(f>, 

.'  0  JO 

*  The  figure  for  the  example  v  =  \,  /J.  =  i,  X  =  i  is  given  by  Schwarz,  Ges.  Werke,  t.  ii,  p.  240; 
and  the  figure  for  the  example  v  =  ^,  ,u.=i,  X  =  7  is  given  in  Klein-Fricke  (p.  370);  both  of  course 
satisfying  the  conditions  'K  +  fi  +  v<l. 

44—2 


692  STEEEOGRAPHIC   PROJECTION  [275. 

K' 

then  w  =  -r^  , 

aK+bK' 
or,  more  generally,  lu  =  —j^ — -yj^, , 

a  relation  between  tu  and  2^  which  gives  the  conformal  representation  of  the 
^6;-triangle  upon  the  ^-half-plane. 

276.  We  now  pass  to  the  consideration  of  the  case  in  which  the  triangle 
with  angles  Xir,  /att,  vtt  has  no  real  orthogonal  circle :  the  other  associated 
triangles  have  therefore  not  a  real  orthogonal  circle.  In  this  case,  the  sum 
of  the  angles  of  the  triangle  is  greater  than  vr,  so  that  we  have 

X  +  /J,  +  V  >  1  from  the  pair  D  and  D', 

—  \  +  /u,  +  v<l  from  the  pair  A  and  A\ 

\  —  fi  +  V  <  1  from  the  pair  B  and  B', 

\  +  fj,  —  V  <  1  from  the  pair  C  and  C", 

as  the  conditions  which  attach  to  the  quantities  X,  /jl,  v.    As  before,  we  invert 

with   respect  to  the  angular  point  X,7r  in   A:    then  the  new  triangle  D  is 

bounded  by  two   straight  lines  and  a  circle,  the 

intersection  of  the  lines  being  in  the  interior  of  the 

circle,  because  the  orthogonal  circle  is  imaginary. 

Let  d  be  distance  of  L  from  the  centre   of  the 

circle,  6  the  angle  OLN,  r  the  radius  of  the  circle : 

then 

d  sin  6  =  —  r  cos  vir,     d  sin  (Xtt  —  6)  =  —  r  cos  /att, 

which  determine  c?  and  ^,    Let  i?^  =  r^  —  dl^  so  that  p-     ^^q 

iR  is  the  radius  of  the  (imaginary)  orthogonal  circle. 

With  L  as  centre  and  radius  equal  to  R  describe  a  sphere :  let  P  be 
the  extremity  of  the  radius  through  L  perpendicular  to  the  plane  Then  P 
can  be  taken  as  the  centre  for  projecting  the  plane  on  the  sphere  stereo- 
graphically*;  so  that,  if  Q  be  a  point  on  the  plane,  Q'  its  projection  on 
the  sphere,  PQ  .  PQ  =  2R\  The  projection  of  LN  is  a  great  circle  through 
P,  the  projection  of  LM  is  another  great  circle  through  P  inclined  at  Xir 
to  the  former :  and  since  PO  is  equal  to  the  radius  of  the  plane  circle,  so 
that  its  diameter  subtends  a  right  angle  at  P,  the  stereographic  projection 
of  that  plane  circle  is  a  great  circle  on  the  sphere,  making  angles  vir  and 
/XTT  with  the  former  great  circles.  There  is  thus,  on  the  sphere,  a  triangle 
bounded  by  arcs  of  great  circles,  that  is,  a  spherical  triangle  in  the  ordinary 
sense,  whose  angles  are  Xtt,  ijltt,  vtt  :  and  this  spherical  triangle  is  conformally 

*  Lachlan,  (I.e.,  p.  688,  note),  p.  43. 


276.]  DIVISION   OF   SPHERICAL  SURFACE  693 

represented  on  the  ^•-half-plane,  its  angular  points  L,  N,  M  finding  their 
homologues  in  z  =  Q,  1,  oo  respectively. 

Just  as  in  the  former  case,  the  successive  passages,  backwards  and 
forwards  across  the  5-axis,  give  in  the  w-plane  new  triangles  with  angles 
Xtt,  yLtTT,  VTT,  all  with  the  same  imaginary  orthogonal  circle  of  radius  iR  and 
centre  L  :  each  of  these,  when  stereographically  projected  on  the  sphere 
with  P  as  the  centre,  becomes  a  spherical  triangle  of  angles  Xtt,  /u-tt,  vtt 
bounded  by  arcs  of  great  circles,  every  triangle  having  one  side  common 
with  its  predecessor :   and  the  triangles  are  equal  in  area. 

Moreover,  the  triangles  thus  obtained  correspond  alternately  to  the 
positive  half  and  the  negative  half  of  the  ^■-plane :  and  it  is  convenient  to 
consider  two  such  contiguous  triangles,  connected  with  the  variable  w,  as 
a  single  combination  for  the  purposes  of  division  of  the  spherical  surface, 
each  combination  corresponding  to  the  whole  of  the  5-plane. 

The  repetition  of  the  analytical  process  leads  to  the  distribution  of  the 
surface  of  the  sphere  into  such  triangles :  and  the  nature  of  the  analytical 
relation  between  w  and  z  depends  on  the  nature  of  this  distribution. 

If  \,  /jb,  or  V  be  incommensurable,  then  the  number  of  triangles  is 
infinite,  so  that  the  relation  is  transcendental  in  w :  and  the  surface  of 
the  sphere  is  covered  an  infinite  number  of  times;  that  is,  corresponding 
to  z  there  is  an  infinite  number  of  sheets,  so  that  the  relation  is  tran- 
scendental in  z.  Thus,  when  X  +  fj,  +  v  is  greater  than  1  and  any  one  of 
the  three  quantities  A,,  /j,,  v  is  incommensurable,  the  integral  relation 
between  lu  and  z,  which  is  equivalent  to 

[^u,z]  =  2l{z), 
is  transcendental  both  in  w  and  in  z. 

If  the  quantities  A,,  fjb,  v  be  commensurable,  the  simplest  possible  cases 
arise  in  connection  with  the  division  of  the  surface  by  the  central  planes 
associated  with  the  inscribed  regular  solids.  These  planes  give  the  divisions 
into  triangles,  which  are  equiangular  with  one  another. 

First,  suppose  that  the  spherical  surface  is  divided  completely  and 
covered  only  once  by  the  two  sets  of  triangles,  corresponding  to  the  upper 
half  and  the  lower  half  of  the  ^•-plane  respectively.  One  of  the  sets,  say 
N  in  number,  will  occupy  one  half  of  the  surface  in  the  aggregate:  and 
similarly  for  the  other  set,  also  N  in  number.     Hence 

R^  {\  +  fjb  +  V  —  1)  IT  ^  the  area  of  a  triangle 

=  ^  (area  of  a  hemisphere;, 

so  that  \ -\-  fji, -{- V  —  1  =  ^ . 


694  SPHERICAL  SURFACE  [276. 

Then,  in  passing  round  an  angular  point,  say  Xtt,  the  triangles  will 
alternately  correspond  to  the  upper  and  the  lower  halves :  hence,  of  the 
whole  angle  27r,  one  half  will  belong  to  one  set  of  triangles  and  the 
other  half  to  the  other  set.     Hence  tt-^Xtt  is  an  integer,  that  is,  \  is  the 

reciprocal  of  an  integer,  say  y .     Similarly  for  /x,  which  must  be  of  the  form 

-  :  and  for  v,  which  must  be  of  the  form  - ;   where  m  and  n  are  integers. 

Thus 

111,2 

J  +  -  +  --  l=lv7- 

i      m      n  JS 

The  only  possible  solutions  of  this  equation  are 

(I.)*     \  =  \,     /J'  =  ^,     ?i  =  any  integer,  N  =  2n 

(II.)     X  =  i,     ,.  =  1,     v  =  l  ,  iY=12 

(IV.)      \  =  i,      ;x=l,      v  =  i  ,  i\^=24 

(VI.)     X  =  i,     /.=  !,     v  =  i  ,  N=60. 

277.  In  each  of  these  cases  there  is  a  finite  number  of  triangles :  with 
each  triangle  a  branch  of  w  is  associated,  so  that  there  is  only  a  finite  number 
of  branches  of  w :  the  sphere  is  covered  only  once,  and  therefore  there  is  only 
a  single  ^^-sheet.  Hence  the  integral  relation  between  w  and  z  is  of  the  first 
degree  in  z :  and  it  is  polynomial  in  w,  of  degrees  2n,  12,  24,  60  respectively. 

The  regular  solids,  with  which  these  sets  of  'triangles  are  respectively 
associated,  are  easily  discerned. 

I.  We  have  A,,  /a,  v  =  ^,  ^,  -.     The  solid  is  a  double  pyramid,  having 

its  summits  at  the  two  poles  of  the  sphere :  the 
common  base  is  an  equatorial  polygon  of  2n  sides : 
the  sides  of  the  various  triangles,  in  the  division  of 
the  sphere,  are  made  by  the  half- meridians  of  longi- 
tude, through  the  angular  points  of  the  polygon  from 
the  respective  poles  to  the  equator,  and  by  arcs  of 
the  equator  subtended  by  the  sides  of  the  polygon. 

II.  We  have  A.,  //,,  v  =  ^,  ^,  ^.  The  solid  is  the 
tetrahedron ;  and  the  division  of  the  surface  of  the 
sphere,  by  the  planes  of  symmetry  of  the  solid,  into 

24  triangles,  12  of  each  set,  is  indicated,  in  fig.  102,  on  the  (visible)  half  of 
the  sphere,  the  other  (invisible)  half  of  the  sphere  being  the  reflexion,  through 
the  plane  of  the  paper,  of  the  visible  half 

*  The  reason  for  the  adoption  of  these  numbers  to  distinguish  the  cases  will  appear  later, 
in  §  279.  , 


277.] 


AND   REGULAR   SOLIDS 


695 


The  angular  summits  of  the  tetrahedron  are  T,  the  middle  points  of  its 
edges  are  S,  the  centres  of  its  faces  are  F:  all 
projected  on  the  surface  of  the  sphere  from  ^ 

the  centre.  If  desired,  the  summits  of  the 
tetrahedron  may  be  taken  at  F :  the  centres 
of  the  faces  are  then  T. 

Each  of  the  angles  at  T  is  ^tt:  each  of 
the  angles  at  ^  is  Jtt:  each  of  the  angles  at 

S  is  ^77. 

The  shaded  triangles  (only  six  of  which 
are  visible,  being  half  of  the  aggregate) 
correspond  to  one  half  of  the  ^-plane ;  and 
the  unshaded  triangles  correspond  to  the 
other  half  of  the  ^-plane. 

IV.  We  have  \,  iju,  v  =^,  \,  \.  The  solid  is  the  cube  or  the  octahedron. 
These  two  solids  can  be  placed  so  as  to  have  the  same  planes  of  symmetry, 
by  making  the  centres  of  the  eight  faces  of  the  octahedron  to  be  the  summits 
of  the  cube.  In  the  figure  (fig.  103),  the  points  0  are  the  summits  of  the 
octahedron  :  the  points  G  are  the  summits  of  the  cube  and  the  centres  of 
the  faces  of  the  octahedron :  and  the  points  >S  are  the  middle  points  of  the 
edges :  all  projected  from  the  centre  of  the  sphere. 

The  shaded  triangles  (the  visible  twelve  being  one  half  of  the  aggregate) 
correspond  to  one  half  of  the  ^-plane ;  the  unshaded  triangles  correspond  to 
the  other  half  of  the  5-plane. 


Each  of  the  angles  at  0  is  ^tt  :  each  of  the  angles  at  C  is  ^tt  :  each  of 
the  angles  at  8  \b  ^ir;  and  it  may  be  noted  that  the  triangles  GOG  are  the 
triangles  in  the  tetrahedral  division  of  the  spherical  surface,  the  point   0 


696  CONSTRUCTION  [2*77. 

in  the  present  triangle  COC  being  the  point  ^  in  a  triangle  STF,  and  the 
two  points  G  being  the  points  F  and  T  in  the  former  figure  (fig.  102). 


0|6 


The  solid  is  the  icosahedron  or  the  dodecahedron.  These  two  solids  can 
be  placed  so  as  to  have  the  same  planes  of  symmetry,  by  making  the  centres 
of  the  twenty  faces  of  the  icosahedron  the  vertices  of  the  dodecahedron.  In 
the  figure  (fig.  104)  the  vertices  of  the  icosahedron  are  the  points  I:  those 
of  the  dodecahedron  are  the  points  D :  and  the  middle  points  of  the  edges 
.  are  the  points  >S^.  The  shaded  triangles  (the  visible  thirty,  six  in  each  lune 
through  the  vertex  of  the  icosahedron,  being  one  half  of  their  aggregate) 
correspond  to  one  half  of  the  ^-plane :  the  unshaded  triangles,  equal  in 
number  and  similarly  distributed,  correspond  to  the  other  half  of  the  ^■-plane. 
The  angles  at  the  vertices  I  of  the  icosahedron  are  ^tt  ;  those  at  the  vertices 
D  of  the  dodecahedron  are  ^v ;  and  those  at  the  middle  points  S  of  the  edges 
(the  same  for  both  solids)  are  -^tt. 

278.  Having  obtained  the  division  of  the  surface,  we  now  proceed  to 
determine  the  functions  which  establish  the  conformal  representation. 

In  all  these  cases,  ^  is  a  rational  function  of  w :  therefore  when  we 
know  the  zeros  and  the  infinities  of  ^  as  a  function  of  w,  each  in  its  proper 
degree,  we  have  the  function  determined  save  as  to  a  constant  factor.  This 
factor  can  be  determined  from  the  value  of  tu  when  z  =  l. 


278.]  OF   TRANSFORMING   RELATIONS  697 

The  variable  tu  belongs  to  the  stereographic  projection  of  the  point  of 
the  spherical  surface  on  the  equatorial  plane,  the  south  pole  being  the  pole 
of  projection.  If  X,  Y,  Z  be  the  coordinates  of  the  point  on  the  spherical 
surface,  the  radius  being  unity,  then 

X  +  iY 

^  =  TTX- 

For  a  point  in  longitude  I  and    latitude  ^tt  —  8,  we   have  X  =  cos  ^  sin  S, 
Y  =  sin 7  sin  8,  Z  =  cos  8 :  so  that,  if  preferable,  another  form  for  w  is 

IV  =  e»'^  tan  ^8. 

In  our  preceding  investigation,  the  angle  at  Xir  was  made  to  correspond 
with  z  =  0,  that  at  vir  with  2=1,  that  at  /j^tt  with  z=  cc . 

Case  I.     We  take  X,=  -,  ix  =  ^,  v  =  I. 

^77"     47r 

For  the  angular  points  /i7r  we  have  8  =  \tt\  1  =  0,  —  ,  — ,  . . . ,  each  pomt 

belonging  to  two  triangles  of  the  same  set,  that  is,  triangles  represented  on 
the  same  half  of  the  plane  :  thus  the  various  ty-points  in  the  plane  are 

2712    ^ 

for  r  =  0,  1, ..,,«  —  1,  each  occurring  twice.     Hence  z  =  cc  ,  when  the  function 

w— 1  '^ 

n  {w-e""  ''y 

r=0 

vanishes,  that  is,  s  =  oo  ,  when  (w"  —  1)'-*  vanishes. 

For  the  angular  points  vir,  we  have  S  =  -^7r;    1=—,  — ,  —  ,  ...,   each 

point  belonging  to  two  triangles  of  the  same  set :  thus  the  various  w-points 
in  the  plane  are 

^\2r  +  l) 

for  r  =  0,  1,  ...,  n  —  1,  each  occurring  twice.     Hence  z  =  l,  when  the  function 

H   [w-e""  Y 

vanishes,  that  is,  z=l,  when  {w''^  +  1)^  vanishes. 

Now  ^  is  a  uniform  function  of  lu :  hence  v^e  can  take 

where    ^   is    a  constant,    easily    seen    to    be    unity :    because,    when    u>  =  0 
(corresponding  to    the    common  vertex   Xtt  at    the   North   pole)  and   when 


698  TETRAHEDRAL   FUNCTION  [278. 

w  =  00  (corresponding  to  the  common  vertex  Xtt  at  the  South  pole),  z  vanishes, 
as  required.     The  relation  is  often  expressed  in  the  equivalent  form 

z  -.2-1  -.1  =  -  4>w''  :  -  (w"  +  1)^  :  (w"  -  l)^ 

which  gives  the  conformation  on  the  half  ^^-plane  of  a  w-triangle  bounded  by 

circular  arcs,   the  angles  being  -,  ^tt,  ^tt.     The  simplest  case  is  that  in 

77" 

which  the  triangle  is  a  sector  of  a  circle  with  an  angle  -  at  the  centre. 
The  preceding  relation  is  a  solution  of  the  equation 

I1--  --1I 


(z-iyz(z-i)]- 

If  we  choose  X  =  ^,  /^  =  ^,  v  =  -;  so  that  z  =  0  when  (w"  +  l)^  vanishes, 

2:  =  00  when  {w'^—Vf  vanishes,  and   z=l   when  w"^  vanishes,  the  relation 
establishing  the  conformal  representation  is 

z  :  z-1  :  1  =  (w'»  +  1)^  :  4^^*  :  {w-^  -  If. 

This  relation  is  a  solution  of  the  equation 


[lU, 


+  T TT„  +  


z-         (z  —  If      z  {z  —  l)j 


Case  II.  We  take  A,  =  I ;  so  that  z  =  0  must  give  the  points  S,  each  of 
them  twice,  since  there  are  two  triangles  of  the  same  set  at  ;S :  yu-  =  ^  (and 
these  are  taken  at  T),  so  that  z  =  co  must  give  the  points  T,  each  of  them 
thrice :  and  v  =  ^  (and  these  are  taken  at  F),  so  that  ^  =  1  must  give  the 
points  F,  each  of  them  thrice. 

Taking  the  plane  of  the  paper  as  the  meridian  from  which  longitudes  are 
measured,  the  coordinates  of  the  four  w-points  in  the  plane,  corresponding  to 
T  by  stereographic  projection,  are 

V2  _V2  .x/2  _  .V2 

V3  V3  'V3  \/3 


V3  V3  '^V3  "^V3 

say  w^,  W2,  Ws,  Wi.     Then  z=oo  gives  each  of  these  points  thrice:  that  is, 
z=  <x> ,  when  {(w  —  Wi)  ...  (w  —  iUi)Y  vanishes,  or  2^  =  00  ,  when 

(iv'  -  2w-  V3  -  1)=^ 
vanishes. 


278.]  OCTAHEDRAL   FUNCTION  699 

The  coordinates  of  the  four  points  corresponding  to  F,  are 


V2 

V2           .  V2              .  \/2 
\/3             ^3               V3 

1^73' 

1  '             1  '             1 

^V3             V3             V3 

Hence  z—1,  when 

(w^  +  2W^  \/3  -  1)' 

vanishes. 

(2j.+i)-:if 

The  coordinates  of  the  six  points  corresponding  to  S  are  0,  e  ^  (for 
?'  =  0,  1,  2,  3)  and  X) :  hence  z  =  Q,  when 

vanishes. 

Moreover,  2;  is  a  uniform  function  of  w :  and  therefore 

_  (w*  +  2w^  V3  -^  1)=^ 
^~(w^-2tt;V3-l)'' 

the  constant  multiplier  on  the  right-hand  side  being  determined  as  unity  by 
the  relation  between  the  points  S  and  the  value  z  =  0. 

The  relation  is  often  expressed  in  the  equivalent  form 

z  :z-\  :  l  =  12V3w2(w*+l)'  :  K  +  2wW3-l)'  :  -{w'-2iu- ^J^-l)\ 

It  gives  the  conformation  on  the  2;-half-plane  of  a  triangle  in  the  w-plane, 
bounded  by  circular  arcs,  the  angles  of  the  triangle 
being  ^-tt,  Jtt,  ^tt. 

The  simplest  case  is  that  of  a  portion  cut  out  ,,--' 

of  a  sector  of  a  circle  of  central  angle  30°,  by  the  ,--'' 

arc  and  two  lines  at  right  angles  to  one  another  ^  

symmetrical  with  respect  to  the  arc.  ^' 

It  has  been  assumed  that  the  plane  of  the  paper  is  the  meridian. 
Another  convenient  meridian  to  take  is  one  which  passes  through  a  point 
8  on  the  equator :  in  that  case,  the  preceding  analysis  applies  if  a  rotation 
through  an  angle  ^tt  be  made.  The  effect  of  this  rotation  is  to  give  the  new 
variable  W  for  any  point  in  the  form 

in 

so  that  'uf  =  —  iW^.     The  relation  then  takes  the  form 
'z  :z-l  :  1 

=  12  V^  W\W'-iy:(W'  +  2W"^^+iy  :  -(W'-2W'^'^  +  iy; 
but  there  is  no  essential  difference  between  the  two  relations. 


700  OCTAHEDRAL   FUNCTION  [278. 

The  lines  by  which  the  w-plane  is  divided  into  triangles,  each  conformally 
represented  on  one  or  other  half  of  the  ^-plane,  are  determined  by  z  =  Zo, 
that  is,  by 

The  figure  is  the  stereographic  projection  of  the  division  of  the  sphere,  and 
it  can  be  obtained  as  in  §  257  (Ex.  20,  Ex.  23). 

Case  IV.  We  take  X  =  i,  so  that  z=0  must  give  the  eight  points  C : 
each  is  given  three  times,  because  at  C  there  are  three  triangles  of  the  same 
set :  we  take  v  =  i,  so  that  z  =  l  must  give  the  six  points  0,  each  four  times : 
and  /i  =  I,  so  that  z  =  cc  must  give  the  twelve  points  S,  each  of  them  twice. 

We  take  the  plane  of  the  paper  as  the  meridian.  The  points  0  are  0,  1, 
i,  —  1,  —  ^,  00  ;  each  four  times.     Hence  z=l,  when  the  function 

vanishes. 

+  1  +  i 
The  points  C  are  the  eight  points     ~     ~      :  the  product  of  the  eight 

i  V  "  ~  1 
corresponding  factors  is 

w^  +  14^4  +  1 : 

and  each  occurs  thrice,  so  that  z  =  0,  when  the  function 

vanishes. 

The  points  S  are  (i)  the  four  points  — =^ — =-  in  the  plane  of  the  paper, 
giving  a  corresponding  product 

w*  -  Qw^  +  1 : 

+  i         . 
(ii)  the  four  points        ~  — -  in  the  meridian  plane,  perpendicular  to  the 

plane  of  the  paper,  giving  a  corresponding  product 

lu^  +  Quj-  +  1:      ■ 

and  (iii)  the  four  points  e*  ,  (for  r  =  Q,  1,  2,  3),  in  the  equator,  giving  a 

corresponding  product 

w*+  1. 

Each  of  these  points  occurs  twice :  and  therefore  .2  =  00,  when  the  function 
that  is,  when  the  function 

(lu'-' -  ssw' -  ssw' +  ly- 

vanishes. 


278.] 

Hence 


z  = 


OCTAHEDRAL   FUNCTION 
(w^  +  14w''  +1)=' 


701 


(wi2-33w«-33w^+l)2' 


the  constant  multiplier  being  determined  as  unity,  by  taking  account  of  the 
value  unity  for  z  :  and 

_  _  108w^  {w^  -  iV 

(w^^  -  33 w«  -  ;i3tt;*  +  1)'-'  ■ 

The  relation  can  be  expressed  in  the  equivalent  form 

z:z-\:\  =  {'ufi-\-  14w*  +  1)^  :  lOSit;^  {w^  -  Vf  :  (w^^  -  33w»  -  33w*  +  Yf. 

It  gives  the  conformation  on  half  of  the  ^-plane  of  a  w-triangle  bounded  by 
circular  arcs  and  having  its  angles  equal  to  \'k,  \ir,  \'k  respectively. 

The  lines,  by  which  the  ?<;-plane  is  divided  into  the  triangles,  are  given 
by  z  =  z^,  that  is,  by 


w*{w*-  ly 


Wo'  {wo*  -ly 


Fig.  106. 

The  division  is  indicated  in  fig.  106,  being  the  stereographic  projection  of 
the  divided  spherical  surface  of  fig.  103,  with  respect  to  the  south  pole, 
taken  to  be  diametrically  opposite  to  the  central  point  0. 

Case   VI.     We  take  \  =  |^,  so  that  z  =  0  must  give  the  twenty  points  D, 
each  of  them  thrice;  v  =  ^,  so  that  z  =  l  must  give  the  twelve  points  I,  each 


702  ICOSAHEDRAL  [278. 

of  them  five  times ;  and  yu.  =  i ,  so  that  ^  =  oo  must  give  the  thirty  points  S, 
each  of  them  twice. 

Let  an  edge  of  the  icosahedron  subtend  an  angle  6  at  the  centre  of  the 

sphere:   then  its  length  is  2rsm^0.     Also,  five  edges  are  the  sides  of  a 

pentagon  inscribed  in  a  small  circle,  distant  d  from  a  summit :  hence  the 

radius  of  this  circle  is  r  sin  6  and  the  length  of  the  edge  is  2rsin  ^siniTr, 

so  that 

2  sin  ^6  =  2  sin  0  sin  ^tt, 

whence  tani6' =  i  (\/5  -  1),     cot  i^  =  i  (V5  +  1). 

2771 

Let  a  denote  e'^^.  Then  the  value  of  w  corresponding  to  the  north  pole  / 
is  0  ;  the  values  of  tu  for  the  projections  on  the  equatorial  plane  of  the  five 
points  /  nearest  the  north  pole  are 

tan  ^6,     aHan^^,     a"  tan  ^^,     aHan^^,     ocHan^^: 

the  values  of  w  for  the  projections  on  the  equatorial  plane  of  the  five  points  / 
nearest  the  south  pole  are 

a  cot  ^  6,     a^  cot  h  6,     o?  cot  \  d,     (x!  cot  \  6,     o?  cot  i  ^ : 

and  for  projection  of  the  south  pole  the  value  of  w  is  infinity.     The  product 
of  the  corresponding  factors  is 

4  4 

w  .   n  (w-a2'-tani6')  .   11  (w  -  a^'^+i  cot  i  ^)  .  1 

=  w{w^-  t&iv'  ^d)  (w'  +  cot^  -|-  6) 

=  w  {v:^°  +  Uw' -  I), 

after  substitution.     Each  point  /  occurs  five  times  ;  and  therefore  z  =  l,  when 
the  function 

vanishes. 

The  points  D  lie  by  fives  on  four  small  circles  with  the  diameter  through 
the  north  pole  and  the  south  pole  for  axis.  The  polar  distance  of  the  small 
circle  nearest  the  north  pole  is  tan  S  =  8  —  V5,  and  of  the  circle  next  to  it  is 
tan  8'  =  3  +  \/5,  so  that 


V1.5-6V5-1                      ,     V15  +  6V5-1 
tan  .^6  = rp ,         tan  16  = ^-- — . 

The  function  corresponding  to  the  projections  of  the  five  points  nearest  the 

north  pole  is 

w^  +  tan'  ^  S, 

and  to  the  projections  of  the  five  nearest  the  south  pole  is 

w'^  —  cot^  ^S; 


278.] 


FUNCTION 


703 


while,  for  the  projections  of  the  other  two  sets  of  five,  the  products  are 

w^  +  tan^iS' 

and  w'-cot'^8' 

respectively.     Each  occurs  thrice.     Hence  z  =  0,  when  the  function 

{{w'  +  tan^S)  (w5  _  cot^  |g)  (^s  +  tan^S')  (W  -  cot^  ^S')Y, 

vanishes,    that   is,    when   (w^-' -  228w'' +  4^Mtv^o  ^  22Siv' +  ly,   which   is   the 
reduced  form  of  the  preceding  product,  vanishes. 


Fig.  107 


The  points  S  lie  by  tens  on  the  equator,  by  fives  on  four  small  circles 
having  the  polar  axis  for  their  axis.  Proceeding  in  the  same  way  with  the 
products  for  their  projections,  it  is  found  that  ^=  oo  ,  when  the  function 

[w^'  +  1  +  522^5  (,^^20  _  1)  _  lOOOotw^o  (w'°  +  1)Y 
vanishes. 


704  THE    FIFTEEN   RATIONAL  [278. 

(w^o  -  228^15  ^  494^1°  +  228w'  +  1  )^ 
Hence        z  =  |,^3o  +  i  +  522m;5  (w^"  -  1)  -  10005w>»  (w"  +  1)1' ' 

the   constant   factor  being  found  to  be   unity,  through  the  value   of  1  —  ^ 

which  IS     l-z  =  1^30  ^_  1  +  522^-5  (lu^-o  _  1)  _  10005m;"  (w^"  +  1)]^ ' 

These  relations  give  the  conformal  representation  on  half  of  the  ^-plane  of  a 
wrtriangle,  bounded  by  circular  arcs  and  having  angles  Itt,  ^tt,  ^tt. 

The  lines,  by  which  the  w-plane  is  divided  into  the  triangles,  are  given 
by  z  =  2o,  that  is,  by 

(w-^«  -  228^15  +  494m;1o  +  228w^  +  1)^  _  (wp^"  -  228<^  +  494wo^«  +  228wo^  +  1)^ 

w' {w'' +  iiiu' -  ly  ~  Wo'{wo''  +  uwo'-iy 

The  division  is  indicated  in  figure  107,  which  is  the  stereographic  projection*  of 
the  divided  spherical  surface  of  figure  104,  with  I^^  as  the  pole  of  projection. 

279.  The  preceding  are  all  the  cases,  in  which  simultaneously  ^  is  a 
uniform  function  of  w,  and  w  is  an  algebraical  function  of  z  :  they  arise  when 
the  surface  of  the  sphere  has  been  completely  covered  once  with  the  two  sets 
of  triangles  corresponding  to  the  upper  half  and  the  lower  half  of  the  ^-plane. 

But  an  inspection  of  the  figures  at  once  shews  that  they  are  not  the 
only  cases  to  be  considered,  if  the  surface  of  the  sphere  may  be  covered 
more  than  once. 

In  the  configuration  arising  through  the  double-pyramid,  the  surface  of 
the  sphere  will  be  covered  completely  and  exactly  m  times,  if  the  angles  at 
the  poles  be  Imirjn,  where  m  is  prime  to  n.  The  corresponding  relation 
between  w  and  z  is  obtained  from  the  simpler  form  by  changing  n  into  nj.m. 

In  the  tetrahedral  configuration  (fig.  ]02)  the  surface  of  the  sphere  will 
be  exactly  and  completely  covered  twice  by  triangles  FFT  (or  by  triangles 
TTF,  it  being  evident  that  these  give  substantially  the  same  division  of  the 
surface).  The  relation  between  w  and  z  will  then  be  of  the  same  degree, 
12,  as  before  in  w,  for  the  number  of  different  triangles  in  the  two  ^y-sheets 
is  still  twelve  of  each  kind :  because  there  are  two  w-sheets  corresponding 
to  the  single  ^^-plane,  that  relation  will  be  of  the  second  degree  in  z.  The 
values  of  the  angles  are  determined  by 

(III.)    \,^.,v  =  l,\,\. 

*  lu  regard  to  all  the  configurations  thus  obtained  as  stereographic  projections  of  a  spherical 
surface,  divided  by  the  planes  of  sjmmetiy  of  a  regular  solid,  Mobius's  "  Theorie  der  symme- 
trischen  Figuren"  [Ges.  Werke,  t.  ii,  especially  pp.  642 — 699),  may  be  consulted  with  advantage  ; 
and  Klein-Fricke,  Elliptische  Modulfunctionen,  vol.  i,  pp.  102 — 106. 


279.]  TRANSFORMING    FUNCTIONS  705 

Again,  in  the  octahedral  configuration,  the  surface  of  the  sphere  will 
be  exactly  and  completely  covered  twice  by  triangles  OCO!  The  relation 
between  w  and  z  will  be  of  degree  24  in  tu  and  degree  2  in  ^ :  and  the  values 
of  the  angles  are  determined  by 

(V.)     X,f.,v=ll,l 

Similarly,  a  number  of  cases  are  obtainable  from  the  icosahedral  configu- 
ration, in  the  following  forms : 

(VII.)     \  /Jb,  V  =  ^,  i,  J  with  triangles  such  as  IiD^D.^; 

(VIII.)    \,f.,v  =  hh^ A/J.; 

(IX.)     \,x,v  =  ^,l^ SJJ,; 

(X.)  \f.,v  =  l,hi DJJ,; 

(XL)  x,/.,  z.  =  |,  f,| IJJ,; 

(XII)  \  /z,  z.  =  f ,  1,  4 /lAAs; 

(XIII.)  \,f,,r;  =  A,i,±  IJJ,,; 

(XIV.)    \,fi,v  =  ^,ii 7AA; 

(XV.)    \, /.,  j.  =  f,i,  1  IJ,D,. 

Other  cases  appear  to  arise  :  but  they  can  be  included  in  the  foregoing, 
by  taking  that  supplemental  triangle  which  has  the  smallest  area.  Thus, 
apparently,  I^DJ^f^  would  be  a  suitable  triangle,  with  ^,  /"-,  ?^  =  f ,  f ,  ^:  it  is 
replaced  by  IioD-zo^iot  an  example  of  case  (X.)  above. 

These,  with  the  preceding  cases  numbered*  (I.),  (II.),  (IV.),  (VI.),  form 
the  complete  set  of  distinct  ways  of  appropriate  division  of  the  surface 
of  the  sphere. 

It  is  not  proposed  to  consider  these  cases  here :  full  discussion  will  be 
found  in  the  references  already  given.  The  nature,  however,  of  the  relation, 
which  is  always  of  the  form 

f(z)  =  F(w), 

where  /  and  F  are  rational  functions,  may  be  obtained  for  any  particular 
case  without  difficulty.     Thus,  for  (III.),  we  have 


[w,  z\  = 
when 


_  1 

2 


"1—1  1_1  l_l-4-i  —  1' 

z^     '^  {1-zf  z{z-\)     ]' 


z■.l-z■.l  =  -li^/^wHw'^-lf:  (w*  +  2w;V3- 1)'  :  (w^  -  2t^V3  -  1)^ 

Again,  if 

z:  1  -  z  :  1  =  (Z  +  iy  :  -  4^Z  :  {Z  -ly, 

*  These  numbers  are  the  numbers  originally  assigned  by  Schwarz,  Ges.  Werke,  t.  ii,  p.  246, 
and  used  by  Cayley,  Camb.  Phil.  Trans.,  vol.  xiii,  pp.  14,  15. 

F     F.  45 


706  POLYHEDRAL   FUNCTIONS  [279. 

a  special  case  of  §  278,  I.,  by  taking  n  =  1,  then 

'dzY 


Hence 


iQ(z  +  iy 


(i-i)^.i(i-i)" 


_  1 

2 


(Z-iy   iii-^y    z{z-i)_ 

"1—1  1—4^  l  —  i-f-l— 1 

9     I        ■'■         9         I     9         9^9         -* 


Z'  ^{Z-iy^  z{Z-i)  J' 

so  that  X.  =  -3-,  v=^,  yU'  =  j.     Hence  the  relation 

{Z+iy:-4>Z:{Z-iy 

=  -  12  V3  w^  (w*  +  1)^-  :  {w'  +  2tv'~  ^S  -iy:(w'-  Iw"  V3  -  1)' 

gives  the  conformation  of  triangles  bounded  by  circular  arcs  and  having 
angles  Jtt,  ivr,  f  tt. 

The  foregoing  are  the  only  cases,  for  A,  +  yu,  +  v  >  1,  in  which  the  integral 
relation  between  w  and  z  is  rational  both  in  w  and  in  z. 

In  all  other  cases  in  which  X,  /x,  v  are  commensurable,  this  integral 
relation  is  rational  in  z  and  transcendental  in  iv. 

It  is  to  be  noticed,  in  anticipation  of  Chapter  XXII.,  that,  since  every 
triangle  in  any  of  the  divisions  of  the  spherical  surface,  or  of  the  plane, 
can  be  transformed  into  another  triangle,  the  functions  which  occur  in 
these  integral  relations  are  functions  characterised  by  a  group  of  substi- 
tutions. When  the  functions  are  rational,  the  groups  are  finite,  and 
the  functions  are  then  the  polyhedral  functions :  when  the  functions  are 
transcendental,  the  groups  are  infinite,  and  the  functions  are  then  of  the 
general  automorphic  type. 

The  case  in  which  \+  /j,+  v  =  1  has  already  been  considered  :  the  spherical 
representation  is  no  longer  effective,  for  the  radius  of  the  sphere  becomes 
infinite  and  the  triangle  is  a  plane  rectilinear  triangle.  The  equation  may 
still  be  used  in  the  form 

{w,z}  =  2I(z), 

with  the  condition  \-\-  ibi  +  v  =  l.  A  special  solution  of  the  equation  is  then 
given  by 

-r-  =  z^-'  (1  -  zY-\ 
dz 

leading  to  the  result  of  §  268,  the  homologue  of  the  angular  point  fjuir  being 

at  ^  =  00  . 

280.  It  is  often  possible  by  the  preceding  methods  to  obtain  a  relation 
between  complex  variables  that  will  represent  a  given  curve  in  one  plane  on 


280.]  ALGEBRAIC   ISOTHERMAL   CURVES  707 

an  assigned  curve  in  the  other :  there  is  no  indication  of  the  character  of  the 
relation  for  an  arbitrary  curve  or  a  family  of  curves.  But  in  one  case,  at  any 
rate,  it  is  possible  to  give  an  indication  of  the  limitations  on  the  functional 
form  of  the  relation. 

Let  there  be  a  family  of  plane  algebraical  curves,  determined  as  potential 
curves  by  a  variable  parameter*:  and  let  their  equation  be 

F  {x,  y,  u)  =  0, 

where  u  is  the  variable  parameter,  which,  when  it  is  expressed  in  terms  of  x 
and  y  by  means  of  the  equation,  satisfies  the  potential-equation 

Since  u  is  a  potential,  it  is  the  real  part  of  a  function  w  oi  x-\-  iy :  and  the 
lines  u  =  constant  are  parallel  straight  lines  in  the  w-plane.  It  therefore 
appears  that  the  functional  relation  between  w  and  z  must  represent  the 
w-plane  conformably  on  the  ^-plane,  so  that  the  series  of  parallel  lines  in  the 
one  plane  is  represented  by  a  family  of  algebraical  curves  in  the  other :  let 
the  relation,  which  effects  this  transformation,  be 

X  {z,  w)  =  0. 

Let  the  algebraical  curve,  which  corresponds  to  some  particular  value  of  u, 

say  u  =  0,  be 

F{x,y,0)=f{x,y)^0, 

which  in  general  is  not  a  straight  line.  Let  a  new  complex  ^  be  determined 
by  the  equation 


f[l 


=  0 


this  equation  is  algebraical,  and  therefore  ^  can  be  regarded  as  a  function  of 
lu,  say  t/t  (w),  between  which  and  z,  regarded  as  a  function  of  w,  say  </>  {w), 
there  is  an  algebraical  equation. 

Now  when  u  —  0,z  describes  the  curve 

f{x,y)=0: 

hence  at  least  one  branch  of  the  function  ^,  defined  by 

*  Such  curves  are  often  called  isothermal,  after  Lame.  The  discussion  of  the  possible  func- 
tional relations,  that  lead  to  algebraical  isothermal  curves,  is  due  to  Schwarz,  Ges.  Werke,  t.  ii, 
pp.  260 — 268:  see  also  Hans  Meyer,  "  Ueber  die  von  geraden  Linien  und  von  Kegelschnitten 
gebildeten  Schaaren  von  Isothermen  ;  so  wis  iiber  einige  von  s-peciellen  Curven  dritter  Ordnung 
gebildete  Schaaren  von  Isothermen,"  (a  Gottingen  dissertation,  Ziivich,  Zlircher  and  Furrer, 
1879);  Cayley,  Quart.  Journ.  Math.,  vol.  xxv,  (1891),  pp.  203—226,  Coll.  Math.  Papers,  vol.  xiii, 
pp.  170 — 191 ;  and  the  memoir  by  Von  der  Miihll,  cited  p.  611. 

45—2 


708  FAMILIES  OF  [280. 

can  be  taken  as  equal  to  x  when  u  =  0,  that  is,  there  is  one  branch  of  the 
function  ^  which  is  purely  real  when  w  is  purely  imaginary. 

The  curves  in  the  ^-plane  are  algebraical :  when  this  plane  is  conformally 
represented  on  the  ^-plane  by  the  foregoing  branch,  which  is  an  algebraical 
function  of  z,  the  new  curves  in  the  ^-plane  are  algebraical  curves,  also 
determined  as  potential  curves  by  the  variable  parameter  u.  And  the  ^-curve 
corresponding  to  i<  =  0  is  (the  whole  or  a  part  of)  the  axis  of  real  quantities. 
In  order  that  the  conformal  representation  may  be  effected  by  the  functions, 
they  must  allow  of  continuous  variation :  hence  lines  on  opposite  sides  of 
u  =  0  correspond  to  lines  on  opposite  sides  of  the  axis  of  real  quantities.  The 
functional  relation  between  ^=  |  +  ^»7  and  w  =  u-\-  iv  is  therefore  such  that 

^  +  ir}  =  '^jr  (u  +  iv), 

^  —  irj^yjr  (—  u  +  iv). 

The  equation  of  the  ^-curves,  which  are  obtained  from  varying  values  of 
u,  is  algebraical :  and  therefore,  when  we  substitute  in  it  for  f  and  77  their 
values  in  terms  of  y^  {u  +  iv)  and  i/r  (—  w  +  iv),  we  obtain  an  algebraical 
equation  between  -^  {u  +  iv)  and  i/r  (—  t^  +  iv),  the  coefficients  of  which  are 
functions  of  u  though  not  necessarily  rational  functions  of  lo.  Let  6  =  —  2u; 
and  let  yjrz,  •\/^3  denote  -^{w),  ylr(w  +  d)  respectively;  then  the  equation  can 
be  represented  in  the  form 

g(ir„ir.,0)  =  0, 

rational  in  1^2  and  yps,  but  not  necessarily  rational  in  6. 

Because  the  functions  allow  continuous  variation,  we  can  expand  -1/^3  in 
powers  of  6 :  hence 

,(^.,^.,.^^,,^*^, ^)=o. 

When  this  equation,  which  is  satisfied  for  all  values  of  w  and  of  6,  where 
w  and  6  are  independent  of  one  another,  is  arranged  in  powers  of  6,  the 
coefficients  of  the  various  powers  of  6  must  vanish  separately.  The  coefficient 
independent  of  6,  when  equated  to  zero,  can  only  lead  to  an  identity,  for  it 
will  obviously  involve  only  yfr^ :  any  non-evanescent  equation  would  determine 
i/to  as  a  constant.  Similarly,  the  coefficient  of  every  power  of  0,  which 
involves  none  of  the  derivatives  of  yjr^,  must  vanish  identically.  The  co- 
efficient of  the  low^est  power  of  6,  which  does  not  vanish  identically,  involves 

"^2.  -y— "   and    constants:    but,   because   the    equation    g  (^jr^,  ■^s,  6)  =  0   is 

rational  in  i/tj,  the  second  and  higher  derivatives  of  yfr^,  associated  with  the 
second  and  higher  powers  of  6  in  the  expansion  of  -v/tj,  cannot  enter  into  the 
coefficient  of  this  power  of  6.     Hence  we  have 


«Vh'>' 


280.]  ALGEBRAIC   ISOTHERMAL   CURVES  709 

an  algebraical  equation  between  o/to  and  -~  ,  the  coefficients  of  which  are 
constants. 

The  coefficient  of  the  next  power  of  6  will  involve  —^ ,  and  so  on  for  the 

powers  in  succession.  Instead  of  using  the  equations,  obtained  by  making 
these    coefficients    vanish,   to  deduce   an    algebraical    equation   between   -v^a 

and  any  one  of  its  derivatives,  we  use  A  =  0.     Thus  for  ~~^ ,  the  equation 

would  be  obtained  by  eliminating  -v/r/  between  the  (algebraical)  equations 

and  so  for  others. 

Returning  now  to  the  equation 

in  which,  as  it  is  rational  in  -v/r,  and  yfr^,  only  a  limited  number  of  co- 
efficients, say  k,  are  functions  of  6,  we  can  remove  these  coefficients  as 
follows.  Let  k  —  1  differentiations  with  regard  to  w  be  effected  :  the  resulting 
equations,  with  g  =  0,  are  sufficient  to  determine  these  k  coefficients  ration- 
ally in  terms  of  -v/^a,  yfra  and  their  derivatives.  But  the  coefficients  are 
functions  of  6  only  and  do  not  depend  upon  w :  hence  the  values  obtained  for 
them  must  be  the  same  whatever  value  be  assigned  to  w.  Let,  then,  a  zero 
value  be  assigned :  yjr^  and  its  derivatives  become  constants ;  yjrs  becomes 
ylr{6),  say  -v/rj,  and  all  its  derivatives  become  derivatives  of  i/tj;  so  that  the 
coefficients  can  be  rationally  expressed  in  terms  of  -x/^j  and  its  derivatives. 
When  these  values  are  substituted  in  g  —  0,  it  takes  the  form 

9i (^2,  i^s,^!,  fi,  -^i",  ■■■)  =  0, 

rational  in  each  of  the  quantities  involved.  But  between  yjr^  and  each 
of  its  derivatives  there  subsists  an  algebraical  equation  with  constant  co- 
efficients :  by  means  of  these  equations,  all  the  derivatives  of  yjri  can  be 
eliminated  from  g^  =  0,  and  the  final  form  is  then  an  algebraical  equation 

involving  only  constant  coefficients.     But 

^fr,=^^lr  (0),      ^|r,=  ^Jr  (w),      f,  =  ir(w  +  d); 
and  therefore  the  function  ^|r  (w)  possesses  an  algebraical  addition-theorem. 
Now  yjr  (w)  and  4>  {w)  are  connected  by  the  algebraical  equation 

therefore  ^{w)  possesses  an  algebraical  addition -theorem.     But,  by  §  151, 


710  FAMILIES   OF  [280. 

when  a  function  </)(w)  possesses  an  algebraical  addition-theorem,  it  is  an 
algebraical  function  either  of  w,  or  of  e'*^,  or  of  an  elliptic  function  of  iv,  the 
various  constants  that  arise  being  properly  chosen:  and  hence  the  only 
equations 

which  can  give  families  of  algebraical  curves  in  the  z-plane  as  the  con  formal 
equivalent  of  the  parallel  lines,  u  =  constant,  in  the  tv-plane,  are  such  that  z 
is  connected  by  an  algebraical  equation  either  with  w,  or  with  a  simply-periodic 
function  of  w,  or  with  a  doubly -periodic  function  of  tu. 

There  are  three  sets  of  fundamental  systems,  as  Schwarz  calls  them,  of 
algebraical  curves  determined  as  potential  curves  by  a  variable  parameter : 
they  are  curves  such  that  all  the  others  can  be  derived  from  them  solely  by 
algebraical  functions. 

The  first  set  is  fundamental  for  the  case  when  z  is  an  algebraical  function 

of  tu  :  it  is  given  by 

u  =  constant, 

being  a  series  of  parallel  straight  lines. 

,  The  second  set  is  fundamental  for  the  case  when  z  is  an  algebraical 
function  of  e'*^" ;  if  W  denote  e'^*",  then  z  is  an  algebraical  function  of  W,  and 
all  the  associated  curves  in  the  ^-plane  are  confoi-mal  representations  of  the 
algebraical  curves  in  the  If -plane.  If  /a  =  a  4-  /3i,  where  a  and  0  are  real, 
then 

(a'  +  ^^^)u=^U  log  (X'  +  F^)  +  /3  tan-i  ~  , 

* 

a  relation  which  can  lead  to  algebraical  curves  in  the  TF-plane  only  if  a  or 
/3  be  zero.  If  a  be  zero,  then  /a  is  a  pure  imaginary,  and  the  TT-curves  are 
straight  lines,  concurrent  in  the  origin :  if  /3  be  zero,  then  yu,  is  real,  and  the 
TT-curves  are  circles  with  the  origin  for  a  common  centre.  Hence  the  set 
of  fundamental  systems  for  the  case,  when  z  is  an  algebraical  function  of  e'^'^, 
consists  of  an  infinite  series  of  concurrent  straight  lines  and  an  infinite  series 
of  concentric  circles,  having  for  their  common  centre  the  point  of  concurrence 
of  the  straight  lines. 

The  third  set  is  fundamental  for  the  case  when  z  is  an  algebraical  function 
of  a  doubly-periodic  function,  say,  of  sn  fjiw. 

Ex.  Prove  that  either  the  modulus  k  is  real  or  an  algebraical  transformation  of 
argument  to  another  elliptic  function  having  a  real  modulus  is  possible  :  and  shew  that 
the  set  of  fundamental  curves  are  quartics,  which  are  the  stereographic  projection  of 
confocal  sphero-conics.  (Schwarz,  Siebeck,  Caylej.) 

We  thus  infer  that  all  families  of  algebraical  curves,  determined  as 
potential  curves  by  a  variable  'parameter,  are  conformal  representations  of 


280.]  ALGEBRAIC   ISOTHERMAL   CURVES  fll 

one  or  other  of  these  sets   of  fundamental  systems,  hy  equations  which  are 
algebraical. 

But  though  it  is  thus  proved  that  the  relation  between  z  and  w  must 

express  z  as  an  algebraical  function  either  of  w,  or  of  e'^'^'^,  or  of  sn  ixw,  in 

order  that  a  family  of  algebraical  curves  may  be  the  conformal  representation 

in  the  ^-plane  of  the  lines  u  =  constant  in  the  w-plane,  the  same  limitation 

does  not  apply,  if  we  take  a  single  algebraical  curve  in  the  2r-plane  as  the 

conformal  representation  of  a  single  line  in  the  w-plane. 

1  —  W 
Let  10  =  - :j^:  then  the  lines  in  the  TT-plane,  which  correspond  to  the 

parallel  lines,   u  =  constant,  in  the  w-plane,  are  the  system  oi  circles 


('^-,7^^)('^-d^)=(T^ 


u  +  \j    (u  +  ly ' 

Now  consider  a  relation 

2K 


IT 


Z  =  STi-^  {Wk' ^), 


where  Z  is  as  yet  some  unspecified  function  of  z :  then 

r*F  =  snC— ^V 

Hence  r  WW,  =  sn  f  ""  z)  sn  ( ^^  Z, 


iw.  =  s„f^^z)snf?^^.,. 


SO  that,  if  W  describe  the  circle  corresponding  to  u  =  0,  we  have 


whence  Z  —  Z, 


1  (2K  „\       f2K  „ 

^  =  sn    —  Z    sn    —  Zoj, 

iirK' 


2K  • 

If  -^=sin~^^,  and  therefore  Zo  =  sin"~^5o.  then 

2x^z  +  Zo  =  2  sin ^{Z+  Z^  cos  ^J^ -  {q^  +  (^)  sin \{Z  +  Z^, 
2iy  =  z-z^  =  2q,o^\{Z^  Zo)  sin  ^^  =  i  (q^^  -  q^)  cos  ^(Z  +  Z^), 


4>K 

x^ 


_  1 

4: 


SO  that  ,  _  1        1. „  "*"  ,  _^       ^.„ 

{q  ^  +  q^y      (q  i^-q^y 

an  ellipse,  agreeing  with  the  result  in  |  257,  Ex.  7.     This  is  obtained  from 

the  relation 

'     il-w  /2K   .     ,   \ 

k  2 =  sn    —  sm  ^  z  ] , 

1+W  \    TT  1 

which    is  not  included   in   the   general   forms  of  relation   obtained  in  the 
preceding  investigation. 


712* 


SUEFACES   OF 


[280. 


But  the  equation 


yfctsnf— ^)  + 


[7I       /2^  „ 


does  not  lead  to  an  algebraical  relation  between  x  and  y  for  a  general  (non- 
zero) value  of  u.  Neither  the  conditions  of  the  earlier  proposition  nor  its 
limitations  apply  to  this  case. 

The  problem  of  determining  the  kinds  of  functional  relation  which  will 
represent  a  single  algebraical  curve  in  the  5-plane  upon  a  single  line  of  the 
w-plane  is  wider  than  that  which  has  just  been  discussed :  it  is,  as  yet, 
unsolved. 


Note  on  §  275  (see  foot-note,  p.  690). 

The  investigations  in  §§  276-279  shew  how  important,  in  the  development 
of  the  polyhedral  functions  connected  with  the  hypergeometric  series,  is  the 
surface  of  a  sphere,  that  is,  a  surface  of  constant  positive  curvature.  All  the 
cases  in  which  algebraic  expression  can  arise  for  \  +  /bu  +  v  >  1  are  thereby 
considered. 

It  might  seem  not  improbable  that  a  corresponding  use  of  surfaces  of 
constant  negative  curvature  could  be  made  for  X  +  fM  +  v  <  1 ;  but  the  whole 
investigation  is  much  more  difficult,  because  the  relation  between  w  and  z  is 
always  transcendental  in  one  of  the  two  variables.  The  following  propositions 
are  worthy  of  record. 

A.  All  surfaces  of  given  constant  (negative)  curvature  are  deformable 
into  one  another,  without  stretching  or  tearing*. 
Consequently,  it  is  sufficient  to  take  any  one  of 
them  as  a  surface  of  reference ;  and  the  simplest, 
as  regards  the  geometrical  property,  appears  to  be 
the  surface  of  revolution  formed  by  the  rotation 
of  the  plane  tractrix — the  symmetrical  involute  of 
a  catenary — about  its  asymptote  which  is  the 
directrix  of  the  catenary.  The  equations  of  the 
generating  curve  ■[-  are 

Xo  =  a  sin  cf),     yQ  =  a  (cos  ^  +  log  tan  ^  cf)) ; 
the  range  of  ^  for  the  upper  part  of  the  (dotted) 
curve  is  tt  to  -|-7r,  and  for  the  lower  part  is  J-tt  to  0. 
The  curve  is  periodic  analytically,  so  far  as  con- 
cerns (f),  with  a  period  27r ;  but  geometrically  it  is 

*  A  well-known  theorem,  originally  due  to  Minding,  on  the  basis  of  a  theorem  proved  by 
Gauss  ;  see  my  Lectures  on  Differential  Geometry,  §§  211,  212. 

t  See  Darboux,  Theorie  generate  des  surfaces,  (t.  iii,  pp.  394  sqq.),  for  a  discussion  of  the 
surface. 


280.]  CONSTANT   NEGATIVE   CURVATURE  713 

imaginary,  and  therefore  also  the  sheet  of  the  surface  is  imaginary,  for  the 
half-period  ir  to  2it.  and  for  every  corresponding  half-period.  And  there  is 
an  infinite  number  of  sheets,  real  and  imaginary. 

The  arc  Sq  of  the  tractrix  from  its  vertex  is  given  by 

So  =  <-i  log  cosec  <^. 

The  arc  of  the  surface  of  revolution  is  given  by 

ds-  =  dSff'  +  x^  d&^ 

=  -,  {dxi-^  +  de% 

where  u  = 


sin<^ ' 

The  surface  is  isothermal  in  terms  of  these  variables  u  and  Q ;  the  range 
of  ^  is  0  to  Stt,  and  the  range  of  u  (for  ^  between  ^tt  and  0)  is  from  1  to  oo . 

The  area  of  the  upper  half  of  the  surface  is  the  same  as  that  of  the  lower 
half;  each  of  them 

=  I      277  Xndsg  =  2'7ra^. 
Jo 

For  the  Gaussian  measure  of  curvature  of  the  surface  of  revolution,  we  have 

p  =  —  a  cot  <p,     n  =  a  tan  (f), 

(the  radius  of  curvature  and  the  normal  of  the  generating  curve,  and  these 
are  in  opposite  directions) :  so  the  measure  is  l/pn,  that  is,  —  1/a^. 

Geodesies  on  the  surface*,  are  given  by  the  equation 

du"^     u  \duj       u  du ' 
the  primitive  of  which  is 

u'  +  {e-af  =  ^^; 

that  is,  in  the  u,  6  plane,  the  representatives  of  the  geodesies  are  circles 
having  their  centres  on  the  straight  line  which  is  the  axis  of  6. 

Thus,  if  we  write 

the  surface  becomes 

ds'  =  -'  (dp  +  dv') ; 

T 
it  is  conformally  represented  on  the  |,  77  plane ;  angles  are  conserved ;  and  its 
geodesies  are  transformed  into  circles  in  this  plane  having  their  centres  on  the 
axis  of  |. 

In  this  conformal  representation  of  the  half-surface  of  constant  negative 
curvature  upon  the  f,  17  plane,  where  |  ranges  from  0  to'27r  (and  in  any 
*  See  my  Lectures  on  Differential  Geometry,  pp.  84,  131. 


714  SPECIAL   CONFORMAL  REPRESENTATIONS  [280. 

periodic  repetition  of  this  range)  while  7;  ranges  from  1  to  cc ,  we  are 
representing  an  infinite  half-closed  rectangle,  bounded  by  a  line  parallel  to 
the  axis  of  ^  and  two  lines  parallel  to  the  axis  of  97. 

B.  (This  statement  is  left  as  an  exercise,  as  well  as  the  discussion 
concerning  the  whole  range  of  the  ^'-plane.)  It  should  be  proved  that,  by 
the  relation 

r  =  sinMi(^-t)}, 

one  sheet  of  the  original  surface  of  revolution  is  represented  upon  half  of 
the  ^'-plane  where  ^  =  ^  +  iv- 

C.  Now,  it  is  a  known  theorem  (originally  due  to  Gauss)  that,  in 
particular,  for  the  surface  of  constant  negative  curvature*,  the  area  of 
,a  geodesic  triangle  having  angles  Xir,  fxir,  vir,  is 

-nhi  (1  —  \  —  fj,  —  v). 

Thus  we  can  associate  the  curvilinear  triangles  in  the  original  plane,  for 
which  X  +  /i  -f  V  <  1,  with  a  surface  of  constant  negative  curvature. 

D.  The  ^,  7]  plane,  or  the  ^-plane,  can  be  changed  into  a  w-plane 
where  w  =  X  +  iY,  by  the  relation 

so  that  the  whole  of  the  interior  of  a  w-circle  of  radius  unity  can  be 
represented  on  the  upper  half  of  the  ^-plane  while  the  circumference  of  the 
circle  is  transformed  into  the  axis  of  real  quantities ;  and  likewise  for  the 
exterior  of  the  u'-circle  into  the  lower  half  of  the  ^-plane  by  the  same 
transformation. 

As  already  stated  (p.  690),  the  whole  subject  concerned  mth  the  restriction 
\-\-  fi  +  v  <1  awaits  much  fuller  investigation. 

*  Ges.  Werke,  t.  iv,  p.  245  ;  my  Lectures  on  Differential  Geometry,  p.  161 ;  Carslaw'f5  trans- 
lation of  Bonola's  work  Non-Euclidean  Geometry,  p.  136. 


CHAPTER   XXI. 

Groups  9F  Linear  Substitutions. 

281.     The  pi-operties  of  the  linear  substitution 

az  +  b     - 

tv  =  J , 

CZ+  a  % 

considered  in  Chap.  XIX.  as  bearing  upon  the  conformal  representation  of  t^yo 
planes,  were  discussed  solely  in  connection  with  the  geometrical  relations  of 
the  conformation  :  but  the  applications  of  these  properties  have  a  significance, 
which  is  wider  than  their  geometrical  aspect. 

The  essential  characteristic  of  singly-periodic  functions  and  of  doubly- 
periodic  functions,  each  with  additive  periodicity,  is  the  reproduction  of  the 
function  when  its  argument  is  modified  by  the  addition  of  a  constant  quantity. 
This  modification  of  argument,  uniform  and  uniquely  reversible,  is  only  a 
special  case  of  a  more  general  modification  which  is  uniform  and  uniquely 
reversible,  viz.,  of  the  foregoing  linear  substitution.  This  substitution  may 
therefore  be  regarded  as  the  most  general  expression  of  linear  periodicity, 
in  a  wider  sense :  and  all  functions,  characterised  by  the  property  in  the 
general  form  or  in  special  forms,  may  be  called  automorphic. 

Our  immediate  purpose  is  the  consideration  of  all  the  points  in  the 
plane,  which  can  be  derived  from  a  given  point  z  and  from  one  another  by 
making  z  subject  to  a  set  of  linear  substitutions.  The  set  may  be  either 
finite  or  infinite  in  number ;  it  is  supposed  to  contain  every  substitution 
which  can  be  formed  by  combining  two  or  more  substitutions.  Such  a  set 
is  called  a  group. 

The  substitution  is  often  denoted  by  S  (z),  or  by   - 

it  is  said  to  be  in  its  normal  form,  when  the  real  part  of  a  (if  a  be  a  complex 
constant)  is  positive  and  ad  —  be  =  1 . 

The  ideas  of  the  theory  of  groups  of  substitutions  are  necessary  for  a  proper  considera- 
tion of  the  properties  of  automorphic  functions.  What  is  contained  in  the  present  chapter 
is  merely  sufficient  for  this  requirement,  being  strictly  limited  to  such  details  as  arise  in 
connection  with  these  special  functions.  Information  on  the  fuller  development  of  the 
theory  of  groups,  which  owes  its  origin  as  a  distinct  branch  of  mathematics  to  Galois, 


716  FUNDAMENTAL   SUBSTITUTIONS  [281. 

will  be  found  in  appropriate  treatises  such  as  those  of  Serret*  Jordan t,  NettoJ,  Klein  §, 
and  Burnsidell;  and  in  memoirs  by  Klein**,  Poincarett,  DyckJJ,  and  Bolza§§.  The 
account  of  the  properties  of  groups  contained  in  the  present  chapter  is  based  upon  the 
works  of  Klein  and  Poincare  just  quoted. 

A  substitution  can  be  repeated;  a  convenient  s3mibol  for  representing 
the  substitution,  that  arises  from  n  repetitions  of  S,  is  S"".  Hence  the  various 
integral  powers  of  S,  considered  in  §  258,  are  substitutions,  indicated  by  the 
symbols /Sf^  >S^  S\  .... 

But  we    have    negative  powers   of  S  also.     The  definition   of   S°  (z)   is 

given  by 

SS'>{z)  =  S(z), 

so  that  So  (z)  =  z  and  it  is  often  called  the  identical  substitution  :  the 
definition  of  S~^  (z)  is  given  by 

SS-^z)  =  S'>{z)  =  z, 

■      {■  -r  a  /   \       Cl^  +  b 

SO  that  8-1  (z)  is  a  substitution  the  inverse  of  S ;  m  fact,  it  w  =  S{z)=       ,  ^  > 

then  z  =  S-Hu^~    ^  And  then,  from  S-'^z,  by  repetition  we  obtain 

cw  —  a 

s-\s-\s-\ .... 

If  some  of  all  the  substitutions  to  which  a  variable  z  is  subject  be 
not  included  in  8  and  its  integral  powers,  then  we  have  a  new  substitution 
T  and  its  integral  powers,  positive  and  negative.  The  variable  is  then 
subject  to  combinations  of  these  substitutions :  and,  as  two  general  linear 
substitutions  are  not  interchangeable,  that  is,  we  do  not  have  T{8z)  =  8{Tz) 
in  general,  therefore  among  the  substitutions  to  which  z  is  subject  there 
must  occur  all  those  of  the  form 

.„8^T^SyT'..., 
where  a,  /3,  ^,8,  ...  are  positive  or  negative  integers. 

If,  again,  there  be  other  substitutions  affecting  z,  that  are  not  included 
among  the  foregoing  set,  let  such  an  one  be  U:  then  there  are  also  powers 
of  U  and  combinations  of  8,  T,  U  (with  integral  indices)  operating  in  any 
order:  and  so  on.  The  substitutions  >S^,  T,  U,  ...  are  called  fundamental  : 
the  sum  of  the  moduli  of  a,  ^S,  7,  8,  ...  of  any  substitution,  compounded  from 

*  Corns  d'Algebre  Superieure,  t.  ii.  Sect,  iv,  (Paris,  Gauthier-Villars). 

f  Traite  des  substitutions,  (lb.,  1870). 

J  Substitutionentheorie  und  ihre  Anwendung  auf  die  Algebra,  (Leipzig,  Teubner,  1882). 

§  Vorlesungen  uber  das  Ikosaeder,  (ib.,  1884). 

II   Theory  of  groups  of  finite  order,  (Cambridge,  University  Press,  2nd  ed.,  1911). 
**  Math.  Ann.,  t.  xxi,  (1883),  pp.  141 — 218,  where  references  to  earlier  memoirs  by  Klein  are 
given. 

ft  Acta  Math.,  t.  i,  (1882),  pp.  1—62,  pp.  193—294;  ib.,  t.  iii,  (1883),  pp.  49—92. 
Xt  Math.  Ann.,  t.  xx,  (1882),  pp.  1—44;  ib.,  t.  xxii,  (1883),  pp.  70—108. 

§§  Amer.  Jottrn.  of  Math.,  vol.  xiii,  (1890),  pp.  59—144. 


281.]  GROUPS   OF   SUBSTITUTIONS  7l7 

the  fundamental  substitutions,  is  called  the  index  of  that  substitution ;  and 
the  aggregate  of  all  the  substitutions,  fundamental  and  composite,  is  the 
group. 

There  may  however  be  relations  among  the  substitutions  of  the  group, 
depending  on  the  fundamental  substitutions ;  they  are,  ultimately,  relations 
among  the  fundamental  substitutions,  though  they  are  not  necessarily  the 
simplest  forms  of  those  relations.  Hence,  as  we  may  have  a  relation  of 
the  form 

...lS^...T^...U'...{z)  =  z, 

the  index  of  a  composite  substitution  is  not  a  determinate  quantity,  being 
subject  to  additions  or  subtractions  of  integral  multiples  of  quantities  of  the 
form  (a)  +  (6)  +  (c)  +  ...,  there  being  one  such  quantity  for  every  relation: 
we  shall  assume  the  index  to  be  the  smallest  positive  integer  thus  obtainable. 

282.  There  are  certain  classifications  which  may  initially  be  associated 
with  such  groups,  in  view  of  the  fact  that  the  arguments  are  the  arguments 
of  uniform  automorphic  functions  satisfying  the  equation 

f{Sz)=f{z): 

in  this  connection,  the  existence  of  such  functions  will  be  assumed  until  their 
explicit  expressions  have  been  obtained. 

Thus  a  group  may  contain  only  a  finite  number  of  substitutions,  that  is, 
the  fundamental  substitutions  may  lead,  by  repetitions  and  combinations,  only 
to  a  finite  number  of  substitutions.  Hence  the  fundamental  substitutions, 
and  all  their  combinations,  are  periodic  in  the  sense  of  §  260,  that  is,  they 
reproduce  the  variables  after  a  finite  number  of  repetitions. 

Or  a  group  may  contain  an  infinite  number  of  substitutions :  these  may 
arise  either  from  a  finite  number  of  fundamental  substitutions,  or  from  an 
infinite  number.  The  latter  class  of  infinite  groups  will  not  be  considered 
in  the  present  connection,  for  a  reason  that  will  be  apparent  (p.  732,  note) 
when  we  come  to  the  graphical  representations.  It  will  therefore  be 
assumed  that  the  infinite  groups,  which  occur,  arise  through  a  finite  number 
of  fundamental  substitutions. 

A  group  may  be  such  as  to  have  an  infinitesimal  substitution,  that  is, 

there  may  be  a  substitution      — -y ,  which  gives  a  point  infinitesimally  near 

to  z  for  every  value  of  z.  It  is  evident  there  will  then  be  other  infinitesimal 
substitutions  in  the  group  ;  such  a  group  is  said  to  be  continuous.  If  there 
be  no  infinitesimal  substitution,  then  the  group  is  said  to  be  discontinuous, 
or  discrete. 

But  among  discontinuous  groups  a  division  must  be  made.  The  definition 
of  group-discontinuity  implies  that  there  is  no  substitution,  which  gives  an 
infinitesimal  displacement  for  every  value  of  z:  but  there  may  be  a  number 


718  '  DISCONTINUOUS   GROUPS  [282. 

of  special  points  in  the  plane  for  regions  in  the  immediate  vicinity  of  which 
there  are  infinitesimal  displacements.  Such  groups  are  called  improperly 
discontinuous  in  the  vicinity,  of  such  points:  all  other  groups  are  called 
properly  discontinvous.     For  instance,  with  the  group  of  real  substitutions 

yz  +  8' 

where  a,  /?,  7,  8  are  integers  such  that  a8  —  ^y  =  l,  it  is  easy  to  see  that,  when 
2-i  a,nd  Z2  are  real,  we  can  make  the  numerical  magnitude  of 

jZi  +  8       yz.2  4-  8 

as  small  a  non-evanescent  quantity  as  we  please  by  proper  choice  of  oc,  /3,  7,  S  : 
thus  the  group  is  improperly  discontinuous,  because  for  real  values  of  the 
variable  it  admits  infinitesimal  transformations.  But  such  infinitesimal 
transformations  are  not  possible,  when  z  does  not  lie  on  the  axis  of  real 
quantities,  that  is,  when  z  is  complex  :  so  that,  for  all  complex  values  of 
z,  the  group  is  properly  discontinuous. 

The  various  points,  derived  from  a  single  point  by  linear  substitutions, 
will,  in  subsequent  investigations,  be  found  to  be  arguments  of  a  uniform 
function.  Continuous  groups  would  give  a  succession  of  points  infinitely 
close  together;  that  is,  for  these  points,  either /(^)  would  be  unaltered  in 
value  for  a  line  or  a  small  area  of  points  and  therefore  constant  everywhere, 
or  else  the  point  would  be  an  essential  singularity,  as  in  §  37.  We  shall 
therefore  consider  only  discontinuous  groups. 

A  group  containing  only  a  finite  number  of  substitutions  is  easily  seen  to 
be  discontinuous  :  hence  the  groups  which  are  to  be  considered  in  the  present 
connection  are  the  discontinuous  groups  which  arise  from  a  finite  number  of 
fundamental  substitutions*. 

The  constants  of  all  linear  substitutions  of  the  form  -^  are  sup- 
posed subject  to  the  relation  ad  —  be  =  1.  This  condition  holds  for  all 
combinations,  if  it  hold  for  the  components  of  the  combination.     For  let 

az  +  ^         rpi     az  +  b 


then  ST  = 


7^  -I-  S  '  cz  +  d' 

(aa  +  ^c)z^ab  +  ^d  _  Az  +  B 


{ya  +  8c)  z  +  yb  +  8d       Cz  +  L  ' 
whence  AD- BC  =  {a8  -  ^y)(ad  -  bc)=  I. 

''*  These  discontinuous,  or  discrete,  groups  will  be  considered  from  the  point  of  view  of  auto- 
morphic  functions.  But  the  theory  of  such  groups,  which  has  many  and  wide  applications  quite 
outside  the  range  of  the  subject  of  this  treatise,  can  be  applied  to  other  parts  of  our  subject. 
Thus  it  has  been  connected  with  the  discussion  of  Riemann's  surfaces  by  Dyck,  Math.  Ann., 
t.  xvii,  (1880),  pp.  473—509,  and  by  Hurwitz  (I.e.,  p.  456,  note). 


282.]  FINITE   GROUPS  •  7l9 

It  is  easy  to  see  that  ST  (=  U)  and  TS  (=  V)  are  of  the  same  class,  that 
is,  they  are  elliptic,  parabolic,  hyperbolic  or  loxodromic  together :  but  there  is 
no  limitation  on  the  class  arising  from  the  character  of  the  component  sub- 
stitutions. 

Moreover,  if  U  =  V,  so  that  S  and  T  are  interchangeable,  then 

a  —  dch 

oi^8 "  7 "  ;s ' 

that  is,  *S*  and  T  have  the  same  fixed  points.  They  can  be  applied  in  any 
order ;  and,  for  any  given  number  of  occurrences  of  8  and  a  given  number  of 
occurrences  of  T,  the  composite  substitution  will  give  the  same  point.  Thus 
if  S  =  z+  CO,  then  T  =  2  +  Qi' ;  it  S  =  kz,  then  T  =  k'z.  The  class  of  func- 
tions, which  have  their  argument  subjecj,  to  interchangeable  substitutions 
of  the  former  category,  have  already  been  considered :  they  are  the  periodic 
functions  with  additive  periodicity.  The  group  is  S'^T'^',  {=  z  +  mw  +  mfo)'), 
for  all  integral  values  of  m  and  of  in. 

The  latter  class  of  functions  have  what  may  be  called  a  factorial 
periodicity,  that  is,  they  resume  their  value  when  the  argument  is  mul- 
tiplied by  a  constant  *. 

283.  Some  examples  have  already  been  given  of  groups  containing  a 
finite  number  of  substitutions!,  in  the  case  of  certain  periodic  elliptic 
substitutions.  The  effect  of  such  substitutions  is  (p.  628)  to  change  a 
crescent-shaped  part  of  the  plane  having  its  angles  at  the  (conjugate)  fixed 
points  of  the  substitution  into  consecutive  crescent-shaped  parts :  and  so  to 
cover, the  whole  plane  in  the  passage  of  a  substitution  through  the  elements 
constituting  its  period.  They  form  the  simplest  discontinuous  group — in 
that  they  have  only  one  fundamental  substitution  and  only  a  finite  number 
of  derived  substitutions. 

The  groups  which  are  next  in  point  of  simplicity  are  those  with  only 
two  substitutions  that  are  fundamental  and  only  a  finite  number  that 
are  composite.  Both  of  the  fundamental  substitutions  must  be  periodic, 
and  therefore  elliptic,  by  §  260.     Taking  one  of  these  groups  as  an  example, 

*  Functions  having  this  property  are  discussed  by  Piucherle,  "  Sulle  funzioni  monodrome 
aventi  un'  equazione  caratteristica,"  Rend.  1st.  Lomb.,  Ser.  2,  t.  xii,  (1879),  pp.  536 — 542.  See 
also  Eausenberger's  Tlieorie  der  periodischen  Functionen,  (Leipzig,  Teubi)er,  1884) :  in  particular, 
Section  VI. 

t  The  complete  theory  of  finite  groups  of  linear  substitutions  is  discussed,  partly  in  its 
geometrical  lelation  with  polyhedral  functions,  by  Klein,  Math.  Ann.,  t  ix,  (1876),  pp.  183—188, 
and,  in  its  algebraical  aspect,  by  Gordan,  Math.  Ann.,  t.  xii,  (1877),  pp.  23 — 46.  A  reference 
to  these  memoirs  will  shew  that  the  previous  chapter  contains  all  the  essentially  distinct  finite 
groups  of  linear  substitutions. 


720  EXAMPLE   OF   FINITE   GROUP  [288. 

one  of  its  fundamental  substitutions  has  ±  1  as  its  fixed  points  and  it  is 
periodic  of  the  second  order :  it  is  evidently 

o        1 

w  =  oz  =  -  . 
z 

The  other  has  ^  and  oo  as  its  fixed  points,  and  it  is  periodic  of  the  second 

order :  it  is  evidently 

xu=Tz  =  \-z. 

Evidently  ^H  =  z,  T'z  =  z,  (S^S-\  T=T-^),  so  that  we  have  already  all  the 
powers  of  the  fundamental  substitutions  taken  separately. 

But  it  is  necessary  to  combine  them.     We  have  Uz  =  STz  = ,  a  new 

substitution :  and  then 

U'z  =  ^^,    mz  =  z, 

z 
so  that  U  is  periodic  of  the  third  order.     Again 

Vz  =  TSz  =  ^-^, 
z 

which  is  not  a  new  substitution,  for  Vz  —  U^z  :  and  it  is  easy  to  see  that  there 
is  only  one  other  substitution,  which  may  be  taken  to  be  either  TUz  ov  8Vz: 
it  gives 

TUz=SVz  =  -^^, 
z  —  1 

again  periodic  of  the  second  order. 

Hence  the  group  consists  of  the  six  substitutions  for  2r«given  by 

1  1         z-  1         z 

z  \ —  z         z         z—1 

taking  account  of  the  identical  substitution. 

These  finite  discontinuous  groups  are  of  importance  in  the  theory  of 
polyhedral  functions  :  to  some  of  their  properties  we  shall  return  later. 

Next,  and  as  the  last  special  illustration  for  the  present,  we  form  a 
discontinuous  group  with  two  fundamental  substitutions  but  containing  an 
infinite  number  of  composite*  substitutions.  As  one  of  the  two  that  are 
fundamental,  we  take 

w=Tz=--, 
z 

which  is  elliptic  and  periodic  of  the  second  order.     As  the  other,  we  take 

w  =  Sz=  z  +  1, 
which  is  parabolic  and  not  periodic.     All  the  substitutions  are  real. 

One  such  group  has  already  occurred  :  its  fundamental  (parabolic)  substitutions  were 

w  —  Sz  =  z  +  o},         iv=Tz=z  +  w'. 


283.]  DIVISION    OF   PLANE  721 

Evidently  T^z  =  z,  so  that  T  =  T~^ :  and  S'^z  =  z  +  m,  where  m  is  any 
integer.  Then  all  the  composite  substitutions  are  either  of  the  form 
...SpTS^'TS'^z  or  of  the  form  ...SpTS^'TS^Tz,  both  of  these  being  included 

in -J ,  where  a,  b,  c,  d  are  integers,  such  that  ad  —  be  =  1. 

CZ  ~r  a 

Ex.     Prove  the  converse— that  the  substitution  -7 ,  where  a,  b,  c,  d  are  integers 

such  that  ad  —  bc=l,  is  compounded  of  the  substitutions  JS  and  T. 

This  group,  again,  is  of  the  utmost  importance :  it  arises  in  the  theory  of 
the  elliptic  modular-functions.  As  with  the  polyhedral  groups,  the  general 
discussion  of  the  properties  will  be  deferred :  but  it  is  advantageous  to 
discuss  one  of  its  properties  now,  because  it  forms  a  convenient  introduction 
to,  and  illustration  of,  the  corresponding  part  of  the  theory  of  groups  of 
general  substitutions. 

284.  In  the  discussion  of  the  functions  with  additive  periodicity,  it  was 
found  convenient  to  divide  the  plane  into  an  infinite  number  of  regions  such 
that  a  region  was  changed  into  some  other  region  when  to  every  point  of  the 
former  Avas  applied  a  transformation  of  the  form  z  +  m&)  +  mfco\  that  is,  a 
substitution :  and  the  regions  were  so  chosen  that  no  two  homologous  points, 
that  is,  points  connected  by  a  substitution,  were  within  one  region,  and  each 
region  contained  one  point  homologous  with  an  assigned  point  in  any  region 
of  reference. 

Similarly,  in  the  case  when  the  variable  is  subject  to  the  substitutions  of 
an  infinite  group,  it  is  convenient  to  divide  the  plane  into  an  infinite  number 
of  regions ;  each  region  is  to  be  associated  with  a  substitution  which,  applied 
to  the^points  of  a  region  of  reference,  gives  all  the  points  of  the  region,  and 
each  region  is  to  contain  one  and  only  one  point  derived  from  a  given  point 
by  the  substitutions  of  the  group.  It  is  a  condition  that  the  complete  plane 
is  to  be  covered  once  and  only  once  by  the  aggregate  of  the  regions. 

When  the  discontinuous  group  has  only  the  two  fundamental  substitutions, 

Sz  =  z  +  1  and    Tz  = ,  the  division  of  the  plane  is  easy  :  the  difficulty  of 

z 

determining  an  initial  region  of  reference  is  slight,  relatively  to  that  which 
has  to  be  overcome  in  more  general  groups*. 

The  ordinates  of  z  and  w  (=  Sz)  are  positive  together  or  negative  together ; 
and  similarly  for  the  ordinates  of  z  and  tu  (=  Tz) :  so  that  it  will  suffice  to 
divide  the  half-plane  on  the  positive  side  of  the  axis  of  real  quantities. 

For  the  repetitions  of  the  substitution  S,  it  is  evidently  sufficient  to  divide 
the  plane  into  a  series  of  strips,  bounded  by  straight  lines  parallel  to  the  axis 
of  y  at  unit  distance  apart. 

*  In  addition  to  the  references  already  given,  a  memoir  by  Hurwitz,  Math.  Ann.,  t.  xviii, 
(1881),  pp.  531—544,  may  be  consulted  for  this  group.  , 

P.  F.  46 


72:2  DIVISION    OF   PLANE   BY  [284. 

For  the  application  of  the  substitution  T,  we  have  to  invert  with  regard 
to  a  circle  of  radius  1  and  centre  the  origin,  and  to  take  the  reflexion  of  the 
inversion  in  the  axis  of  y. 

In  these  circumstances,  we  can  choose  as  an  initial  region  of  reference,  the 
space  bounded  by  the  conditions 

1  1         ,        ,     . 

It  is  sufficient  to  prove  that  any  point  in  this  region  when  subjected  to  a 

substitution  of  the  group,  necessarily  of  the  form  — -^  ,  where  a,  b,  c,  d  are 

integers   such   that   ad  —  bc  =  l,  is  transformed   to   some  point  without  the 
region,  and  that  the  aggregate  of  the  regions  covers  the  half-plane. 

If  c  be  0,  then  a=l  =d  and  the  transformation  is  only  some  power  of  *S', 
which  transforms  the  point  out  of  the  region. 

If  c  be  +  1,  then,  since  ad  —  hc  =  1,  we  have 

1 

z  +  d 

a  and  d  being  integers.     For  any  point  z  within  the  region,  |i^  +  cZ|,  which  is 
the  distance  of  the  point  from  some  point  0,  ±1,  ±  2,  ...  on  the  axis  of  x,  is 

>  1 :  hence 

jw  —  a|  <  1, 

that  is,  the  distance  of  w  from  some  point  0,  +1,  +  2,  ...  on  the  axis  is  <  1, 
and  therefore  the  transformed  point  is  without  the  region. 

Similarly,  if  c  be  —  1. 

-rn ,   ,  -,         ^     .  all 

11  c  be  >  1,  then  w  —  =  — 


c         c-        d' 
z+- 
c 


d 

z+- 
c 

VS. 

^  2  • 

V3 
2 

a 

w 

c 

As  z  is  within  the  region,    z  +-   ^-^  '•  and  therefore 

1      1 

^  c^  ^  4  ' 


so  that 


Hence  the  distance  of  w  from  some  point  0,  +  1,  +  2,  ...  on  the  axis  of  x 
is  <  ^  \/S,  that  is,  the  transformed  point  is  without  the  region. 

The  exceptions  are  points  on  the  boundary  of  the  region.  The  boundary 
x  =  —  ^  is  transformed  by  S  to  x  =  +  ^:  the  boundary  x"^  -f  ^Z"-  =  1  is  trans- 
formed by  T  into  itself:  but  all  other  points  are  transformed  into  others 
without  the  region. 


284.] 


ELLIPTIC   MODULAR-FUNCTION   GROUP 


723 


We  now  apply  the  substitutions  S  and  T  to  this  region  and  to  the 
resulting  regions.  Each  substitution  is  uniform  and  is  reversible :  so  that 
to  a  given  point  in  the  initial  region  there  is  one,  and  only  one,  point  in  each 
other  region. 

The  accompanying  diagram  (Fig.  108)  gives  part  of  the  division  of  the 
plane  into  regions,  the  substitutions  associated  with  each  region  being 
placed  in  the  region  in  the  figure ;  it  is  easy  to  see  that  the  aggregate  of 
regions  completely  covers  the  half-plane.  All  the  linear  boundaries  of  S'^, 
for  different  integral  values  of  n,  are  changed  by  the  substitution  T  into 
circles  having  their  centres  on  the  axis  of  x  and  touching  at  A  :  thus  the 
boundary  between  S  and  S'"  is  transformed  into  the  boundary  between 
TS  and  TS^.  All  the  lines  which  bound  the  regions  are  circles  having 
their  centres  on  the  axis  of  x  or  are  straight  lines  perpendicular  to  that 
axis ;  and  the  configuration  of  each  strip  is  the  same  throughout  the 
diagram. 


Fig.  108. 

It  will  be  noticed  that  in  one  region  there  are  two  symbols,  viz.,  S~^TS~^ 
and  TST :  the  region  can  be  constructed  either  by  S'^  applied  to  TS-^  or  by 
T  applied  to  ST.     It  therefore  follows  that 

TST  =  S-'TS-\ 
Hence  S .  TST.  S=S.  S'^TS'' .S=T, 

or,  since  T^  =  1 ,  we  have     S  TST  ST  =  1  =  TSTSTS, 

a  relation  among  the  fundamental  substitutions.     Thus  the  symbol  of  any 
region  is  not  unique :  and,  as  a  matter  of  fact,  if  we  pass  clockwise  in  a  small 

46—2 


724  FUCHSIAN  GROUPS  [284. 

circuit  round  0  from  the  initial  region,  we  find  the  regions  to  be  1,  T,  TS,  TST, 
TST8,  TSTST,  TST8TS,  the  seventh  being  the  same  as  the  first  and  giving 
the  above  relation. 

By  means  of  this  relation  it  will  be  found  possible  to  identify  the  non- 
unique  significations  of  the  various  regions.  At  each  point  there  are  six 
regions  thus  circulating  always,  either  in  the  form  ®S,  ©ST,  SSTS,  ...  or  in 
the  form  @T,  @TS,  @T8T,....  And  by  successive  transformations,  the  space 
towards  the  axis  of  on  is  distributed  into  regions. 

The  decision  of  the  region  to  which  a  boundary  should  be  assigned  will 
be  made  later  in  the  general  investigation ;  it  will  prove  a  convenient  step 
towards  the  grouping  of  edges  of  a  region  in  conjugate  pairs. 

Note.  It  may  be  proved  in  the  same  way  that,  for  any  discontinuous 
group  of  substitutions,  the  plane  of  the  variable  can  be  divided  into  regions 
of  a  similar  character.  As  will  subsequently  appear,  there  is  considerable 
freedom  of  choice  of  an  initial  region  of  reference,  which  may  be  called  a 
fundamental  region. 

285.  We  now  pass  to  the  consideration  of  the  more  general  discontinuous 
groups,  based  on  the  composition  of  a  finite  number  of  fundamental  substitu- 
tions. By  means  of  these  groups  and  in  connection  with  them,  the  plane  of 
the  variable  can  be  divided  into  regions,  one  corresponding  to  each  substitu- 
tion of  the  group.  The  regions  are  said  to  be  congruent  to  one  another : 
the  infinite  series  of  points,  one  in  each  of  the  congruent  regions,  which  arise 
from  z  when  all  the  substitutions  of  the  group  are  applied  to  z,  are  said  to 
be  corresponding  or  homologous  points :  and  the  point  in  Mq  of  the  series  is 
the  irreducible  point  of  the  series.  As  remarked  before,  the  correspondence 
between  two  regions  is  uniform :  interiors  transform  to  interiors,  boundaries 
to  boundaries. 

/  Two  regions  are  said  *  to  be  contiguous,  when  a  part  of  their  boundaries  is 
commbri  to  both.  Each  region,  lying  entirely  in  the  finite  part  of  the  plane, 
is  closed:  the  boundary  is  made  up  of  a  succession  of  lines  whicb  may  for 
convenience  be  called  edges,  and  the  meeting-point  of  two  edges  may  for  con- 
venience be  called  a  corner. 

Such  a  group,  when  all  the  substitutions  are  real,  is  called f  Fuchsian, 
by  Poincare ;  the  preceding  example  will  furnish  a  simple  illustration,  useful 
for  occasional  reference.     All  the  substitutions  are  of  the  form 

ds^  +  bs  .      . 

CgZ  +  ds ' 

*  Poincare  uses  the  term  limitrophes. 

t  Math.  Ann.,  t.  xix,  p.  554,  t.  xx,  pp.  52,  53:  Acta  Math.,  t.  i,  p.  62.  The  same  term  is 
applied  to  a  less  limited  class  of  groups;   see  p.  740,  note. 


285;]  CATEGORIES  725 

which  form  will  be  denoted  by  fs{z).  We  shall  suppose  that  an  infinite 
group  of  real  substitutions  is  given,  and  that  it  is  known  independently  to 
be  a  discontinuous  group:  we  proceed  to  consider  the  characteristic  properties 
of  the  associated  division  of  the  plane,  which  is  to  be  covered  once  and  only 
once  by  the  aggregate  of  the  regions.  The  fundamental  region  is  denoted 
by  R^:  the  region,  which  results  when  the  substitution  fm{z)  is  applied 
to  the  points  of  Rq,  will  be  denoted  by  Rm. 

So  long  as  we  deal  with  real  substitutions,  it  is  sufficient  to  divide  the 
half-plane  above  the  axis  of  x  into  regions  :  and  this  axis  may  be  looked  upon 
as  a  boundary  of  the  plane.  Since  the  group  is  infinite,  the  division  into 
regions  must  extend  in  all  directions  in  the  plane  to  its  finite  or  infinite 
boundaries  :  for  we  should  otherwise  have  infinitesimal  transformations.  Thus 
the  edge  of  a  region  is  either  the  edge  of  a  contiguous  region,  and  then  it  is 
said  to  be  of  the  first  hind ;  or  it  is  a  part  of  the  boundary  of  the  plane,  that 
is,  in  the  present  case  it  is  a  part  of  the  axis  of  x :  and  then  it  is  said  to  be  of 
the  second  kind.  Since  all  real  substitutions  transform  a  point  above  the  axis 
of  X  into  another  point  above  the  axis  of  x,  it  follows  that  all  edges  congruent 
with  an  edge  of  the  first  kind  (an  edge  lying  ofi"  the  axis  of  x)  themselves 
lie  ofi"  the  axis  of  x,  that  is,  are  of  the  first  kind :  and  similarly  all  edges  con- 
gruent with  an  edge  of  the  second  kind  are  themselves  of  the  second  kind. 

The  corners,  being  the  extremities  of  the  edges,  are  of  three  categories. 
If  a  corner  be  an  extremity  of  two  edges  of  the  first  kind  and  not  on  the 
axis  of  X,  then  it  is  of  the  first  category :  and  the  infinite  series  of  corners 
homologous  with  it  are  of  the  first  category.  If  it  be  common  to  two 
edges  of  the  first  kind  and  lie  on  the  axis  of  x,  then  it  is  of  the  second 
category:  and  the  infinite  series  of  corners  homologous  with  it  are  of  the 
second  category.  If  it  be  common  to  two  edges,  one  of  the  first  and  one  of 
the  second  kind,  it  is  of  the  third  category ;  of  course  it  lies  on  the  axis 
of  X  and  the  infinite  series  of  corners  homologous  with  it  are  of  the  third 
category.  We  do  not  consider  two  edges  of  the  second  kind  as  meeting : 
they  would,  in  such  a  case,  be  regarded  as  a  single  edge. 

Each  edge  of  the  first  kind  belongs  to  two  regions.  We  do  not  assign 
such  an  edge  to  either  of  the  regions,  but  we  use  this  community  of 
region  to  range  edges  as  follows.  Let  the  edge  be  Bp,  common  to  Rq 
and  Rp ;  then,  making  the  substitution  inverse  to  fp  (z),  say  fp~'^  (z),  Rp 
becomes  Rq,  Rq  becomes  R-p,  and  Ep  becomes  fp~'^(Ep),  which  is  necessarily 
an  edge  of  the  first  kind  and  is  common  to  the  new  regions  R_p  and  R^, 
that  is,  it  is  an  edge  of  Rq.  Let  it  be  Ep  :  then  Ep  and  Ep'  may  be 
the  same  or  they  may  be  different. 

If  Ep  and  Ep  be  different,  then  we  have  a  pair  of  edges  congruent  to 
one  another :  two  such  congruent  edges  of  the  same  region  are  said  to  be 
conjugate.     Since  the  substitutions  are  of  the  linear  type,  the  correspondence 


726  FUNDAMENTAL   SUBSTITUTIONS  [285. 

being  uniform,  not  more  than  one  edge  of  a  region  can  be  conjugate  with 
a  given  edge  of  that  region. 

If  Ep  and  Ep  be  the  same,  then  the  substitution  transforms  Ep  into 
itself:  hence  some  point  on  Ep  must  be  transformed  into  itself  As  the  edge 
is  of  the  first  kind  so  that  the  point  is  above  the  axis  of  X,  the  substitution 
is  elliptic  and  has  this  point  as  the  fixed  point  of  the  substitution  in 
the  positive  half-plane.  The  two  parts  of  Ep  can  be  regarded  as  two 
edges:  and  the  common  point  as  the  corner,  evidently  of  the  first  category. 
Because  the  directions  of  the  edges  measured  aAvay  from  the  point  are 
inclined  at  an  angle  tt,  it  follows  that  the  multiplier  of  the  elliptic  sub- 
stitution is  6"^  or  —1.  An  illustration  of  this  occurs  in  the  special 
example  of  §  284,  where  the  circular  boundary  of  the  initial  region  of 
reference  is  changed  into  itself  by  the   fundamental  substitution  wz  =  —  \, 

that  is, 

w  —  i  z  —  i 

w  +  i         z  +  i' 

Hence  the  edges  of  the  first  kind  are  even  in  number  and  can  he  arranged 
in  conjugate  pairs. 

Further,  a  point  on  an  edge  of  the  first  kind  is  transformed  into  a 
point  on  the  conjugate  edge — uniquely,  unless  the  point  be  a  corner,  when 
it  belongs  to  two  edges.  Hence  points  on  edges  of  the  first  kind  other  than 
corners  correspond  in  pairs. 

An  edge  of  the  second  kind  is  transformed  into  one  of  the  second  kind, 
but  belonging  to  a  different  polygon :  there  is  no  correspondence  between 
points  on  edges  of  the  second  kind  belonging  to  the  same  polygon. 

Each  corner,  as  the  point  common  to  two  edges,  belongs  to  at  least  three 
regions.  As  a  point  of  one  edge,  it  will  have  as  its  homologue  an  extremity 
of  the  conjugate  edge  :  as  a  point  of  another  edge,  it  will  have  as  its  homologue 
an  extremity  of  the  edge  conjugate  to  that  other :  and  these  homologues  may 
be  the  same  or  they  may  be  different.  Hence  several  corners  of  a  given 
region  may  he  homologous :  the  set  of  homologous  coymers  of  a  given  region  is 
called  a  cycle.  Since  points  of  a  series  homologous  with  a  given  point  all 
belong  to  one  category,  it  is  convenient  to  arrange  the  cycles  in  connection 
with  the  categories  of  the  component  parts. 

The  number  of  edges  of  the  first  kind  is  even,  say  2?i :  and  they  can  be 
arranged  in  pairs  of  conjugates,  say  E^,  En+i ;  Ez,  En+2 ;  • . .  •  Then  since  En+p 
is  the  conjugate  of  Ep,  and  fn+p  (z)  is  the  substitution  which  changes  Rq  into 
Rn+p,  fn+p{z)  is  a  substitution  changing  Ep  into  E^-^p.  After  the  preceding 
explanation,  /j5~^  {z)  is  also  a  substitution  changing  Ep  into  its  conjugate  En+p  : 
hence  we  have 

fn+p  (^)  =/p    ^  i^)' 


285.]  FUNDAMENTAL   SUBSTITUTIONS  72? 

Hence  for  a  division  of  the  plane,  each  region  of  which  has  2n  edges  of  the 
first  kind,  the  group  contains  n  fundamental  substitutions :  the  remaining  n 
substitutions,  necessary  to  construct  the  remaining  contiguous  regions,  are 
obtained  by  taking  the  first  inverses  of  the  fundamental  substitutions. 

The  edge  Ep  has  been  taken  as  the  edge  common  to  Ro  and  Rp,  the  region 
derived  from  R^  by  the  substitution  fp  (z).  Every  region  will  have  an  edge 
congruent  to  Ep :  if  Ri  be  one  such  region,  then  the  region,  on  the  other  side 
of  that  line  and  having  that  line  for  an  edge  (the  edge  is,  for  that  other 
region,  the  congruent  of  the  conjugate  of  Ep),  is  obtainable  from  Rq  by  the 
substitution  y^- 1/^(5)}.  We  thus  have  an  easy  method  of  determining  the 
substitution  to  be  associated  with  the  region,  by  considering  the  edges  which 
are  crossed  in  passing  to  the  region :  and,  conversely,  when  the  substitutions 
are  associated  with  the  regions,  the  correspondence  of  the  edges  is  known. 

As  in  the  special  example,  there  are  relations  among  the  fundamental 
substitutions.  The  simplest  mode  of  determining  them  is  to  describe  a  small 
circuit  round  each  corner  of  Rq  in  succession  :  in  the  description  of  the  circuit, 
the  symbol  of  each  new  region  can  be  derived  by  a  knowledge  of  the  edge  last 
crossed  and  when  the  circuit  is  closed  the  last  symbol  is  the  symbol  also  of  R^, 
so  that  a  relation  is  obtained.  « 

286.  The  only  limitations  as  yet  assigned  to  the  initial  region  (and  there- 
fore to  each  of  the  regions)  of  the  plane  are  (i)  that  it  contains  only  one  point 
homologous  with  z,  and  (ii)  that  the  even  number  of  edges  of  the  first  kind 
can  be  arranged  in  congruent  conjugate  pairs.     But  now, 

without  detracting  from  the  generality  of  the  division,  we        q      E___^_^ 

can  modify  the  initial  region  in  such  a  way  that  all  the  /^---il^ — -^ 

edges  of  the  first  kind  are  arcs  of  circles  with  their  centres 
on  the  axis  of  x.     For  let  C. ..AFB.  ..DGG  be  a  region  with        \  \ 

CGD  and  AFB  for  conjugate  edges;  join  CD  by  an  arc  of        \  ^       \ 

a  circle  CED  with  its  centre  on  the  axis  of  x  :  and  apply  to       aV^^^^^o^J^  B 

CED  the  substitution  inverse  to  that  which  gives  the  region  ^ 

.  .  Fi".  109 

in  which  E  lies  :  let  AHB  be  the  result,  being  also  (§  258) 

an  arc  of  a  circle  with  its  .centre  on  the  axis  of  x.     Then  the  part  AFBHA, 

say  So,  is  transformed  to  CGD  EG,  say  So,  by  the  substitution  which  causes  a 

passage  from  Rq  across  CGD  into  another  region:   every  point  in  *S„  has  a 

homologue  in  >So' :  and  there  is,  by  the  hypothesis  that  Ro  is  the  initial  region, 

no  homologue  in  Ro  of  a  point  in  So  except  the  point  itself.     If,  then,  we  take 

away  ^0  from  Ro  and  add  >S^o')  we  have  a  new  region 

Ro'  =  Ro  +  So  —  So- 

It  satisfies  all  the  conditions  which  apply  to  the  regions  so  far  obtained  :  there 
is  no  point  in  Ro  homologous  with  a  point  in  it,  and  the  conjugate  edges 
CGD   and   AFB  are  replaced  by  conjugate  edges   CED,  AHB  congruent 


728  CONVEXITY  OF  [286. 

by  the  same  substitution  as  the  former  pair.  And  the  new  conjugate 
edges  are  circles  having  their  centres  on  the  axis  of  x. 

Proceeding  in  this  way  with  each  pair  of  conjugate  edges  that  are  not 
arcs  of  circles  having  their  centres  on  the  axis  of  x,  and  replacing  it  by  a  pair 
of  conjugate  edges  congruent  by  the  same  substitution  and  consisting  of 
arcs  of  circles  having  their  centres  on  the  axis  of  x,  we  ultimately  obtain  a 
region  in  which  all  the  edges  of  the  first  kind  are  arcs  of  circles  having  .their 
centres  on  the  axis  of  x.  These  can,  of  course,  be  arranged  in  conjugate  pairs, 
congruent  by  the  assigned  fundamental  substitutions.  Straight  lines  perpen- 
dicular to  the  axis  of  x  count  as  circles  with  centres  at  x=<X)  on  that  axis : 
all  other  straight  lines,  not  being  parts  of  the  axis  of  x,  can  be  replaced  by 
circles. 

The  edges  of  the  second  kind  are  left  unaltered. 

A  region,  thus  bounded,  is  called  a  normal  polygon. 

Further,  this  normal  polygon  may  be  taken  convex,  that  is,  edges  do  not 
cross  one  another.  If  the  preceding  reduction  of  a  region  to  the  form  of 
a  normal  polygon  should  lead  to  a  cross  polygon,  then,  as  is  usual  in 
dealing  with  the  area  of  such  cross  figures,  part  of  the  area  is  to  be 
considered  negative :  and  therefore,  for  every  point  in  this  negative  part, 
there  must  be  two  points  in  the  positive  part.  Hence, 
in  the  positive  part,  there  are 

(i)  points,  none  of  which  has  a  homologue  in 
the  negative  part,  or  in  the  positive 
part  except  itself:  their  aggregate  gives 
a  normal  polygon  Q : 

(ii)  two  sets  of  points,  each  set  of  which  consists 
of  the  homologues  of  points  in  the  nega- 
tive part,  and  makes  up  a  positive  normal 
polygon;  let  the  polygons  be  T^  and  T^. 

The  negative  part  is  a  normal  polygon  T,  to  which  T^  and  T^  are  each  congruent. 

We  now  change  R  by  adding  a  normal  polygon  T  and  subtracting  a 
normal  polygon  T^ :  thus  for  the  new  region  we  have  a  positive  (that  is,  a 
convex)  polygon  Q,  and  a  positive  (convex)  polygon  T^.  No  point  in  Q  has  a 
homologue  in  Tgi  hence  T2  and  Q  together  make  up  a  region  such  that 
homologues  of  all  points  within  it  lie  outside :  this  region  is  a  normal 
polygon,  and  it  is  convex.  Hence  we  may  take  as  the  initial  region  of 
reference  a  normal  convex  polygon,  that  is,  a  convex  polygon  bounded  by  arcs  of 
circles  having  their  centres  on  the  axis  of  x,  or  by  portions  of  the  axis  of  x :  the 
nwnnber  of  arc-edges  is  even,  and  they  can  be  arranged  in  conjugate  pairs. 

Simplicity  is  obtained  by  securing  that  the  curves,  which  compose  the 
boundary,  are  as  like  one  another  in  character  as  possible.     The  substitutions 


286.]  NORMAL  POLYGON  729 

are  linear  and  they  change  boundaries  into  boundaries :  the  whole  plane  is  to 
be  covered :  and  there  are  no  gaps  between  a  bounding  edge  and  the  homo- 
logue  of  the  conjugate  bounding  edge.  The  only  curves,  which  satisfy  this 
condition  of  leaving  no  gaps,  and  which  are  of  the  same  character  after  any 
number  of  linear  transformations,  are  circles  and  straight  lines. 

287.     We  have  seen  that  two  (or  more  than  two)  corners  of  a  convex 
polygon  may  be  homologous :    it  is 


now  necessary  to  arrange  all  the 
corners  in  their  cycles.  Let  AB  and 
ED  be  two   conjugate   edges   of  a 

normal   polygon,  and   let  ^  be 

^    ^^  cz  +  d  * 

the  substitution  which  changes  AB  Fig.  m. 

into  BD;  then,  as  usual,  we  have 


a  _      ad  —  be     1  11 

c  c'^  d         c^        d' 

Z+-  2+- 

c  c 

so  that  arg.  Uv ]  +  arg.  (z  +  -)=='ir. 

a  cL 

This  at  once  shews  that,  whatever  be  the  value  of-  and  of-,  the  points  A, 

c  c 

E  are  homologous,  and  likewise  the  points  B,  D.  Hence  to  obtain  a  corner 
homologous  to  a  given  corner  we  start  from  the  corner,  describe  the  edge  of 
the  polygon  beginning  there,  then  describe  in  the  same  direction*  the  conju- 
gate edge  :  the  extremity  of  that  edge  is  a  homologous  corner. 

The  process  may  now  be  reapplied,  beginning  with  the  last  point ;  and  it 
can  be  continued,  each  stage  adding  one  point  to  the  cycle,  until  we  either 
return  to  the  initial  point  or  until  we  are  met  by  an  edge  of  the  second  kind. 
In  the  former  case  we  have  a  completed  cycle,  which  may  be  regarded  as  a 
closed  cycle.  In  the  latter  case  we  can  proceed  no  further,  as  edges  of  the 
second  kind  are  not  ranged  in  conjugate  pairs ;  but,  resuming  at  the  initial 
point  we  apply  the  process  with  a  description  in  the  reverse  direction  until 
we  again  arrive  at  an  edge  of  the  second  kind :  again  we  have  a  cycle,  which 
may  be  regarded  as  an  open  cycle. 

In  the  case  of  a  closed  cycle,  if  one  of  the  included  points  be  of  the  first 
category,  then  all  the  points  are  of  the  first  category :  the  cycle  itself  is  then 
said  to  be  of  the  first  category.  If  one  of  the  points  be  of  the  second  category, 
then  since  no  edge  of  the  second  kind  is  met  in  the  description,  all  the  edges 
met  are  of  the  first  kind ;  and  therefore  all  the  points,  lying  on  the  axis  of  x 

*  This  is  necessary :  the  direction  is  easily  settled  for  a  complete  polygon  the  sides  of  which 
are  described  in  positive  or  in  negative  direction  throughout. 


730 


CLOSED  CYCLES  AND 


[287. 


and  being  the  intersections  of  edges  of  the  first  kind,  are  of  the  second 
category :   the  cycle  itself  is  then  said  to  be  of  the  second  category. 

Open  cycles  will  contain  points  of  the  third  category :  they  may  also 
contain  points  of  the  second  category,  because  points  both  of  the  second  and 
of  the  third  categories  lie  on  the  axis  of  x,  and  homology  of  the  points  does 
not  imply  conjugacy  of  all  edges  of  which  they  are  extremities.  Such  cycles 
are  said  to  be  of  the  third  category. 

It  thus  appears  that  the  cycles  can  be  derived  when  the  arrangement  in 
conjugate  pairs  of  edges  of  the  first  kind  is  given ;  and  it  is  easy  to  see  that 
the  number  of  open  cycles  is  equal  to  the  number  of  edges  of  the  second 
kind. 

We  may  take  one  or  two  examples.  For  a  quadrilateral,  in  which 
the  conjugate  pairs  are  1,  4  ;  2,  3 — the  numbers  being 
as  in  the  figure — we  have  by  the  above  process  A,  AB, 
DA,  A  :  that  is,  ^  is  a  cycle  by  itself  Then  B,  BG,  CD, 
D,  DA,  AB,  B:  that  is,  B  and  D  form  a  cycle ;  and  then 
G,  CD,  BG,  G,  that  is,  C  is  a  cycle  by  itself  The  cycles 
are  therefore  three,  namely.  A;  B,  D;  G. 

For  a  hexagon,  in  which  the  conjugate  pairs  are  1,  5 ;    2 
cycles  are  two,  namely.  A,  F,  D,  G  and  B,  E.     If  the  conjugate  pairs  be 


4;    3,  6,  the 


Fig.  113. 

1,  6;  2,  5  ;  3,  4,  the  cycles  are  four,  namely,  A;  B,  F;  G,  E;  D.  If  the 
conjugate  pairs  be  1,  4 ;  2,  5 ;  3,  6,  the  cycles  are  two,  namely.  A,  C,  E ; 
B,  D,  F. 

For  a  pentagon,  with  one  edge  of  the  second  kind  as  in  the  figure  and 


287.]  OPEN    CYCLES  t31 

having  1,  3;    4,  5  as  the  conjugate  pairs,  the  cycles  are  three,  namely, 
E]    A,  D;    B,  G;    the  last  being  open  and  of  the  third  category. 

For  a  quadrilateral  as  in  the  figure,  having  three  corners  on  the  axis  of  oc 
and  1,  2 ;  3,  4  as  the  arrangement  of  its  conjugate 
pairs,  the  cycles  are  D;    A,  G ;    B ;   the  last  two 
being  of  the  second  category. 

We  have  now  to  consider  the  angles  of  the 
polygons  taken  internally.  It  is  evident  that  at 
any  corner  of  the  second  category,  the  angle  is 
zero,  for  it  is  the  angle  between  two  circles  meeting 
on  their  line  of  centres ;  and  that  at  any  corner  of 
the  third  category  the  angle  is  right.  There  therefore  remain  only  the 
angles  at  corners  of  the  first  category.  Let  A-^,  A^,  •■■,  An  be  the  corners 
in  a  cycle  of  the  first  category  and  denote  the  angles  by  the  same  letters. 

Since  A^  and  Az  are  homologous  corners,  they  are  extremities  of  conjugate 
edges.  Apply  to  the  plane,  in  the  vicinity  of  Aq,  the  substitution  which 
changes  the  edge  ending  in  A2  to  its  conjugate  ending  in  A^ :  then  the 
point  A2  is  transferred  to  the  point  A-^;  one  edge  at  A2  coincides  with  its 
conjugate  at  J-i  and  the  other  edge  at  ^2  makes  an  angle  ^ 
A2  with  it,  because  of  the  substitution  which  conserves       -^^^L^g' 


angles.     The  latter  edge  was  the  edge  which  followed  A 2  ^\ 

in  the  cycle  for  the  derivation  of  J.3 :  we  take  its  conju- 
gate ending  in  As,  and  treat  these  and  the  points  A^  and  '^" 
A3  as  before  for  A^  and  A^  and  their  conjugate  edges,  namely,  by  using  the 
substitutions  transforming  conjugate  edges  and  passing  from  A3  to  A2  and 
then  those  from  A2  to  A^. 

Proceeding  in  this  way  round  the  cycle,  we  shall  have 

(1)  .  a  series  of  lines  at  the  point,  each  line  between  two  angles  being 

one  of  the  conjugate  edges  on  which  the  two  comers  lie  : 

(2)  the  angles  corresponding  to  the  corners  taken  in  cyclical  order. 
Hence  after  n  such  operations  we  shall  again  reach  an  angle  A^.     If  the  edge 
do  not  coincide  with  the  first  edge,  we  repeat  the  set  of  n  operations :  and 
so  on. 

Now  all  these  substitutions  lead  to  the  construction  of  the  various  regions 
meeting  in  A,  which  are  to  occupy  all  the  plane  round  A,  and  no  two  of 
which  are  to  contain  a  point  which  does  not  lie  on  an  edge.  Hence 
after  the  completion  of  some  set  of  operations,  say  the  ^th  set,  the 
edges  of  ^^  will  coincide  with  their  edges  of  the  first  angle  A^;  and 
therefore 

_p  (4l  +  J.2  +  ...  +  ^n)  =  27r, 

/        .  .  27r         . 

so  that  Ai  +  A2+...+An=—-. 


732  PEOPERTIES   OF   A  [287. 

Hence  the  sum  of  the  angles  at  the  corners,  in  a  cycle  of  the  first  category, 
is  a  submulti'ple  of  ^ir. 

Further,  if  q  be  the  number  of  polygons  at  J.,  we  have 

np  =  q. 

Corollary  1.  For  a  cycle  of  the  second  category — it  is  a  closed  cycle — 
both  p  and  q  are  infinite. 

The  cycle  contains  only  a  finite  number  of  comers,  because  the  polygon 
has  only  a  finite  number*  of  edges :  as  each  corner  is  of  the  second  category, 
the  angle  is  zero :  and  therefore  the  repetition  of  the  set  of  operations  can  be 
effected  without  limit.  Hence  p  is  infinite ;  and,  as  n  polygons  at  a  corner 
are  given  by  each  set  of  operations,  the  number  q  of  polygons  is  infinite. 

Corollary  2.  Corresponding  to  every  cycle  of  the  first  category,  there  is 
a  relation  among  the  fundamental  substitutions  of  the  group. 

Let  /i2  be  the  substitution  interchanging  the  conjugate  edges  through  A^ 
and  A2;  fis  the  substitution  interchanging  the  conjugate  edges  through  A.2 
and  A3 ;  and  so  on.     Let  U  denote 

/12      •J2Z      •/s4      •■■•Jn-i,n      wj 

then  UP  {2)  =  z. 

For  U  is  the  substitution  which  reproduces  the  polygon  with  the  angle 
J-x  at  Ai ;  and  this  substitution  is  easily  seen,  after  the  preceding  explanation, 
to  be  periodic  of  order  p.     Moreover,  this  substitution  U  is  elliptic. 

288.  The  following  characteristics  of  the  fundamental  region  have  now 
been  obtained : 

(i)  It  is  a  convex  polygon,  the  edges  of  which  are  either  arcs 
of  circles  with  their  centres  on  the  axis  of  x  or  are  portions 
of  the  axis  of  ^ : 

(ii)  The  edges  of  the  former  kind  are  even  in  number  and  can  be 
arranged  in  conjugate  pairs :  there  is  a  substitution  for  which 
the  edges  of  a  conjugate  pair  are  congruent;  if  this  sub- 
stitution change  one  edge  a  of  the  pair  into  a',  it  changes 
the  given  region  into  the  region  on  the  other  side  of  a': 

(iii)  The  corners  of  the  polygon  can  be  arranged  in  cycles  of  one  or 
other  of  three  categories : 

*  If  the  number  be  infinite,  the  edges  must  be  infinitesimal  in  length  unless  the  perimeter  of 
each  of  the  polygons  is  infinite :  each  of  these  alternatives  is  excluded. 

The  reason  for  finiteness  (§  282)  in  the  number  of  fundamental  substitutions  in  the  group 
is  now  obvious  :  their  number  is  one-half  of  the  number  of  edges  of  the  first  kind. 


288.]  FUNDAMENTAL    REGION  733 

(iv)  The  angles  at  corners  in  a  cycle  of  the  second  category  are  zero : 
each  of  the  angles  at  corners  in  a  cycle  of  the  third  category 
is  right :  the  sum  of  the  angles  at  corners  in  a  cycle  of  the 
first  category  is  a  submultiple  of  27r. 

Let  there  be  an  infinite  discontinuous  group  of  substitutions,  such  that  its 
fundamental  substitutions  are  characterised  by  the  occurrence  of  the  fore- 
going properties  in  the  edges  and  the  angles  of  the  geometrically  associated 
region :  and  let  the  whole  group  of  substitutions  be  applied  to  the  region. 

Then  the  half-plane  on  the  positive  side  of  the  axis  of  x  is  covered :  no 
part  is  covered  more  than  once,  and  no  part  is  unassigned  to  regions.  It  is 
easy  to  see  in  a  general  way  how  this  given  condition  is  satisfied  by  the 
various  properties  of  the  regions.  Since  the  edges  of  the  first  kind  in 
the  initial  region  can  be  arranged  in  conjugate  pairs,  it  is  so  with  those 
edges  in  every  region :  and  the  substitution,  which  makes  them  congruent, 
makes  one  of  them  to  coincide  with  the  homologue  of  the  other  for  the 
neighbouring  region,  so  that  no  part  is  unassigned.  No  part  is  covered 
twice,  for  the  initial  region  is  a  normal  convex  polygon  and  therefore  every 
region  is  a  normal  convex  polygon :  the  edges  are  homologous  from  region  to 
region,  and  form  a  common  boundary.  The  angle  of  intersection  with  a 
given  arc  is  sufficient  to  fix  the  edge  of  the  consecutive  polygon :  for  an  arc 
of  a  circle,  making  on  one  side  an  assigned  angle  with  a  given  arc  and  having 
its  centre  on  the  axis,  is  unique.  At  every  corner  of  any  polygon,  there  will 
be  a  number  of  polygons :  the  corners  which  coincide  there  are,  for  the 
different  polygons,  the  corners  homologous  with  a  cycle  in  the  original 
region :  and  the  angles  belonging  to  those  corners  fill  up,  either  alone  or 
after  an  exact  number  of  repetitions,  the  full  angle  round  the  point. 

We  have  seen  that  the  substitution,  which  passes  from  a  polygon  at  a 
point  to  the  same  polygon,  after  n  polygons,  reproduces  the  angular  point 
at  the  same  time  as  it  reproduces  the  polygon;  the  point  is  a  fixed  point 
of  an  elliptic  substitution.  Similarly,  if  the  point  belong  to  a  cycle  of  the 
second  category,  n  is  infinite  and  the  substitution  does  not  change  the  point, 
which  is  therefore  a  fixed  point  of  the  substitution ;  as  the  fixed  point  is  on 
the  axis,  the  substitution  is  parabolic  (§  292). 

The  preceding  are  the  essential  properties  of  the  regions,  which  are 
sufficient  for  the  division  of  the  half-plane  when  a  group  is  given,  and 
therefore  by  reflexion  through  the  axis  of  os,  they  are  sufficient  for  the 
division  of  the  other  half-plane. 

The  position  of  corners  of  the  first  category,  and  the  orientation  of  edges 
meeting  in  those  corners,  are  determinate  when  the  group  is  supposed 
given :  within  certain  limits,  half  of  the  corners  of  the  third  category  can 
be  arbitrarily  chosen. 


734 


EXAMPLE 


[289. 


289.  In  the  preceding  investigation,  the  group  has  been  supposed  given  : 
the  problem  was  the  appropriate  division  of  the  plane.  The  converse  problem 
occurs  v^^hen  a  fundamental  region,  with  properties  appropriate  for  the 
division  of  the  half-plane,  is  given :  it  is  the  determination  of  the  group. 
The  fundamental  substitutions  of  the  group  are  those  which  transform  an  edge 
into  its  conjugate,  and  they  are  to  be  real— conditions  which,  by  §  258, 
are  sufficient  for  their  construction.  The  whole  group  of  substitutions  is 
obtained  by  combining  those  that  are  fundamental.  The  complete  division 
of  the  half-plane  is  effected,  by  applying  to  each  polygon  in  succession  the 
series  of  fundamental  substitutions  and  of  their  first  inverses. 

It  is  evident  that  a  given  division  of  the  plane  into  regions  determines 
the  group  uniquely :  but,  as  has  already  been  seen  in  the  general  ex- 
planation, the  existence  of  a  group  with  the.  requisite  properties  does  not 
imply  a  unique  division  of  the  plane. 

As  an  example,  let  the  fundamental  substitutions  be  required  when  a  quadrilateral  as 
in  Fig.  112,  having  1,  2 ;  3,  4  for  the  conjugate  pairs  of  edges,  is  given  as  a  fundamental 
region.     The  cycles  of  the  corners  are  B;  D;  A,  C;  so  that 


B-. 


D-. 


A  +  C=' 


277 

I  m 

where  I,  m,  n  are  integers. 

The  simplest  case  has  already  been  treated,  §  284:  there,  /  =  2,  m  =  <xi,  n-- 
the  region  is  a  triangle,  really  a  quadrilateral  with  two 
edges  as  conterminous  arcs  of  the  same  circle.  We  shall 
therefore  suppose  this  case  excluded;  we  take  the  case 
next  in  point  of  simplicity,  viz.  l  =  %  A  =  C.  Then  AB 
and  BG  are  conterminous  arcs  of  one  circle :  we  shall 
take  the  centre  of  this  circle  to  be  the  origin,  its  radius 
unity  and  B  on  the  axis  of  y ;  then  ^  is  a  fixed  point 
of  the  substitution,  which  changes  AB  into  BC.     The 

substitution  is 

1 


=  3,  A  =  C; 


it  is  one  of  the  two  fundamental  substitutions. 


Fig.  117. 


Evidently  A=  —  ,  ADB  =  —  .     Let  ^be  the  centre  of  the  circle  AD,  and  p  its  radius: 

•'         n  m  '  r 

then  OAE=-,  OI)E=Z  -  -,  and  so 

p2-t- 1  -  2p  cos -p  0£'2=p2  cos'^  -  , 


whence 


p  sin^  — = cos  —  +  I  cos2 sin^  - 


the  negative  sign  of  the  radical  corresponding  to  the  case  when  D  lies  below  ABC.     The 
radius  p  must  be  real  and  therefore 

n      in      " 
we  omit  the  case  of  m  =  oo  ,  and  therefore  « >  2. 


289.]  EXAMPLE  735 

The  fundamental  substitution,  which  changes  AD  into  CZ),  has  D  and  the  complex 
conjugate  to  B  for  its  fixed  points:  these  points  are  +^psin  — .     The  argument  of  the 

9 

multiplier  is  — ,  being  the  angle  ADC:   hence  the  substitution  is 

w  —  ip  Sin  —     z  —  ip  sin  —  ^ 
= e 


which  reduces  to 


w  +  ip  sin—      z  +  ip  sin  — 

VI  '^  Vfl 


z  cos  — Vp  sin^  — 
w  — 


1-cos  — 

p  v% 

where  p  has  the  value  given  by  the  above  equation. 

This  substitution,  and  the  substitution  w=  — ,  are  the  fundamental  substitutions  of 

z 

the  group.     The  special  illustration  in  §  284  gives 

m  =  00  ,  p  =  00 ,  ?i  =  3,  p  sin^  —  =  2  cos  —  =  1 : 

VI  11 

the  special  form  therefore  is 

w=z  +  \. 

Taking  cos— =  a,  cos- =  6,   /S.  =  (a'^  +  b'^  —  \)^,  we   have   p  (l-a^)  =  h  +  ^\    the   second 

VI  11  r  \  / 

fundamental  substitution  is 

az+A  +  ft 


i  =  Sz= 


(A-6)3  +  a" 
It  is  easy  to  see  that 

where  Tz=  —  ;   the  complete  figure  can  be  constructed  as  in  §  284. 

An  interesting  figure  occurs  for  m=4,  n  =  Q. 

In  the  same  way  it  may  be  proved  that,  if  an  elliptic  substitution  have  re    *  for  its 
common  points  and  29  for  the  argument  of  its  multiplier,  its  expression  is 

__Az+B 

^~Gz  +  D' 

,  ,     sin(<9-e)  „       sine  ^        1  sin  6  ^     sin((9  +  e) 

where  A  = ^ — -^-^  ,        B=r—. — ^r ,         C= -. — ^,         D= ^ — ^    , 

sm  6  sm  ^  /•  sm  ^  sm  6 

Taking  now  the  more   general  case  where  B  =  ^ri  D  =  — ,  A  +  C=  —  ,  let  B  (in 

figure  112)  be  the  point  6e^\  and  A  the  point  ae*'.    Then  the  substitution  which  transforms 
AB  into  BC  is  the  above,  when  ^  =  /3,  r  =  h,  e=B,  so  that,  if  C  be  ce^*. 


yj     g  sin  (/3-^)e°''  +  &  sing 
giving  two  relations  among  the  constants. 


■^sin5e"'  +  sin(/3  +  5) 

0 


736  FUNDAMENTAL  [289. 

Similarly,  two  more  relations  will  arise  out  of  the  substitution  which  transforms  CD 
into  DA.  And  three  relations  are  given  by  the  conditions  that  the  sum  of  the  angles  at 
A  and  C  is  an  aliquot  part  of  Stt,  and  that  each  of  the  angles  B  and  D  is  an  aUquot  part 
of  27r. 

290.  All  the  substitutions  hitherto  considered  have  been  real :  we  now 
pass  to  the  consideration  of  those  which  have  complex  coefficients.     Let 

75  +  S 

be  such  an  one,  supposed  discontinuous :  then  the  effect  on  a  point  is  obtained 
by  displacing  the  origin,  inverting  with  respect  to  the  new  position,  reflecting 
through  a  line  inclined  to  the  axis  of  a;  at  some  angle,  and  again  displacing 
the  origin.  The  displacements  of  the  origins  do  not  alter  the  character  of 
relations  of  points,  lines,  and  curves :  so  that  the '  essential  parts  of  the 
transformation  are  an  inversion  and  a  reflexion. 

Let  a  group  of  real  substitutions  of  the  character  considered  in  the 
preceding  sections  be  transformed  by  the  foregoing  single  complex  substitu- 
tion :  a  new  group 


az  +  ^        70  +  S 


+  6 


<yz  -\-h         oiz  -\-  8       , 

'  c 4.  +  o5 

ryZ  +  0 

will  thus  be  derived.     The  geometrical  representation  is  obtained  through 
transforming  the  old  geometrical  representation  by  the  substitution 

so  that  the  new  group  is  discontinuous. 

The  original  group  left  the  axis  of  x  unchanged,  that  is,  the  line  z  =  Zo 
was  unchanged ;  hence  the  substitutions 

a.z  +  ^      ,  "^ 
az  +  p         yz  +  b 

ryz  +  6 

will    leave    unchanged    the    line    which    is    congruent    with    z  =  Zo   by   the 

lis  line  is 
Bz  +  ^     -So 


substitution  (— — ^  ,  z] .     This  line  is 

\yz  +  6       J 


yz  —  a. 
or  it  may  be  taken  in  the  form 


,     .-8z  +  0      . 
imaginary  part  01 =  0. 


290.]  ciKCLE  737 

It  is  a  circle,  being  the  inverse  of  a  line  ;  it  is  unaltered  by  the  substitutions 
of  the  new  group,  and  it  is  therefore  called*  the  fundamental  circle  of  this 
group.     The  group  is  still  called  Fuchsian  (p.  740,  note). 

The  half-planes  on  the  two  sides  of  the  .axis  of  x  are  transformed  into  the 
two  parts  of  the  plane  which  lie  within  and  without  the  fundamental  circle 
respectively:  let  the  positive  half-plane  be  transformed  into  the  part  within 
the  circle. 

With  the  group  of  real  substitutions,  points  lying  above  the  axis  of  x 
are  transformed  into  points  also  lying  above  the  axis  of  x,  and  points  below 
into  points  below  :  hence  with  the  new  group,  points  within  the  fundamental 
circle  are  transformed  into  points  also  within  the  circle,  and  points  without 
into  points  without. 

The  division  of  the  half-plane  into  curvilinear  polygons  is  changed  into  a 
division  of  the  part  within  the  circle  into  curvilinear  polygons.  The  sides  of 
the  polygons  either  are  circles  having  their  centres  on  the  axis  of  x,  that  is, 
cutting  the  axis  orthogonally,  or  they  are  parts  of  the  axis  of  x:_  hence  the 
sides  of  the  polygons  in  the  division  of  the  circle  either  are  arcs  of  circles 
cutting  the  fundamental  circle  orthogonally  or  they  are  arcs  of  the  funda- 
mental circle. 

The  division  of  the  part  of  the  plane  without  the  circle  is  the  trans- 
formation of  the  half-plane  below  the  axis  of  x,  which  is  a  mere  reflexion 
in  the  axis  of  x  of  the  half-plane  above :  thus  the  division  is  characterised  by 
the  same  properties  as  characterise  the  division  of  the  part  within  the 
fundamental  circle.  But  when  the  division  of  the  part  within  the  circle 
is  given,  the  actual  division  of  the  part  without  it  can  be  more  easily 
obtained  by  inversion  with  the  centre  of  the  fundamental  circle  as  centre 
and  its  radius  as  radius  of  inversion. 

This  process  is  justified  by  the  proposition  that  conjugate  complexes  are 
transformed  by  the  substitution  ( ^  ,  z  j  into  points  which  are  the  in- 
verses of  one  another  with  regard  to  the  fundamental  circle.  For  a  system 
of  circles  can  be  drawn  through  two  conjugate  complexes,  cutting  the  real 
axis  orthogonally :  when  the  transformation  is  applied,  we  have  a  system  of 
circles,  orthogonal  to  the  fundamental  circle  and  passing  through  the  two 
corresponding  points.  The  latter  are  therefore  inverses  with  regard  to  the 
fundamental  circle. 

This  proposition  can  also  be  proved  in  the  following  elementary  manner. 

Let  OC,  the  axis  of  x,  be  inverted,  with  A  as  the  centre  of  inversion,  into  a  circle: 
P  and  Q  be  two  conjugate  complexes,  and  let  AP  cut  the  axis  of  ^  in  C :  let  CQ  cut  the 
diameter  of  the  circle  in  R.  Since  OC  bisects  PQ,  it  bisects  AR;  and  therefore  the  centre 
of  the  circle  is  the  inverse  of  R. 

*  Klein  uses  the  word  Hauptkreis.  .  ' 

F.  F.  47 


738 


GROUPS   CONSERVING 


[290. 


Let  p  and  q  be  the  inverses  of  P  and  Q :  join  pq,  qr.    Then  the  angle  pqQ=  CPQ=GQP, 
and  Aqr=  CRO :  thus  pqr  is  a  straight  line. 


Also 


and 


qr 
Aq  = 

QR 
'  AR 

AP 
AR 

Ar 
^  Ap' 

pr 
Ap 

PE 
^  AR^ 

AQ 
AR' 

Ar 

~-Aq' 

P/ 

C 

I          '"           / 

0                                 R 

Fig.  118. 


so  that  rp  .rq  =  Ar^. 

Thus  p  and  q  are  inverses  of  each  other, 
relative  to  r  and  with  the  radius  of  the 
fundamental  circle  as  radius.  Transference 
of  origin  and  reflexion  in  a  straight  line  do 
not  alter  these  properties :  and  therefore  p 
and  q,  the  transformations  of  the  conjugate 
P  and  Q,  are  inverses  of  one  another  with  regard  to  the  fundamental  circle. 

Hence  with  the  present  group,  constructed  from  an  infinite  discontinuous 
group  of  real  substitutions  transformed  by  a  single  complex  substitution,  the 
fundamental  circle  has  the  same  importance  as  the  axis  of  real  quantities 
in  the  group  of  real  substitutions.  It  is  of  finite  radius,  which  will  be  taken 
to  be  unity :  its  centre  will  be  taken  to  be  the  origin.  The  area  within  it  is 
divided  into  regions  congruent  with  one  another  by  the  substitutions  of  the 
group :  the  whole  of  the  area  is  covered  by  the  polygons,  but  no  part  is 
covered  more  than  once. 

All  the  points,  homologous  with  a  given  point  z  within  the  circle,  lie 
within  the  circle  :  each  polygon  contains  only  one  of  such  a  sefc  of  homologous 
points. 

The  angular  points  of  a  polygon  can  be  arranged  in  cycles  which  are 
of  three  categories.  The  sum  of  the  angles  at  points  in  a  cycle  of  the  first 
category  is  unchanged  by  the  substitution ;  it  is  equal  to  an  aliquot  part  of 
lir.  At  points  in  a  cycle  of  the  second  category  each  angle  is  zero  :  at  points 
in  a  cycle  of  the  third  category  each  angle  is  right. 

In  fact,  all  the  properties  obtained  for  the  division  of  the  plane  into 
polygons  now  hold  for  the  division  of  the  circle  into  polygons  associated 
with  the  group 

OiZ-\- , 


az  +  /3         jz  +  S 
yz  +  S 


+  b 


az  +  ^ 

C TT+f^ 

jz  +  b 

provided  we  make  the  changes  that  are  consequent  on  the  transformation  of 
the  axis  of  x  into  the  fundamental  circle. 


The  form  of  the  substitution 


az  +  13 
'yz  +  8'' 


290.]  A   FUNDAMENTAL   CIRCLE  739 

which  secures  that  the  fundamental  circle  in  the  ?<;-plane  shall  be  of  radius  unity  and 
centre  the  origin,  is  easily  obtained. 

It  has  been  proved  that  inverse  points  with  respect  to  the  circle  correspond  to  conjugate 
complexes;  hence  w=0  and  w  =  <x)  correspond  to  two  conjugate  complexes,  say  X  and  Xq? 
and  therefore 

2-X 

Z-Aq 

where  |  k  |  =  1  because  the  radius  of  the  fundamental  circle  is  to  be  unity.  The  presence 
of  this  factor  k  is  equivalent  to  a  rotation  of  the  ^y-plane  about  the  origin.  As  the  origin 
is  the  centre  of  the  fundamental  circle,  the  circle  is  unaltered  by  such  a  change  :  and 
therefore,  without  affecting  the  generality  of  the  substitution,  we  may  take  k  =  1,  so  that 
now 

s-\ 

where  X  is  an  arbitrary  complex  constant.  The  substitution  is  not  in  its  canonical  form, 
which  however  can  at  once  be  deduced. 

291.  It  has  been  seen,  in  §  260,  that,  when  any  real  substitution  is  para- 
bolic or  hyperbolic,  then  practically  an  infinite  number  of  points  coincide 
with  the  fixed  point  when  the  substitution  is  repeated  indefinitely,  whatever 
be  the  point  z  initially  subjected  to  the  transformation;  this  fixed  point  lies 
on  the  axis, of  x,  and  is  called  an  essential  singularity  of  the  substitution. 
When  we  consider  such  points  in  reference  to  automorphic  functions,  which 
are  such  as  to  resume  their  value  when  their  argument  is  subjected  to 
the  linear  substitutions  of  the  group,  then  at  such  a  point  the  function 
resumes  the  value  which  it  had  at  the  point  initially  transformed ;  that  is, 
in  the  immediate  vicinity  of  such  a  fixed  point  of  the  substitution,  the 
function  acquires  any  number  of  different  values  :  such  a  point  is  an  essential 
singularity  of  the  function.  Hence  the  essential  singularities  of  the  group 
are  the  essential  singularities  of  the  corresponding  fiinction. 

Now  all  the  essential  singularities  of  a  discontinuous  group  lie  on  the 
axis  of  X  when  the  group  is  real ;  the  line  may  be  or  may  not  be  a  con- 
tinuous line  of  essential  singularity.  If,  for  example,  x  be  any  such  point 
for  the  group  of  §§  283,  284  which  is  characteristic  of  elliptic  modular- 
functions,  then  all  the  others  for  that  group  are  given  by 

ax  +  h 
cx  +  d' 

where  a,  h,  c,  d  are  integers,  subject  to  the  condition  ad  —  be  =  1 :  and 
therefore  all  the  essential  singularities  are  given  by  rational  linear  trans- 
formations. For  points  on  the  real  axis,  this  group  is  improperly  dis- 
continuous :  and  therefore  for  this  group  the  axis  of  «  is  a  line  of  essential 
singularity. 

Hence  when  we  use  the  transformation  ( ^ ,  ^  1  to  deduce  the  division 

of  the  fundamental  circle  into  regions,  the  essential  singularities  of  the  new 

47—2 


740  FAMILIES   OF  [291. 

group  are  points  on  the  circumference  of  the  fundamental  circle  :  the  cir- 
cumference is  or  is  not  a  line  of  essential  singularity  for  the  function  or 
the  group*,  according  as  the  group  is  properly  or  improperly  discontinuous 
for  the  circle. 

Ex.     Shew  that  for  all  figures,  congruent  to  a  closed  simply  connected  figure  by  the 

transformation 

az  +  c 

where  a,  a^  and  c,  Cq  are  conjugate  constants  such  that  aaQ«-cco  =  l,  the  quantities 

/"  I  c^z  I  r  /"  rdrdd 

where  z=re^^,  are  invariable.  (Poincare.) 

Let  ABC  be  a  triangle,  having  for  its  sides  arcs  of  circles  that  are  orthogonal  to  the 
fundamental  circle  of  the  substitution ;  and  denote  by  a,  b,  c  the  quantity  L  for  the  three 
sides  respectively.     Prove  that 

cosh  c  =  cosh  a  cosh  b  —  sinh  a  sinh  b  cos  C.  (Kapteyn.) 

292.  It  is  convenient  to  divide  the  groups  into  families,  the  discrimin- 
ation adopted  by  Poincare  being  made  according  to  the  categories  of  cycles  of 
angular  points  in  the  polygons  into  which  the  group  divides  the  plane.  The 
group  is  of  the 

1st  family,  if  the  polygon  have  cycles  of  the  1st  category  only, 

2nd  2nd  

3rd  3rd  

4th   2nd  and  3rd  

5th  1st  and  3rd 

6th  1st  and  2nd  

7th   : all  three  categories. 

Thus  in  the  polygons  associated  with  groups  of  the  1st,  the  2nd,  and  the  6th 
families,  all  the  edges  are  of  the  first  kind ;  in  the  polygons  associated  with 
groups  of  the  remaining  families,  edges  of  the  second  kind  occur. 

A  subdivision  of  some  of  the  families,  is  possible.  It  has  been  proved  that 
the  sum  of  the  angles  in  a  cycle  of  the  first  category  is  a  submultiple  of  27r. 
If  the  sum  is  actually  27r,  the  cycle  is  said  to  belong  to  the  first  sub-category: 
if  it  be  less  than  'Itt  (being  necessarily  a  submultiple),  the  cycle  is  said  to 
belong  to  the  second  sub-category.  And  then,  if  all  the  cycles  of  the  polygon 
belong  to  the  first  sub-category,  the  group  is  said  to  belong  to  the  first  order 
in  the  first  family :  if  the  polygon  have  any  cycle  belonging  to  the  second 

*  Poincar(^  calls  the  group  Fuchsian,  both  when  all  the  coefficients  are  real  and  when  they 
arise  from  the  transformation  of  such  an  infinite  group  by  a  single  substitution  that  has  imaginary 
coefficients.  A  convenient  r^sum^  of  his  results  is  given  by  him  in  a  paper,  Math  Ann.,  t.  xix, 
(1882),  pp.  553—564. 


292.]  GROUPS  741 

sub-category,  the  group  is  said  to  belong  to  the  second  order  in  the  first 
family. 

It  has  been  proved  in  §  288  that  a  corner  belonging  to  a  cycle  of  the 
second  category  is  not  changed  by  the  substitution  which  gives  the  conti- 
guous polygons  in  succession  ;  the  corner  is  a  fixed  point  of  the  substitution, 
so  that  the  substitution  is  either  parabolic  or  hyperbolic.  In  his  arrange- 
ment of  families,  Poincare  divided  the  cycles  of  the  second  category  into 
cycles  of  two  sub-categories,  according  as  the  substitution  is  parabolic  or 
hyperbolic:  but  Klein  proved*  that  there  are  no  cycles  for  hyperbolic 
substitutions,  and  therefore  the  division  is  unnecessary.  The  families  of 
groups,  the  polygons  associated  with  which  have  cycles  of  the  second 
category,  are  the  second,  the  fourth,  the  sixth  and  the  seventh. 

There  is  one  very  marked  difference  between  the  set  of  families,  con- 
sisting of  the  first,  the  second  and  the  sixth,  and  the  set  constituted  by 
the  remainder. 

No  polygon  associated  with  a  real  gToup  in  the  former  set  has  an  edge  of 
the  second  kind :  and  therefore  the  only  points  on  the  axis  taken  account  of 
in  the  division  of  the  plane  are  the  essential  singularities  of  the  group. 
The  domain  of  any  ordinary  point  on  the  axis  in  the  vicinity  of  each  of  the 
essentia]  singularities  is  infinitesimal :  and  therefore  the  axis  of  x  is  taken 
account  of  in  the  division  of  the  plane  only  in  so  far  as  it  contains  essential 
singularities  of  the  group  and  the  functions.  This,  of  course,  applies  equally 
to  the  transformed  configuration  in  which  the  conserved  line  is  the  funda- 
mental circle :  and  therefore,  in  the  division  of  the  area  of  the  circle,  its 
circumference  is  taken  account  of  only  in  so  far  as  it  contains  essential 
singularities  of  the  groups  and  the  functions. 

But  each  polygon  associated  with  a  real  group  in  the  second  set  of 
families  has  an  edge  of  the  second  kind :  the  groups  still  have  all  their 
essential  singularities  on  the  axis  of  x  (or  on  the  fundamental  circle) 
and  at  least  some  of  these  are  isolated  points ;  so  that  the  domain  of  an 
ordinary  point  on  the  axis  is  not  infinitesimal.  Hence  parts  of  the  axis  of 
X  (or  of  the  circumference  of  the  fundamental  circle)  fall  into  the  division  of 
the  bounded  space. 

293.  There  is  a  method  of  ranging  groups  which  is  of  importance  in 
connection  with  the  automorphic  functions  determined  by  them. 

The  upper  half  of  the  plane  of  representation  has  been  divided  into 
curvilinear  polygons ;  it  is  evident  that  the  reflexion  of  the  division,  in  the 
axis  of  real  quantities,  is  the  division  of  the  lower  half  of  the  plane.  Let  the 
polygon  of  reference  in  the  upper  half  be  Rq  and  in  the  lower  half  be  R^', 
obtained  from  R^  by  reflexion  in  the  axis  of  real  quantities.     Then,  if  the 

*  Math.  Ann.,  t.  xl,  (1892),  p.  132. 


742  GENUS  [293. 

group  belong  to  the  set,  which  includes  the  first,  the  second  and  the  sixth 
families,  Rq  and  RJ  do  not  meet  except  at  those  isolated  points,  which  are 
polygonal  corners  of  the  second  category.  But  if  the  group  belong  to  the 
set  which  includes  the  remaining  families,  then  Rq  and  Rq  are  contiguous 
along  all  edges  of  the  second  kind,  and  they  may  be  contiguous  also  at 
isolated  points  as  before. 

In  the  former  case  Ro  and  Rq  may  be  regarded  as  distinct  spaces, 
each  fundamental  for  its  own  half-plane.  Let  Rq  have  2n  edges  which  can 
be  arranged  in  n  conjugate  pairs,  and  let  q  be  the  number  of  cycles  all 
of  which  are  closed ;  each  point  in  one  edge  corresponds  to  a  single  point  in 
the  conjugate  edge.  Let  the  surface  included  by  the  polygon  R^  be  deformed 
and  stretched  in  such  a  manner  that  conjugate  edges  are  made  to  coincide  by 
the  coincidence  of  corresponding  points.  A  closed  surface  is  obtained.  For 
each  pair  of  edges  in  the  polygon  there  is  a  line  on  the  surface,  and  for  each 
cycle  in  the  polygon  there  is  a  point  on  the  surface  in  which  lines  meet ;  and 
the  lines  make  up  a  single  curvilinear  polygon  occupying  the  whole  surface. 
The  process  is  reversible ;  and  therefore  the  connectivity  of  the  surface  is  an 
integer  which  may  properly  be  associated  with  the  fundamental  polygon. 

When  two  consecutive  edges  are  conjugate,  their  common  corner  is  a 

cycle  by  itself     The  line,  made  up  of  these  two  edges  after  the  deformation, 

ends  in  the  common  corner  which  has  become  an  isolated  point ;   this  line 

can   be   obliterated  without   changing  the  connectivity.     The  obliteration 

annuls  two  edges  and  one  cycle  of  the  original  polygon  :  that  is,  it  diminishes 

n  by  unity  and  q  by  unity.     Let  there  be  r  such  pairs  of  consecutive  edges. 

The  deformed  surface  is  now  occupied  by  a  single  polygon,  with  n  —  r  sides 

and  q  —  r  angular  points ;   so  that,  if  its  connectivity  be  2N  +  1,  we  have 

(§  165) 

2N^2  +  {n-r)-l-(q-r) 

=  n  +  l-q. 

The  group  is  said  to  be  of  genus  iV. 

In  the  latter  case,  the  combination  of  R^  and  Rq  may  be  regarded  as 
a  single  region,  fundamental  for  the  whole  plane.  Let  -Ro  have  2n  edges  of 
the  first  kind  and  m  of  the  second  kind,  and  let  q  be  the  number  of  closed 
cycles :  the  number  of  open  cycles  is  m.  Then  Rq  has  2?i  edges  of  the  first 
kind  and  q  closed  cycles;  it  has,  in  common  with  Rq,  the  m  edges  of  the 
second  kind  and  the  m  open  cycles.  The  correspondence  of  points  on  the 
edges  of  the  first  kind  is  as  before.  Let  the  surface  included  by  Rq  and 
Rq  taken  together  be  deformed  and  stretched  in  such  a  manner  that  con- 
jugate edges  coincide  by  the  -coincidence  of  corresponiiing  points  on  those 
edges.  A  closed  surface  is  obtained.  As  the  process  is  reversible,  the 
connectivity  of  the  surface  thus  obtained  'is  an  integer  which  may  properly 
be  associated  with  the  fundamental  polygon. 


293.]  OF  GROUPS  743 

This  integer  is  determined  as  before.  For  each  pair  of  edges  of  the  first 
kind  in  either  polygon,  a  line  is  obtained  on  the  surface ;  so  that  2?2  lines  are 
thus  obtained,  n  from  R^  and  n  from  Eq.  Each  of  the  common  edges  of  the 
second  kind  is  a  line  on  the  surface,  so  that  m  lines  are  thus  obtained.  The 
total  number  of  lines  is  therefore  2n  +  m.  For  each  of  the  closed  cycles 
there  is  a  point  on  the  surface  in  which  lines,  obtained  through  the  deform- 
ation of  edges  of  the  first  kind,  meet :  their  number  is  2q,  each  of  the 
polygons  providing  q  of  them.  For  each  of  the  open  cycles  there  is  a  point 
on  the  surface  in  which  one  of  the  m  lines  divides  one  of  the  n  lines  arising 
through  i?o  from  the  corresponding  line  arising  through  R^  :  the  number  of 
these  points  is  m.     The  total  number  of  points  is  therefore  2q  +  m. 

The  total  number  of  polygons  on  the  surface  is  2.  Hence,  if  the  con- 
nectivity be  2iV  +  l,  we  have  (§  165) 

2iV^  =  2  +  2n+  m  -(2q  +  m)-2 

=  2n  -  2q. 

The  group  is  said  to  be  of  genus  N. 

Thus  for  the  generating  quadrilateral  in  figure  112  (p.  730),  the  genus  of 
the  group  is  zero  when  the  arrangement  of  the  conjugate  pairs  is  1,  2 ;  3,  4 : 
and  it  is  unity  when  the  arrangement  of  the  pairs  is  1,  3 ;  2,  4.  For  the 
generating  hexagon  in  figure  113  (p.  730),  the  genus  of  the  group  is  zero  when 
the  arrangement  of  the  conjugate  pairs  is  1,  6 ;  2,  5 ;  3,  4 :  and  it  is  unity 
when  the  arrangement  of  the  pairs  is  1,  4  ;  2,  5 ;  3,  6.  For  the  generating 
pentagon  in  figure  114  (p.  730),  the  genus  of  the  group  is  zero  when  the 
arrangement  of  the  conjugate  pairs  is  1,3;  4,  5 :  and  it  is  two  when  the 
arrangement  of  the  pairs  is  1,  4;  3,  5.  For  a  generating  polygon,  bounded 
by  2n  semi-circles  each  without  all  the  others  and  by  the  portions  of  the 
axis  of  oc,  the  number  of  closed  cycles  is  zero:  hence  N=n. 

294.  In  all  the  groups,  which  lead  to  a  division  of  a  half-plane  or  of  a 
circle  into  polygons,  the  substitutions  have  real  coefficients  or  are  composed 
of  real  substitutions  and  a  single  substitution  with  complex  coefiicients : 
and  thus  the  variation  in  the  complex  part  of  the  coefficients  in  the  group  is 
strictly  limited.     We  now  proceed  to  consider  groups  of  substitutions 

f      az  +  ^\ 

in  which  the  coefficients  are  complex  in  the  most  general  manner :   such 
groups,  when  properly  discontinuous,  are  called  Kleinian,  by  Poincare. 

The  Fuchsian  groups  conserve  a  line,  the  axis  of  x,  or  a  circle,  the  funda- 
mental circle:  the  Kleinian  groups  do  not  conserve  such  a  line  or  circle, 
common    to    the    group.     Every    substitution   can    be    resolved    into    two 


744  KLEINIAN  [294. 

displacements  of  origin,  an  inversion  and  a  reflexion,  as  in  §  258.  The  inver- 
sion has  for  its  centre  the  point  -  S/7,  being  the  origin  after  the  first  displace- 
ment ;  the  reflexion  is  in  the  line  through  this  point  making  with  the  real 
axis  an  angle  tt  —  2  arg.  7.  The  only  line  left  unaltered  by  these  processes  is 
one  which  makes  an  angle  ^ir  -  arg.  7  with  the  real  axis  and  passes  through 
the  point ;  and  the  final  displacement  to  the  point  a/7  will  in  general  displace 
this  line.  Moreover,  arg.  7  is  not  the  same  for  all  substitutions ;  there  is 
therefore  no  straight  line  thus  conserved  common  to  the  group. 

Similar  considerations  shew  that  there  is  no  fundamental  circle  for  the 
group,  persisting  untransformed  through  all  the  substitutions. 

Hence  the  Kleinian  gi-oups  conserve  no  fundamental  line  and  no  funda- 
mental circle :  when  they  are  used  to  divide  the  plane,  the  result  cannot  be 
similar  to  that  secured  by  the  Fuchsian  groups.  As  will  now  be  proved, 
they  can  be  used  to  give  relations  between  positions  in  space,  as  well  as 
relations  between  positions  merely  in  a  plane. 

The  lineo-linear  relation  between  two  complex  variables,  expressed  as  a 
linear  substitution,  has  been  proved  (§  261)  to  be  the  algebraical  equivalent 
of  any  even  number  of  inversions  with  regard  to  circles  in  the  plane  of  the 
variables.  This  analytical  relation,  when  developed  in  its  geometrical  aspect, 
can  be  made  subservient  to  the  correlation  of  points  in  space. 

Let  spheres  be  constructed  which  have,  as  their  equatorial  circles,  the 
circles  in  the  system  of  inversions  just  indicated  ;  let  inversions  be  now  carried 
out  with  regard  to  these  spheres,  instead  of  merely  with  regard  to  their 
equatorial  circles.  It  is  evident  that  the  consequent  relations  between  points 
in  the  plane  of  the  variable  z  are  the  same  as  when  inversion  is  carried  out 
with  regard  to  the  circles  :  but  now  there  is  a  unique  transformation  of  points 
that  do  not  lie  in  the  plane.  Moreover,  the  transformation  possesses  the 
character  of  conformal  representation,  for  it  conserves  angles  and  it  secures 
the  similarity  of  infinitesimal  figures :  points  lying  above  the  plane  of  z 
invert  into  points  lying  above  the  plane  of  z,  so  that  the  plane  of  z  is 
common  to  all  these  spherical  inversions  and  therefore  common  to  the  sub- 
stitutions, the  analytical  expression  of  which  is  to  be  associated  with  the 
geometrical  operation ;  and  a  sphere,  having  its  centre  in  the  plane  of  the 
complex  z  is  transformed  into  another  sphere,  having  its  centre  in  that  plane, 
so  that  the  equatorial  circles  correspond  to  one  another. 

Through  any  point  P  in  space,  let  an  arbitrary  sphere  be  drawn,  having 
its  centre  in  the  plane  of  the  complex  variable,  say,  that  of  the  coordinates 
^,  7],  It  will  be  transformed,  by  the  various  inversions  indicated,  into  another 
sphere,  having  its  centre  also  in  the  plane  of  |,  tj  and  passing  through  the 
point  Q  obtained  from  P  as  the  result  of  all  the  inversions  ;  and  the  equatorial 
planes  will  correspond  to  one  another. 


294,]  GROUPS  745 

Let  the  sphere  through  Q  he 

or  ^'2  +  T)'-^  +  ^'2 -  2af  -  2br}'  +  k=0. 

Hence,  if  Q  be  determined  by 

this  equation  is  .  p'^  +  h^z'  +  hz^'  +  k  =  0, 

where  —  h,  —1iq=  a  +  ib,  a  —  ib  respectively.  The  equatorial  circle  of  this 
sphere  is  evidently  given  by  ^'  =  0,  so  that  its  equation  is 

z'zq'  +  hoz  +  hz„'  +  k  =  0; 

this  circle  can  be  obtained  from  the  equatorial  circle  of  the  sphere  through  P 

by  the  substitution  z'  = ^  •     Hence  the  latter  circle,  by  §  258,  is  given  by 

2^0  (affo  +  Ao«7o  +  ha^J  +  hjo)  +  ^o  («oyS  +  h^Jo  +  huoB  +  kjoB) 

+  z  (a/8o  +  kc^So  +  hSoi  +  %5o)  +  y3/3o  +  Ih^K  +  ^/SoS  +  khh,  =  0 ; 
and  therefore  the  equation  of  the  sphere  through  P  is 

p-  {aao  +  /;o«7o  +  ha^y  +  kyy^)  +  ^o  ("o^  +  K^Jo  +  hoc,S  +  %§) 

+  z  (a/3o  +  JhdSo  +  A/3o7  +  h^o)  +  yS/3o  +  K^^o  +  h/3oS  +  kS8,  =  0. 

The  quantities  h,  h^,  k  are  arbitrary  quantities,  subject  to  only  the  single 
condition  that  the  sphere  passes  through  the  point  Q:  there  is  no  other 
relation  that  connects  them.  Hence  the  equation  of  the  sphere  through  P 
must,  as  a  condition  attaching  to  the  quantities  h,  ho,  k,  be  substantially  the 
equivalent  of  the  former  condition  given  by  the  equation  of  the  sphere 
through  Q.  In  order  that  these  two  equations  may  be  the  same  for  h,  h^,  k, 
the  variables  p'^,  z' ,  z^  of  the  point  Q  and  those  of  P,  being  p^,  z,  z^,  must  give 
practically  the  same  coefficients  of  h,  Jiq,  k  in  the  two  equations,  and  therefore 

•  p^-  :  p^aa^  +  z.a^^  +  za^^  +  ^^o 

=  z   :  p-ajo  +  Zo^jo  +  zaSo  +  ^8o 

=  Zo  :  p2ao7  +  ^0^0^  +  ^/5o7  +  /^qS 

=  1  :  pVTo  +  -^o7oS  +  ^7^0  +  BSq. 

These  are  evidently  the  equations  which  express  the  variables  of  a  point  Q  in 
space  in  terms  of  the  variables  of  the  point  P,  when  it  is  derived  from  P  by 
the  generalisation  of  the  linear  substitution 

,      aw  +  8 

w  =  ^  : 

yw  +  6 


746  COEEESPONDENCE   OF   POINTS   IN   SPACE  [294. 

they  may  be  called  the  equations  of  the  substitution.     It  is  easy  to  deduce 
that 

r^ 1 

which  may  be  combined  with  the  preceding  equations  of  the  substitution. 

Also,  the  magnification  for  a  single  inversion  is  dsjds,  or  rjr,  where  r^ 
and  r  are  the  distances  of  the  arcs  from  the  centre  of  the  sphere  relative  to 
which  the  inversion  is  effected.  But  rjr  =  ^J^,  where  ^i  and  ^  are  the 
heights  of  the  arcs  above  the  equatorial  plane ;  hence  the  magnification  is 
^i/f,  for  a  single  inversion.  For  the  next  inversion  it  is  ^a/^u  and  therefore  it 
is  ^2/^  for  the  two  together ;  and  so  on.  Hence  the  final  magnification  m 
for  the  whole  transformation  is 


m  = 


r^ 1 

t      ^^77o  +  (7^  +  S)  (70^0  +  So) 
_  1 

a  quantity  that  diminishes  as  the  region  recedes  from  the  equatorial  plane. 

It  is  justifiable  to  regard  the  equations  obtained  as  merely  the  generalisa- 
tion of  the  substitution :  they  actually  include  the  substitution  in  its  original 
application  to  plane  variables.  When  the  variables  are  restricted  to  the  plane 
of  ^,  V>  we  have  p^  =  zzq,  and  therefore 

/  ^  ^■^00:70  +  Zq^jo  +  zaSp  +  /38o  _  az  +  ^ 
^^o77o  +  ^o7oS  +  zy^o  +  8S0      yz  +  B' 

on  the  removal  of  the  factor  70^0  +  ^o  common  to  the  numerator  and  the 
denominator;  and  ^'  vanishes  when  ^=0.  The  uniqueness  of  the  result  is 
an  a  posteriori  justification  of  the  initial  assumption  that  one  and  the  same 
point  Q  is  derived  from  P,  whatever  be  the  inversions  that  are  equivalent  to 
the  linear  substitution. 

Ex.  1.     Let  an  elliptic  substitution  have  ti  and  v  as  its  fixed  points. 

Draw  two  circles  in  the  plane,  passing  through  u  and  v  and  intersecting  at  an  angle 
equal  to  half  the  argument  of  the  multiplier.  The  transformation  of  the  plane,  caused  by 
the  substitution,  is  equivalent  to  inversions  at  these  cii'cles  ;  the  corresponding  transforma- 
tion of  the  space  above  the  plane  is  equivalent  to  inversions  at  the  spheres,  having  these 
circles  as  equatorial  circles.  It  therefore  fallows  that  every  point  on  the  line  of  intersection 
of  the  spheres  remains  unchanged :  hence  when  a  Kleinian  substitution  is  elliptic,  evet'y 
point  on  the  circle^  in  a  plane  perpendicular  to  the  plane  of  x,  y  and  having  the  line  joining 
the  common  points  of  the  substitution  as  its  diameter,  is  unchanged  by  the  substitution. 
Poincare  calls  this  circle  C  the  double  (or  fixed)  circle  of  the  elliptic  substitution. 

.  Ex.  2.  Prove  that,  when  a  Kleinian  substitution  is  hyperbolic,  the  only  points  in 
space,  which  are  unchanged  by  it,  are  its  double  points  in  the  plane  of  x,  y ;  and  shew 
that  it  changes  any  circle  through  those  points  into  itself  and  also  any  sphere  through 
those  points  into  itself. 


294.]  KLEINIAN   GEOUPS  747 

Ex.  3.  •  Prove  that,  when  the  substitution  is  loxodromic,  the  circle  C,  in  a  plane 
perpendicular  to  the  plane  x,  y  and  having  as  its  diameter  the  line  joining  the  common 
points  of  the  substitution,  is  transformed  into  itself,  but  that  the  only  points  on  the 
circumference  left  unchanged  are  the  common  points. 

Ex.  4.     Obtain  the  corresponding  properties  of  the  substitution  when  it  is  parabolic. 
(All  these  results  are  due  to  Poincare.) 

295.  The  process  of  obtaining  the  division  of  the  ^^-plane  by  means  of 
Kleinian  groups  is  similar  to  that  adopted  for  Fuchsian  groups,  except 
that  now  there  is  no  axis  of  real  quantities  or  no  fundamental  circle 
conserved  in  that  plane  during  the  substitutions :  and  thus  the  whole 
plane  is  distributed.  The  polygons  will  be  bounded  by  arcs  of  circles  as 
before :  but  a  polygon  will  not  necessarily  be  simply  connected.  Multiple 
connectivity  has  already  arisen  in  connection  with  real  groups  of  the  third 
family  by  taking  the  plane  on  both  sides  of  the  axis. 

As  there  are  no  edges  of  the  second  kind  for  polygons  determined  by 
Kleinian  groups,  the  only  cycles  of  corners  of  polygons  are  closed  cycles  ; 
let  A^,  A-^,  ...,  An-i  in  order  be  such  a  cycle  in  a  polygon  R^.  Round  Aq 
describe  a  small  curve,  and  let  the  successive  polygons  along  this  curve  be 
Ra,  R^,  ...,  Rn-1,  Rn,  ••••  The  corner  ^o  belongs  to  each  of  these  polygons: 
when  considered  as  belonging  to  R^,  it  will  in  that  polygon  be  the  homologue 
of  Am  as  belonging  to  R^,  if  m<  n;  but,  as  belonging  to  Rn,  it  will,  in  that 
polygon,  be  the  homologue  of  Aq  as  belonging  to  R^.  Hence  the  substitution, 
which  changes  Ra  into  Rn,  has  Aq  for  a  fixed  point. 

This  substitution  may  be  either  elliptic  or  parabolic,  (but  not  hyperbolic, 
I  292):  that  it  cannot  be  loxodromic  may  be  seen  as  follows.  Let  pe*"  be 
the  multiplier,  where  (§  259)  p  is  not  unity  and  a  is  not  zero :  and  let 
2o  denote  the  aggregate  of  polygons  Rq,  R^,  ...,  Rn-i,  2i  the  aggregate 
Rn,  ...,  Rm-i,  and  so  on.  Then  Xo  is  changed  to  2i,  2i  to  Sa,  and  so  on, 
by  the  substitution.  Let  p  be  an  integer  such  that  ptw  ^  27r ;  then,  when 
the  substitution  has  been  applied  p  times,  the  aggregate  of  the  polygons 
is  1p,  and  it  will  cover  the  whole  or  part  of  one  of  the  aggregates  2o,  Sj,  .... 
But,  because  p^  is  not  unity,  Xp  does  not  coincide  with  that  aggregate  or  the 
part  of  that  aggregate :  the  substitution  is  not  then  properly  discontinuous, 
contrary  to  the  definition  of  the  group.  ,  Hence  there  is  no  loxodromic 
substitution  in  the  group.  If  the  substitution  be  elliptic,  the  sum  of  the 
angles  of  the  cycle  must  be  a  submultiple  of  27r ;  when  it  is  parabolic,  each 
angle  of  the  cycle  is  zero. 

In  the  generalised  equations  whereby  points  of  space  are  transformed 
into  one  another,  the  plane  of  x,  y  is  conserved  throughout :  it  is  natural 
therefore  to  consider  the  division  of  space  on  the  positive  side  of  this  plane 
into  regions  Po,  Pi,  ...,  such  that  Pq  is  changed  into  all  the  other  regions  in 
turn  by  the  application  to  it  of  the  generalised  equations.     The  following 


748  DIVISION  OF  SPACE  [295. 

results  can  be  obtained  by  considerations  similar  to  those  before  adduced  in 
the  division  of  a  plane*. 

The  boundaries  of  regions  are  either  portions  of  spheres,  having  their 
centres  in  the  plane  of  x,  y,  or  they  are  portions  of  that  plane :  the 
regions  are  called  polyhedral,  and  such  boundaries  are  called  faces.  If  the 
face  is  spherical,  it  is  said  to  be  of  the  first  kind  :  if  it  is  a  portion  of 
the  plane  of  oc,  y,  it  is  said  to  be  of  the  second  kind.  Faces  of  the 
second  kind,  being  in  the  plane  of  x,  y  and  transformed  into  one  another, 
are  polygons  bounded  by  arcs  of  circles. 

The  intersections  of  faces  are  edges.  Again,  an  edge  is  of  the  first 
kind,  when  it  is  the  intersection  of  two  faces  of  the  first  kind :  it  is  of 
the  second  kind,  when  it  is  the  intersection  of  a  face  of  the  first  kind 
with  one  of  the  second  kind.  An  edge  of  the  second  kind  is  a  circular 
arc  in  the  plane  oi  x,  y :  an  edge  of  the  first  kind,  being  the  intersection 
of  two  spheres  with  their  centres  in  the  plane  of  x,  y,  is  a  circular  arc, 
which  lies  in  a  plane  perpendicular  to  the  plane  of  x,  y  and  has  its 
centre  in  that  plane. 

The  extremities  of  the  edges  are  corners  of  the  polyhedra.  They  are 
of  three  categories : 

(i)  those  which  are  above  the  plane  of  x,  y  and  are  the  common 
extremities  of  at  least  three  edges  of  the  first  kind : 

(ii)  those  which  lie  in  the  plane  of  x,  y  and  are  the  common  extremities 
of  at  least  three  edges  of  the  first  kind  : 

(iii)  those  which  lie  in  the  plane  of  x,  y  and  are  the  common  extremities 
of  at  least  one  edge  of  the  first  kind  and  of  at  least  two  edges  of 
the  second  kind. 

Moreover,  points  at  which  two  faces  touch  can  be  regarded  as  isolated  corners, 
the  edges  of  which  they  are  the  intersections  not  being  in  evidence. 

Faces  of  a  polyhedron,  which  are  of  the  first  kind,  are  conjugate  in  pairs: 
two  conjugate  faces  are  congruent  by  a  fundamental  substitution  of  the  group. 

Edges  of  the  first  kind,  being,  the  limits  of  the  faces,  arrange  themselves 
in  cycles,  in  the  same  way  as  the  angles  of  a  polygon  in  the  division  of  the 
plane.  If  E^,  E^,  ...,  En-i  be  the  n  edges  in  a  cycle,  the  number  of  regions 
which  have  an  edge  in  Eo  is  a  multiple  of  n :  and  the  sum  of  the  dihedral 
angles  at  the  edges  in  a  cycle  (the  dihedral  angle  at  an  edge  being  the 
constant  angle  between  the  faces,  which  intersect  along  the  edge)  is  a 
submultiple  of  27r. 

The  relation  between  the  polyhedral  divisions  of  space  and  the  polygonal 
divisions  of  the  plane  is  as  follows.     Let  the  group  be  such  as  to  cause  the 

*  See,  in  particular,  Poincar^,  Acta  Math.,  t.  iii,  pp.  66  et  seq. 


295.]  FUNDAMENTAL   POLYHEDRA   AND   POLYGONS  749 

fundamental  polyhedron  Pq  to  possess  n  faces  of  the  second  kind,  say  ^oi, 
^02,  •••,  F^n-  Every  congruent  polyhedron  will  then  have  n  faces  of  the 
second  kind;  let  those  of  Pg  be  jP^i,  -^82,  •••,  Fm-  Every  point  in  the  plane 
of  X,  y  belongs  to  some  one  of  the  complete  set  of  faces  of  the  second  kind : 
and,  except  for  certain  singular  points  and  certain  singular  lines,  no  point 
belongs  to  more  than  one  face,  for  the  proper  discontinuity  of  the  group 
requires  that  no  point  of  space  belongs  to  more  than  one  polyhedron. 

Then  the  plane  of  x,  y  is  divided  into  n  regions,  say  Z)i,  Dg,  •••,  Dn',  each 
of  these  regions  is  composed  of  an  infinite  number  of  polygons,  consisting  of 
the  polygonal  faces  F.  Thus  D,.  is  composed  of  F^^.,  F-^^,  F.^r,  ... ;  and  these 
polygonal  areas  are  such  that  the  substitution  Sg  transforms  -For  into  Fg^. 
Hence  it  appears  that,  by  a  Kleinian  group,  the  whole  plane  is  divided  into 
a  finite  number  of  regions ;  and  that  each  region  is  divided  into  an  infinite 
number  of  polygons,  which  are  congruent  to  one  another  by  the  substitutions 
of  the  group. 

296.  The  preceding  groups  of  substitutions,  that  have  complex  co- 
eflficients,  have  been  assumed  to  be  properly  discontinuous. 

JSx.  Prove  that,  if  any  group  of  substitutions  with  complex  coefficients  be  improperly 
discontinuous,  it  is  improperly  discontinuous  only  for  points  in  the  plane  of  x,  y. 

(Poincare.) 

One  of  the  simplest  and  most  important  of  the  improperly  discontinuous 
groups  of  substitutions,  is  that  compounded  from  the  three  fundamental 
substitutions 

z  =Sz  =  z-^\,     z  =-Tz  = ,     z'  =Yz  =  z  +  %, 

z 

where  %  has  the  ordinary  meaning.  All  the  substitutions  are  easily  proved  to 
be  of  the  form 

a^  +  /3 
7^  -f-  S ' 

where  aS  —  ^^7  =  1,  and  a,  y8,  7,  S  are  complex  integers,  that  is,  are  represented 
by  m  +  ni,  where  m  and  n  are  integers.  This  is  the  evident  generalisation  of 
the  modular-function  group:  consequently  there  is  at  once  a  suggested 
generalisation  to  a  polyhedron  of  reference,  bounded  by 

which  will  thus  have  one  spherical  and  four  (accidentally)  plane  faces. 

The  following  method  of  consideration  of  the  points  included  by  the 
polyhedron  of  reference  differs  from  that  which  was  adopted  for  the  polygon 
of  reference  in  the  plane. 

If  possible,  let  a  point  (|,  77,  ^)  lying  within  the  above  region  be  transformed 
by  the  equations  generalised  from  some  one  substitution  of  the  group,  say 


750  EXAMPLE   OF   AN   IMPROPERLY  [296. 

from  "^     \, ,  into  another  point  of  the  region,  say  ^',  rj',  ^'.     Then  we  have 
<yz  +  o 

i>l>-i>    4>^>-i,    r  +  ^^  +  r>i- 

From  the  last,  it  follows  that  ^  >  ^^ :  and  similarly  for  |',  r]',  ^',  by  the 

\/  z 

hypothesis  that  the  point  is  in  the  region.     Now 

^' 1 1 

and  therefore  1/(?D  =  l7p  +  p  |7^  +  ^P- 

Hence,  as  ^  and  ^'  are  both  >  -7^ ,  we  have  |7|^  <  2 :   so  that,  because  7  is 

a  complex  integer,  we  have 

7  =  0,     +  1,     +  i 
as  the  only  possible  cases. 

If  ry  =  0,  then  since  aS  —  ^y  —  1,  we  have  a8  =  1  and  a,  S  are  complex 
integers :   thus  either 

a=l)  a=:_l]  a=     i]  a  —  —  i 


For  the  first  of  these  sub-cases  we  have,  from  the  equations  of  the  substitu- 
tion, 

where  yS  is  a  complex  integer :  if  the  new  point  lie  within  the  region,  then 
/3  =  0,  and  we  have 

z'  =  z,     K'=t 
which  is  merely  an  identity. 

For  the  second,  we  have  z  =  z  —  ^  :  leading  to  the  same  result. 

For  the  third,  we  have,  since  So  =  i, 

z  =  —  z-\- 1/3. 

But  as  j  I'  1 ,  1 7;'  1 ,  11^  1 ,  1 77 1  are  all  less  than  \,  we  have  /3  =  0,  and  so 

f  =  -|,  v'  =  -v;   and  ^'=C 
For  the  fourth  case,  we  have 

z'  =  —  z  —  i^, 

leading  to  the  same  result  as  the  third.     Hence,  if  7  =  0,  the  only  point  lying 
within  the  region  is  given  by 

determined  by  the  substitution  w'  =  — . ,  which  is  TVT~^  V~^TV. 


296.]  DISCONTINUOUS   GROUP  751 

If  I7I  =  1,  that  is,  77o=  1,  then 

,         p  =  p''  +  2;oyo8  +  zyBo  +  8So. 

Of  the  two  quantities  ^  and  f ',  one  will  be  not  greater  than  the  other :  we 
choose  ^  to  he  that  one  and  consider  the  accordingly  associated  substitution*. 
Thus  ^/r^l,  p2>l,  andso 

^o7oS  4-  ^7^0  +  5^0  <  0, 

say  ^„l  +  ^io^Sao^()_ 

7         To       7  7o 

Now  I7I  =  1,  so  that  -  is  of  the  form  p  +  iq,  where  p  and  q  are  integers  :  thus 
Ave  have 

p''  +  q^+  2p^  +  2qr)  <  0, 

which  is  impossible  because  2f  <  1,  2?;  <  1. 

Hence  it  follows  that  within  the  region  there  are  only  two  equivalent 
points,  derived  by  the  generalised  equations  from  the  substitution 


f       %w 
w  = -. 

—  I 


and  that  all  points  within  the  region  can  be  arranged  in  equivalent  pairs 

^,  77,  ^     and     -^,-v,^- 

If  the  region  be  symmetrically  divided  into  two,  so  that  the  boundaries  of 
a  new  region  are 

then  no  point  within  the  new  region  is  equivalent  to  any  other  point  in  the 
regionf.  As  in  the  division  of  the  plane  by  the  modular  group,  it  is  easy 
to  see  that  the  whole  space  above  the  plane  of  ^,  ij  is  divided  by  the  group : 
therefore  the  region  is  a  polyhedron  of  reference  for  the  group  composed  of  the 
fundamental  substitutions  S,  T,  V. 

The  preceding  substitutions,  with  complex  integers  for  coefficients,  are  of  use  in  appli- 
cations to  the  discussion  of  binary  quadratic  forms  in  the  theory  of  numbers.  The  special 
division  of  all  space  corresponds,  of  course,  to  the  character  of  the  coefficients  in  the 
substitutions  :  other  divisions  for  similar  groups  are  possible,  as  is  proved  in  Poincare's 
memoir  already  quoted. 

*  Were  it  f ',  all  that  would  be  necessary  would  be  to  take  the  inverse  substitution, 
t  Bianchi,  Math.  Ann.,  t.  xxxviii,  (1891),  pp.  313—324,  t.  xl,  (1892),  pp.  332—412;  Picard,  ib  , 
t.  xxxix,  (1891),  pp.  142—144;  Mathews,  Quart.  Journ.  Math.,  vol.  xxv,  (1891),  pp.  289—296. 


752  EXAMPLE  [296. 

These  divisions  all  presuppose  that  the  group  is  infinite :  but  similar  divisions  for  only- 
finite  groups  (and  therefore  with  only  a  finite  number  of  regions)  are  possible.  These  are 
considered  in  detail  in  an  interesting  memoir  by  Goursat* ;  the  transformations  conserve 
an  imaginary  sphere  instead  of  a  real  plane  as  in  Poincare's  theory. 

Ex.     Shew  that,  for  the  infinite  group  composed  of  the  fundamental  substitutions 

z  = ,     2=0  +  1,     z=z  +  e, 

z 

where  e  is  a  primitive  cube  root  of  unity,  a  fundamental  region  for  the  division  of  space 
above  the  plane  of  z,  corresponding  to  the  generalised  equations  of  the  group,  is  a  sym- 
metrical third  of  the  polyhedron  extending  to  infinity  above  the  sphere 

and  bounded  by  the  sphere  and  the  six  planes 

2£=±1,     |  +  W3=±1,     |-W3=±1-  (Bianchi.) 

*  "  Sur  les  substitutions  orthogonales  et  les  divisions  regulieres  de  I'espace,"  Ann.  de  VEc. 
Norm.  Sup.,  3'"'=  Ser.,  t.  vi,  (1889),  pp.  9—102.     See  also  Schonflies,  Math.  Ann.,  t.  xxxiv,  (1889),  , 
pp.  172 — 203  :  other  references  are  given  in  these  papers. 


CHAPTEE   XXII. 

AuTOMORPHic  Functions.  , 

297.  As  was  stated  in  the  course  of  the  preceding  chapter,  we  are 
seeking  the  most  general  form  of  the  arguments  of  functions  which  secures 
the  property  of  periodicity.  The  transformation  of  the  arguments  of  trigo- 
nometrical and  of  elliptic  functions,  which  secures  this  property,  is  merely  a 
special  case  of  a  linear  substitution :  and  thus  the  automorphic  functions  to 
be  discussed  are  such  as  identically  satisfy  the  equation 

where  Si  is  any  one  of  an  assigned  group  of  linear  substitutions  of  which  only 
a  finite  number  are  fundamental. 

Various  references  to  authorities  will  be  given  in  the  present  chapter,  in  connection 
with  illustrative  examples  of  automorphic  functions  :  but  it  is,  of  course,  beyond  the  scope 
of  the  present  treatise,  deahng  only  with  the  generalities  of  the  theory  of  functions,  to 
enter  into  any  detailed  development  of  the  properties  of  special  classes  of  automorphic 
functions  such  as,  for  instance,  those  commonly  called  polyhedral  and  those  commonly 
called  elliptic-modular.  Automorphic  functions,  of  types  less  special  than  those  just 
mentioned,  are  called  Ftichsian  functions  by  Poincare,  when  they  are  determined  in 
association  with  a  Fuchsian  group  of  substitutions,  and  Kleinian  functions,  when  they 
are  determined  in  association  with  a  Kleinian  group :  as  our  purpose  is  to  provide  only 
an  introduction  to  the  theory,  the  more  general  term  automorphic  will  be  adopted. 

The  establishment  of  the  general  classes  of  automorphic  fimctions  is  eflfected  by 
Poincare  in  his  memoirs  in  the  early  volumes  of  the  A  eta  Mathematica,  and  by  Klein  in 
hie  memoir  in  the  21st  volume  of  the  Mathematische  Annalen :  these  have  been  already 
quoted  (p.  716,  note) :  and  Poincare  gives  various  historical  notes*  on  the  earlier  scattered 
occurrences  of  automorphic  functions  and  discontinuous  groups.  Other  memoirs  that  may 
be  consulted  with  advantage  are  those  of  Von  Mangoldtt,  Weber |,  Schottky§,  Stahl||, 

*  Acta  Math.,  t.  i,  pp.  61,  62,  293:  ib.,  t.  iii,  p.  92.  Poincare's  memoirs  occur  in  the  first, 
third,  fourth  and  fifth  volumes  of  this  journal :  a  great  part  of  the  later  memoirs  is  devoted  to 
their  application  to  linear  differential  equations. 

t  Gbtt.  Nadir.,  (1885),  pp.  313—319;  ib.,  (1886),  pp.  1—29. 

J  Gbtt.  Nachr.,  (1886),  pp.  359—370. 

§  Crelle,  t.  ci,  (1887),  pp.  227—272. 

II  Math.  Ann.,  t.  xxxiii,  (1889),  pp.  291—309. 

F.  F.  4S 


754  ANHARMONIC  GROUP 'AND   FUNCTION  [297. 

Schlesinger*  and  Rittert :  and  there  are  two  by  BurnsideJ,  of  special  interest  and 
importance  in  connection  with  the  third  of  the  seven  families  of  groups  (§  292).  Finally, 
reference  may  be  made  to  the  comprehensive  treatise**  by  Fricke  and  Klein. 

298.  We  shall  first  consider  functions  associated  with  finite  discrete 
groups  of  linear  substitutions. 

There  is  a  group  of  six  substitutions 

1  1  2-1       ^_ 

z  \  —  z         z         z—\ 

which  (§  283)  is  complete.      Forming  expressions  z  —  x,  z ,  z  —  ^Y—x), 

1  QC  *^  1  OC 

z  — ,  z ,  z z.  and  multiplying  them  together,  we  can  express 

X  ~"  *ju  OG  3G  "^  ±. 

their  product  in  the  form 

(,-> _  ,y  [(^^-^+1)^ _  {x^-x  +  m 
^        ^  \    {z'-zf  {a?-xy    I' 

so  that  A  {z)  =  ^-— -^ 

{z^  —  zf 

is  a  function  of  z  which  is  unaltered  by  any  of  the  transformations  of  its 

variable  given  by  the  six  substitutions  of  the  group.     The  function  is  well 

known,  being  connected  with  the  six  anhannonic  ratios  of  four  points  in  a 

line  which  can  all  be  expressed  in  terms  of  any  one  of  them  by  means  of  the 

substitutions. 

Another  illustration  of  a  finite  discrete  group  has  already  been  furnished 

in   the   periodic   elliptic    transformation   of  §  258,    whereby  a   crescent   of 

the  plane  with  its  angle  a  subijiultiple  of  27r  was  successively  transformed, 

ultimately  returning  to  itself:  so  that  the  whole  plane  is  divided  into  portions 

equal  in  number  to  the  periodic  order  of  the  substitution. 

If  a  stereographic  projection  of  the  plane  be  made  with  regard  to  any 
external  point,  we  shall  have  the  whole  sphere  divided  into  a  number  of 
triangles,  each  bounded  by  two  small  circles  and  cutting  at  the  same  angle. 
By  choice  of  centre  of  projection,  the  common  corners  of  the  crescents  can  be 
projected  into  the  extremities  of  a  diameter  of  the  sphere :  and  then  each  of 
the  crescents  is  projected  into  a  lune.  The  effect  of  a  substitution  on  the 
crescent  is  changed  into  a  rotation  round  the  diameter  joining  the  vertices 
of  a  lune  through  an  angle  equal  to  the  angle  of  the  lune. 

299.  This  is  merely  one  particular  illustration  of  a  general  correspondence 
between  spherical  rotations  and  plane  homographies,  as  we  now  proceed  to 
shew.  The  general  correspondence  is  based  upon  the  following  proposition 
due  to  Cayley  : — 

*  CreMe,  t.  cv,  (1889),  pp.  181—232. 
t  Math.  Ann.,  t.  xli,  (1892),  pp.  1—82. 

X  Loud.  Math.  Soc.  Proc,  vol.  xxiii,  (1892),  pp.  48—88,  ib.,  pp.  281—295. 
**  Vorlesungen  ilber  die  Theorie  der  automorphen  Functionen,  (Leipzig,  Teubner,  Bd.  i,  1897). 


299.]  HOMOGRAPHY   AND   ROTATIONS  755 

When  a  sphere  is  displaced  hy  a  rotation  round  a  diameter,  the  variables  of 
the  stereographic  projections  of  any  point  in  its  original  position  and  in  its 
displaced  position  are  connected  hy  the  relation 

,  _{d-\-  ic)  z  —  {h  —  ia) 
{b  4-  ia)  z  +  (d  -  ic)  ' 

where  a,  b,  c,  d  are  7'eal  quantities. 

Rotation  about  a  given  diameter  through  an  assigned  angle  gives  a 
unique  position  for  the  displaced  point:  and  stereographic  projection,  which 
is  a  conformal  operation  in  that  it  preserves  angles,  also  gives  a  unique  point 
as  the  projection  of  a  given  point.  Hence  taking  the  stereographic  projec- 
tion on  a  plane  of  the  original  position  and  the  displaced  position  of  a  point 
on  the  sphere,  they  will  be  uniquely  related :  that  is,  their  complex  variables 
are  connected  by  a  lineo-linear  relation,  which  thus  leads  to  a  linear  substitu- 
tion for  the  plane-transformation  corresponding  to  the  spherical  rotation. 

Now  the  extremities  of  the  axis  are  unaltered  by  the  rotation ;  hence  the 
projections  of  these  points  are  the  fixed  points  of  the  substitution.  If  the 
points  be  ^,  ??,  ^  and  —  f ,  —  77,  —  ^,  on  a  sphere  of  radius  unity,  and  if  the 
origin  of  projection  be  the  north  pole  of  the  sphere,  the  fixed  points  of  the 
substitution  are 

^^^''^    and    -l+h- 

so  that  the  substitution  is  of  the  form 


,      ^  +  ir]  ^  +  ir)' 

To  determine  the  multiplier  K,  we  take  a  point  P  very  near  C,  one  extremity 
of  the  axis  :  let  P'  be  the  position  after  the  rotation,  so  that  CF'  =  CP.  Then, 
in  the  stereographic  projection,  the  small  arcs  which  correspond  to  OP  and 
CP'  are  equal  in  length,  and  they  are  inclined  at  an  angle  a.     Hence  the 

multiplier  K  is  e^'* :  for  when  z,  and  therefore  z',  is  nearly  equal  to  —  ■-,      y  ,  a 

fixed  point  of  the  substitution,  the  magnification  is  |  ^  |  and  the  angular 
displacement  is  the  argument  of  K,  which  is  a. 

Inserting    the   value    of   K,   solving    for    z     and    using    the    condition 
^^  +  7}^  +  ^^=  1,  we  have 

,  _{d  +  ic)  z  —  (b-  ia) 
~  {b  +  ia)  z  +  (d-  ic) ' 

where  a  =  |^ sin  |a,     b  =  rj  sin  \a,     c=  ^ sin -|a,     d  =  cos ^a, 

so  that  a^  +  ¥  +  c''  +  d^  =  1, 

the  equivalent  of  the  usual  condition  to  which  the  four  coefficients  in  any 

48—2 


756  HOMOGENEOUS  SUBSTITUTIONS  [299. 

linear  substitution  are  subject :  it  is  evident  that  the  substitution  is  elliptic. 

The  proposition*  is  thus  proved. 

When  the  axis  of  rotation  is  the  diameter  perpendicular  to  the  plane,  we 

have,  by  §  256, 

z  =  ke-^+i^,        z'  =  A;e-^+^<*+»', 

so  that  /  =  ^e*", 

agreeing  with  the  above  result  by  taking  ^=0  =  77,  ^=1,  so  that  a  =  0  =  6, 

c  =  sin  |a,  d  =  cos  ^a. 

It  should  be  noted  that  the  formula  gives  two  different  sets  of  coefficients 
for  a  single  rotation :  for  the  effect  of  the  rotation  is  unaltered  when  it  is 
increased  by  27r,  a  change  in  a  which  leads  to  the  other  signs  for  all  the 
constants  a,  b,  c,  d. 

It  thus  appears  that  the  rotation  of  a  sphere  about  a  diameter  interchanges 
pairs  of  points  on  the  surface,  the  stereographic  projections  of  which  on  the 
plane  of  the  equator  are  connected  by  an  elliptic  linear  substitution :  hence, 
in  the  one  case  as  in  the  other,  the  substitution  is  periodic  when  a,  the 
argument  of  the  multiplier  and  the  angle  of  rotation,  is  a  submultiple  of  27r. 

In  the  discussion  of  functions  related  in  their  arguments  to  these  linear 
substitutions,  it  proves  to  be  convenient  to  deal  with  homogeneous  variables, 
so  that  the  algebraic  forms  which  arise  can  be  connected  with  the  theory  of 
invariants.  We  take  zz^  =  z^ :  the  formulae  of  transformation  may  then  be 
represented  by  the  equations 

z^  =  K  {az^  +  ^z^,  z^  =  K  {'yz^  +  hz^, 
for  the  substitution  z'  =  {az  +  ,8)/(yz  +  8).  As  we  are  about  to  deal  with 
invariantive  functions  of  position  dependent  upon  rotations,  it  is  important 
to  have  the  determinant  of  homogeneous  transformation  equal  to  unity. 
This  can  be  secured  only  if  k  =  +  1  or  if  «  =  —  1 :  the  two  values  correspond 
to  the  two  sets  of  coefficients  obtained  in  connection  with  the  rotation. 
Hence,  in  the  present  case,  the  formulae  of  homogeneous  transformation  are 

z-i'  =^{d  +  ic)  Zi  —  {b  —  ia)  z^,     z^  =  (6  +  ia)  z-^-\-id  —  ic)  z^, 
where  a^  +  6^  +  c^  +  rf^,  being  the  determinant  of  the  substitution,  =  1 ;  every 
rotation  leads  to  two  pairs  of  these  homogeneous  equations  f.     Each  pair  of 
equations  will  be  regarded  as  giving  a  homogeneous  substitution. 

Moreover,  rotations  can  be  compounded :  and  this  composition  is,  in  the 
analytical  expression  of  stereographically  projected  points,  subject  to  the  same 
algebraic  laws  as  is  the  composition  of  linear  substitutions.     If,  then,  there 

*  Cayley,  Math.  Ami.,  t.  xv,  (1879),  pp.  238—240;  Klein's  Vorlesungen  ilber  das  Ikosaeder, 
pp.  32—34. 

+  The  succeeding  account  of  the  polyhedral  functions  is  based  on  Klein's  investigations,  which 
are  collected  in  the  first  section  of  his  Vorlesungen  ilber  das  Ikosaeder  (Leipzig,  Teubner,  1884):  see 
also  Cayley,  Camb.  Phil.  Trans.,  vol.  xiii  (1883),  pp.  4—68;  Coll.  Math.  Papers,  vol.  xi,  pp.  148—216. 

It  will  be  seen  that  the  results  are  intimately  related  to  the  results  obtained  in  §§  271 — 279, 
relative  to  the  eonformal  representation  of  figures,  bounded  by  circular  arcs,  on  a  half-plane. 


299.]  GROUPS  FOR  THE  REGULAR  SOLIDS  757 

be  a  complete  group  of  rotations,  that  is,  a  group  such  that  the  composition 
of  any  two  rotations  (including  repetitions)  leads  to  a  rotation  included  in  the 
group,  then  there  will  be  associated  with  it  a  complete  group  of  linear 
homogeneous  substitutions.  The  groups  are  finite  together,  the  number  of 
members  in  the  group  of  homogeneous  substitutions  being  double  of  the 
number  in  the  group  of  rotations :  and  the  substitutions  can  be  arranged  in 
pairs  so  that  each  pair  is  associated  with  one  rotation. 

300.  Such  groups  of  rotations  arise  in  connection  with  the  regular  solids. 
Let  the  sphere,  which  circumscribes  such  a  solid,  be  of  radius  unity :  and  let 
the  edges  of  the  solid  be  projected  from  the  centre  of  the  sphere  into  arcs  of 
great  circles  on  the  surface.  Then  the  faces  of  the  polyhedron  will  be  repre- 
sented on  the  surface  of  the  sphere  by  closed  curvilinear  figures,  the  angular 
points  of  which  are  summits  of  the  polyhedron.  There  are  rotations,  of  proper 
magnitude,  about  diameters  properly  chosen,  which  displace  the  polyhedron 
into  coincidence  (but  not  identity)  with  itself,  and  so  reproduce  the  above- 
mentioned  division  of  the  surface  of  the  sphere  :  when  all  such  rotations  have 
been  determined,  they  form  a  group  which  may  be  called  the  group  of  the 
solid.  Each  such  rotation  gives  rise  to  two  homogeneous  substitutions,  so 
that  there  will  thence  be  derived  a  finite  group  of  discrete  substitutions : 
and  as  these  are  connected  with  the  stereographic  projection  of  the  sphere, 
they  are  evidently  the  group  of  substitutions  which  transform  into  one 
another  the  divisions  of  the  plane  obtained  by  taking  the  stereographic 
projection  of  the  corresponding  division  of  the  surface  of  the  sphere.  For 
the  construction  of  such  groups  of  substitutions,  it  will  therefore  be  sufficient 
to  obtain  the  groups  of  rotations,  considered  in  reference  to  the  surface  of 
the  sphere. 

I.  The  Dihedral  Group.  The  simplest  case  is  that  in  which  the  solid, 
hardly  a  proper  solid,  is  composed  of  a  couple  of  coincident  regular  polygons 
of  n  sides*  :  a  reference  has  already  been  made  to  this  case.  We  suppose  the 
polygons  to  lie  in  the  equator,  so  that  their  corners  divide  the  equator  into 
n  equal  parts :  one  polygon  becomes  the  upper  half  of  the  spherical  surface, 
the  other  the  lower  half  The  two  poles  of  the  equator,  and  the  middle 
points  of  the  n  arcs  of  the  equator,  are  the  corners  of  the  corresponding  solid. 

Then  the  axes,  rotations  about  which  can  bring  the  surface  into  such 
coincidence  with  itself  that  its  partition  of  the  spherical  surface  is  topo- 
graphically the  same  in  the  new  position  as  in  the  old,  are 

(i)    the  polar  axis, 

(ii)   a  diameter  through  each  summit  on  the  equator, 

(iii)  a  diameter  through  each  middle  point  of  an  edge : 

the  last  two  are  the  same  or  are  different  according  as  n  is  odd  or  is  even. 

*  The  solid  may  also  be  regarded  as  a  doable  pyramid. 


758  DIHEDEAL  GROUP   AND   FUNCTION  [300. 

For  the  polar  axis,  the  necessary  angle  of  rotation  is  an  integral  multiple 

27- 
of  — .     Thus  we  have  ^  =  0  =  v,  ^=1,  and  therefore    ■ 
n 

a  =  0  =  6,     c  =  sm  -  ,     ct  =  cos  — , 
n  n 

the  substitutions  are 

iirr  JTT 

Z-i  =  6      Zi,       Z^  =  6  Z<^ , 

for  r  =  0,  1,  ...,n  —  1,  and 

iirr  iirr 

z(r=  —  e'^z^,     %'  =  — e    "•^2, 
for  the  same  values  of  r.     These  are  included  in  the  set 

inr  inr 

2r/=e^^i,     zl  =  e    "^  z^, 
forr  =  0,  1,  2,  ...,  2^1  — 1,  being  ^n  in  number:   the  identical  substitution  is 
included   for  the  same  reason  as  before,  when  we  associated  a  region  of 
reference  in  the  ^r-plane  with  the  identical  substitution. 

For  each  of  the  axes  lying  in  the  equator,  the  angle  of  rotation  is 
evidently  tt.  Let  an  angular  point  of  the  polygon  lie  on  the  axis  of  |,  say  at 
I  =  1,  77  =  0,  ^=0.     Then  so  far  as  concerns  (ii)  in  the  above  set,  if  we  take 

the  axis  through  the  (r  +  l)th  angular  point,  we  have  |  =  cos ,  77  =  sin , 

^=  0 ;  hence,  as  a  is  equal  to  tt,  we  have,  for  the  corresponding  substitutions, 

z^'  =  ie     "^  ^2,     z^=  ie     '^  z^, 
for  r  =  0,  1,  . . .,  n  —  1,  and 

2nri  2rTci 

z-i  =  —  ie     "■  Z2,     Z2  =  —  ie     '^  z^, 
for  the  same  values  of  r- 

And  so  far  as  concerns  (iii)  in  the  above  set,  if  we  take  an  axis  through 
the  middle  point  of  the  rth  side,  that  is,  the  side  which  joins  the  rth  and  the 

(2r-l)7r            .    (2r-l)7r 
(r  +  l)th  points,  then  ^  =  cos  ^ ,  77  =  sm — ,  ^=  0  :  hence  as  a 

is  equal  to  tt,  we  have,  for  the  corresponding  substitutions, 

{2r-l)ni  (2r-l)  Tri 

Zj'=ie       *"      Z2,     Z2=ie        **     z^, 
for  r  =  0,  1,  . . .,  71  —  1,  and 

(2r-l)Tri  {.2r-l)m 

Zi  =  —  ie       ^      Zz,     Z2  =  —  ie        '^      Z:^, 
for  the  same  values  of  r. 

If  n  be  even,  the  set  of  substitutions  associated  with  (ii)  are  the  same  in 
pairs,  and  likewise  the  set  associated  with  (iii) ;  if  w  be  odd,  the  set  associated 
with  (ii)  is  the  same  as  the  set  associated  with  (iii).  Thus  in  either  case  there 
are  2n  substitutions :  and  they  are  all  included  in  the  form 


VKT 

%Tvr 

Z^    ^  '2'^         -^2? 

■^2  ^~  "^^          ^\  3 

rr-O.  1,  . 

..,  2n-l. 

300:] 


TETRAHEDRAL  GROUP 


759 


Thus  the  whole  group  of  4w  substitutions,  in  their  homogeneous  form,  is 


■  e '"  ^1 


Zn     ^^   6  Zf} 


zirr 

z(  =  ie  ^  z^ 

_77rr 


for  r  =  0,  . . . ,  2w  —  1 :  and  in  the  non-homogeneous  form,  the  group  is 


z'  =  e~>^  z. 


e~n' 


Z  = 


where  r  =  0,  1,  ...,  n—\  for  each  of  them.  The  non-homogeneous  expres- 
sions are  not  in  their  normal  form  in  which  the  determinant  of  the  coefficients 
in  the  numerator  and  denominator  is  unity.  Each  expression  gives  two 
homogeneous  substitutions. 

It  is  easy  geometrically  to  see  that  all  the  axes  have  been  retained :  and 
that  they  form  a  group,  that  is,  composition  of  rotations  about  any  two  of  the 
axes  is  a  rotation  about  one  of  the  axes.     The  period  for  each  of  the  equatorial 

axes  is  2 ;  the  period  for  a  rotation about  the  polar  axis  depends  on  the 

r 
reducibilitv  of  - . 

"^        n 

Before  passing  to  the  construction  of  the  functions  which  are  unaltered 
for  the  dihedral  group  of  substitutions,  we  shall  obtain  the  tetrahedral  group 
and  construct  the  tetrahedral  functions,  for  the  explanations  in  regard  to  the 
dihedral  functions  arise  more  naturally  in  the  less  simple  case. 

II.  The  Tetrahedral  Group.  We  take  a  regular  cube  as  in  the  figure. 
Then  ABGD  is  a  tetrahedron,  A'B'C'D'  is  the  polar  tetrahedron. 


Fig.  119. 
It  is  easy  to  see  that  the  axes  of  rotation  for  the  tetrahedron  are 
(i)     the  four  diagonals  of  the  cube  AA',  BB',  CC,  DD' ; 


%^  TETRAHEDRAL  GROUP  -  [300. 

(ii)    the  three  lines  joining  the  middle  points  of  the  opposite  edges  of 
the  tetrahedron. 

The  latter  pass  through  the  centre  of  the  cube  and  are  perpendicular  to 
pairs  of  opposite  faces.  When  the  sphere  circumscribing  the  cube  is  drawn, 
the  three  axes  in  (ii)  intersect  the  sphere  in  six  points  which  are  the  angles 
of  a  regular  octahedron.  Thus,  though  the  axes  of  rotation  for  the  three 
solids  are  not  the  same,  the  tetrahedron,  the  cube,  and  the  octahedron  may 
be  considered  together:  in  fact,  in  the  present  arrangement  whereby  the 
surface  of  the  sphere  is  considered,  the  cube  is  merely  the  combination  of  the 
tetrahedron  and  its  polar. 

For  each  of  the  diagonals  of  the  cube,  the  necessary  angle  of  rotation 
for  the  tetrahedron  is  0  or  f tt  or  |7r :  the  first  of  these  gives  identity,  and 
the  others  give  two  rotations  for  each  of  the  four  diagonals  of  the  cube,  so 
that  there  are  eight  in  all. 

For  each  of  the  diagonals  of  the  octahedron,  the  angle  of  rotation  for 
the  tetrahedron  is  tt  :    there  are  thus  three  rotations. 

With  these  we  associate  identity.  Hence  the  number  of  rotations  for  the 
tetrahedron  is  (8  +  3  +  1  =)  12  in  all. 

There  are  two  sets  of  expressions  for  the  tetrahedron  according  to  the 
position  of  the  coordinate  axes  of  the  sphere.  One  set  arises  when  these  are 
taken  along  Ox,  Oy,  Oz,  the  diagonals  of  the  octahedron;  the  other  arises 
when  a  coordinate  plane  is  made  to  coincide  with  a  plane  of  symmetry  of  the 
tetrahedron  such  as  B'DBD'. 

Let  the  axes  be  the  diagonals  of  the  octahedron.  The  results  are 
obtainable  just  as  before,  and  so  may  now  merely  be  stated: 

For  OB',  ^  =  V=^=^^'>  when  a  =  f7r,  the  substitution  is 


,  _z  +  i 

~  z  —  i' 


and  when  a  =  ^ir,  the  substitution  is 

,      .z  +  \ 

z  =1 ^  . 

z  —  \ 

For  OA,  ^=  —  7]  =  ^=  --;  when  a  =  |7r,  the  substitution  is 

.z  +  1 

and  when  a  =  |7r,  the  substitution  is 

,      z-i 
z  = -. . 

Z  +  l 


300.]  OF   SUBSTITUTIONS  761 

For  OG,  —  ^  =  '7  =  ?=-7q5  when  a  =  |7r,  the  substitution  is 

z  =% , 

z  +  l 

and  when  a.  =  ^tt,  the  substitution  is 

,         z  +  i 

z  = -. . 

z  —  t 

For  OD',  —  ^  =  —  '?  =  ^=  -7^ ;  when  a  =  |7r,  the  substitution  is 

,         z  —  i 
z  +  i' 

and  when  a  =  ^ir,  the  substitution  is 

.z-1 

Z  =-l -. 

z+1 
For  Ox,  1  =  1,  'n  =  (),  ^=0  and  a  =  tt  :  the  substitution  is 

Z 

For  Oy,  f  =  0,  ■/7  =  1,  ^=0,  and  a  =  tt :  the  substitution  is 

1 

z  = . 

z 

For  Oz,  f  =  0,  7?  =  0,  ^=1  and  a  =  tt  :  the  substitution  is 

z'  =  -z. 
And  identity  is  ^'  =  z. 

Hence  the  group  of  tetrahedral  non-homogeneous  substitutions  is 

,  1  .z—1  .z  +  l  z  —  i  z  +  i 

z  =  -^  z,     +-,     +z =■ ,     +  I =- ,     +  — --. ,     ± . , 

-  z       -    z+l  z-1  z  +  l  z-% 

when  the  axes  of  reference  in  the  sphere  are  the  diameters  bisecting  opposite 
edges  of  the  tetrahedron.  Each  of  these  substitutions  gives  rise  to  two  homo- 
geneous substitutions,  making  24  in  all. 

To  obtain  the  transformations  in  the  case  when  the  plane  of  xz  is  a  plane 
of  symmetry  of  the  tetrahedron  passing  through  one  edge  and  bisecting  the 
opposite  edge,  such  as  B'DBD'  in  the  figure,  it  is  sufficient  to  rotate  the 
preceding  configuration  through  an  angle  ^tt  about  the  preceding  O^f-axis, 
and  then  to  construct  the  corresponding  changes  in  the  preceding  formulae. 

For  this  rotation  we  have,  with  the  preceding  notation  of  §  299,  ^  =  0  =  ?;, 
^=1,  a  =  i7r:  then  a  =  0  =  b,  c  =  sini7r,  c?  =  cosi7r,  so  that  d +  ic  =  e^^'^^ : 
and  therefore  the  ^'  of  the  displaced  point  in  the  stereographic  projection  is 
connected  with  the  ^  of  the  undisplaced  point  in  the  stereographic  projection 
by  the  equation 

y,_d  +  ic  in_l+i^y 


762  TETRAHEDRAL  SUBSTITUTIONS  [300. 

If  then  Z  be  the  variable  of  the  projection  of  the  undisplaced  point  and  Z' 
that  of  the  projection  of  the  displaced  point  with  the  present  axes,  and  z 
and  /  be  the  corresponding  variables  for  the  older  axes,  we  have 

1+z  1+*    ' 

that  IS,  z  =  —j^  z  ,        z  =  —^  Zi. 

Taking  now  the  twelve  substitutions  in  the  form  of  the  last  set  and  substi- 
tuting, we  have  a  group  of  tetrahedral  non-homogeneous  substitutions  in  the 

.form 

i  Z^^-{l+i)  ^V2  +  (l+0 

^-±A     ±^,     -\l-i)z+^Jr     -\\-i)z-^r 

.  Z^/2  +  (1  -  i)  .  Z^/2  - (1  -0 

-\l+i)Z-^2'       -\l+i)Z+^f2' 

when  one  of  the  coordinate  planes  is  a  plane  through  one  edge  of  the 
tetrahedron  bisecting  the  opposite  edge:  each  of  these  gives  rise  to  two 
homogeneous  substitutions,  making  24  in  all. 

301.  The  explanations,  connected  with  these  groups  of  substitutions, 
implied  that  certain  aggregates  of  points  remain  unchanged  by  the  operations 
corresponding  to  the  substitutions.  These  aggregates  are  (i)  the  summits  of 
the  tetrahedron,  (ii)  the  summits  of  the  polar  tetrahedron — these  two  sets 
together  make  up  the  summits  of  the  cube  :  and  (iii)  the  middle  points  of  the 
edges,  being  also  the  middle  points  of  the  edges  of  the  polar  tetrahedron — 
this  set  forms  the  summits  of  an  octahedron. 

When  these  points  are  stereographically  projected,  we  obtain  aggregates 
of  points  which  are  unchanged  by  the  substitutions.  We  therefore  project 
stereographically  with  the  extremity  z  of  the  axis  Oz  for  origin  of  projection : 
and  then  the  projections  of  x,  x,  y,  y ,  z,  z  are  1,  —  1,  i,  —  ^,  oo ,  0,  which  are 
the  variables  of  these  points. 

Instead  of  taking  factors  z—1,  z-\-l,  ...,  we  shall  take  homogeneous 
forms  z-^  —  z^,  Z1+Z2,  z^  —  iz^,  z^  +  iz^,  Z2,  ■^i  5  the  product  of  all  these  factors 
equated  to  zero  gives  the  six  points.     This  product  is 

t  =  ZiZo^  (z^*  -  zi). 

1      —  1      1 

For  the  tetrahedron  ABCD,  the  summits  A,  B,  G,  D  are  —r^,  — -,  —r^; 

\/S     1^0     v^ 

_j^    _i_    __L    _J_   J^    A    J_    Ji_   zl 

V3'       V3'       V3'        V3'    V3'    V3'    \/3'   V3'    V3 '    ^^^P^^^^^^^^  •    ^^^ 
therefore  the  variables  of  the  points  in  the  stereographic  projection  are 


^3  _  1 '  "^  ^'  ^3  + 1 '  '  V3  - 1 '  '  V3  +  1  ■ 


301.]  TETRAHEDRAL  FUNCTIONS  763 

Forming  homogeneous  factors  as  before,  the  product  of  the  four  equated  to 
zero  gives  the  stereographic  projections  of  the  four  summits  of  the  tetra- 
hedron ABGD.     This  product  is 

Similarly  for  the  tetrahedron  A'B'G'D';  the  product  of  the  factors 
corresponding  to  the  stereographic  projections  of  its  four  summits  is 

^  =  ^1^  +  2  V^^i  V  +  Z2*- 
And  the  product  of  the  eight  points  for  the  cube  is  <I>'^,  that  is, 

Tr=5/  +  i42iV  +  .^2'. 

All  these  forms  t,  <l>,  ^  are,  by  their  mode  of  construction,  unchanged 
(except  as  to  a  constant  factor,  which  is  unity  in  the  present  case)  by  the 
homogeneous  substitutions  :  and  therefore  they  are  invariantive  for  the  group 
of  24  linear  homogeneous  substitutions,  derived  from  the  group  of  12  non- 
homogeneous  tetrahedral  substitutions.  If  '^  be  taken  as  a  binary  quartic, 
then  <l>  is  its  Hessian  and  t  is  its  cubicovariant :  the  invariants  are  numerical 
and  not  algebraical :  and  the  syzygy  which  subsists  among  the  system  of 
concomitants  is 

a  relation  easily  obtained  by  reference  merely  to  the  expressions  for  the  forms 
<|),  ^,  t. 

The  object  of  this  investigation  is  to  form  Z,  the  simplest  rational 
function  of  z  which  is  unaltered  by  the  group  of  substitutions.  For  this 
purpose,  it  will  evidently  be  necessary  to  form  proper  quotients  of  the 
foregoing  homogeneous  forms,  of  zero  dimensions  in  z-^  and  z^.  Let  R 
be  any  rational  function  of  z,  which  is  unaltered  by  the  tetrahedral 
substitutions.  These  substitutions  give  a  series  of  values  of  z,  for  which 
Z  has  only  one  value :  hence  R  and  Z,  being  both  functions  of  z  and 
therefore  of  one  another,  are  such  that  to  a  value  of  Z  there  is  only  one 
value  of  R,  so  that  ii  is  a  rational  function  of  ^. 

In  particular,  the  relation  between  R  and  Z  may  be  lineo-linear :  thus  Z 
is  determinate  except  as  to  linear  transformations.  This  unessential  indeterm- 
inateness  can  be  removed,  by  assigning  three  particular  conditions  to 
determine  the  three  constants  of  the  linear  transformation. 

The  number  of  substitutions  in  the  ^-group  is  12.  As  there  will  thus 
be  a  group  of  12  ^r-points  interchanged  by  the  substitutions,  the  simplest 
rational  function  of  Z  will  be  of  the  12th  degree  in  z,  and  therefore  the 
numerator  and  the  denominator  of  the  fraction  for  Z,  in  their  homogeneous 
forms,  are  of  the  12th  degree.     The  conditions  assigned  will  be 

(i)      Z  must  vanish  at  the  summits  of  the  given  tetrahedron : 
(ii)     Z  must  be  infinite  at  the  summits  of  the  polar  tetrahedron : 
(iii)     Z  must  be  unity  at  the  middle  points  of  the  sides. 


764 


TETRAHEDRAL 


[301. 


As  ^  is  a  fractional  function  with  its  numerator  and  its  denominator  each 
of  the  12th  degree  and  composed  of  the  functions  ^,  '^,  t,  it  must,  with 
the  foregoing  conditions,  be  given  by 


z  = 


<|)3 


By  means  of  the  syzygy,  we  have 

Z:^-  1  : 1  =  ^3 :  -  Us/^St' :  ^^ 

which  is  Klein's  result.     Removing  the  homogeneous  variables,  we  have 

Z:  Z-1  :  l={z'-  2\/^2^  +  If  :  -  l2^/^z'  (^  -  1)^  :  (^  +  2 V^^^  +  1)^ ; 

and  then  ^  is  a  function  of  z  which  is  unaltered  by  the  group  of  12  tetra- 
hedral  substitutions  of  p.  761.  And  every  such  function  is  a  rational  function 
of  ^. 

This  is  one  form  of  the  result,  depending  upon  the  first  position  of  the 
axes.  For  the  alternate  form  it  is  necessary  merely  to  turn  the  axes  through 
an  angle  of  ^tt  round  the  2^-axis,  as  was  done  in  §  300  to  obtain  the  new 
groups.  The  result  is  that  a  function  Z,  unaltered  by  the  group  of  12 
substitutions  of  p.  762,  is  given  by 

^  :  ^ -  1  :  1  =  (^*  -  2^/Sz'  -  1)='  :  -  12^/Sz'  (^  +  1)^  :  (z^  +  2  VS^^  _  ly^ 

It  still  is  of  importance  to  mark  out  the  partition  of  the  plane  corre- 
sponding to  the  groups,  in  the  same  manner  as  was  done  in  the  case  of  the 
infinite  groups  in  the  preceding  chapter.  This  partition  of  the  plane  is  the 
stereographic  projection  of  the  partition  of  the  sphere,  a  partition  effected  by 
the  planes  of  symmetry  of  the  tetrahedron.  Some  idea  of  the  division  may 
be  gathered  from  the  accompanpng  figure,  which  is  merely  a  projection  on 
the  circumscribing  sphere  from  the  centre  of  the  cube.     The  great  circles 


Fig.  120. 


301.]  FUNCTIONS  765 

meet  by  threes  in  the  summits  of  the  tetrahedron  and  its  polar,  being  the 
sections  by  the  three  planes  of  symmetry,  which  pass  through  every  such 
summit,  and  the  circles  are  equally  inclined  to  one  another  there :  they  meet 
by  twos  in  the  middle  points  of  the  edges  and  they  are  equally  inclined  to 
one  another  there.  They  divide  the  sphere  into  24  triangles,  each  of  which 
has  for  angles  ^-tt,  ^tt,  Jtt.     (See  case  II.,  §  278.) 

The  corresponding  division  of  the  plane  is  the  stereographic  projection  of 
this  divided  surface.     Taking  A  as  the  pole  of  projection,  which  is  projected 


Fig.  121. 

to  infinity,  then  A'  is  the  origin :  the  three  great  circles  through  A'  become 
three  straight  lines  equally  inclined  to  one  another;  the  other  three  great 
circles  become  three  circles  with  their  centres  on  the  three  lines  concurrent 
in  the  origin.  The  accompanying  figure  shews  the  projection:  the  points  in 
the  plane  have  the  same  letters  as  the  points  on  the  sphere  of  which  they 
are  the  projections :  and  the  plane  is  thus  divided  into  24  parts.  There  are, 
in  explicit  form,  only  12  non-homogeneous  substitutions:  but  each  of  these 
has  been  proved  to  imply  two  homogeneous  substitutions,  so  that  we  have 
the  division  of  the  plane  corresponding  to  the  24  substitutions  in  the  group. 
The  fundamental  polygon  of  reference  is  a  triangle  such  as  GA'x. 

302.     It  now  remains  to  construct  the  function  for  the  dihedral  group. 
The  sets  of  points  to  be  considered  are  : — 

(i)     the  angular  points  of  the  polygon  :  in  the  stereographic  projection, 
these  are 

2Trsi 

e  '^  ,  for  s=0,l,  ...,n-l; 


766  DIHEDEAL   FUNCTION  [302. 

(ii)     the  middle  points  of  the  sides :    in  the  stereographic  projection, 
these  are 

e     "■      ,  for  5  =  0,  1,  . . . ,  w  -  1 ;  and 
(iii)    the  poles  •  of  the  equator  which  '  are  unaltered  by  each  of  the 
rotations :  in  the  stereographic  projection,  these  are  0  and  oo  , 
Forming  the  homogeneous  products,  as  for  the  tetrahedron,  we  have,  for  (i), 

for(ii),  V  =  2,^+z,^; 

and,  for  (iii),  W  =  z^z^  \ 

these  functions  being  connected  by  a  relation 

Because  the  dihedral  group  contains  2n  non-homogeneous  substitutions, 
the  rational  function  of  z,  say  Z,  must,  in  its  initial  fractional  form,  be  of 
degree  2n  in  both  numerator  and  denominator ;  and  it  must  be  constructed 
from  U,  V,  W. 

The  function  Z  becomes  fully  determinate,  if  we  assign  to  it  the  following 
conditions : 

(i)      Z  must  vanish  at  points  corresponding  to  the  summits  of  the 

polygon, 
(ii)     Z  must  be  infinite  at  points  corresponding  to  the  poles  of  the 

equator, 
(iii)     Z  must  be  unity  at  points  corresponding  to  the  middle  points  of 
the  edges : 
and  then  we  find 

Z:Z-1.1  =  {i(^--  l)r-  :  {i(^'^  +  l)p  :-z-, 
which  gives  the  simplest  rational  function  of  z  that  is  unaltered  by  the 
substitutions  of  the  dihedral  group. 

The  discussion  of  the  polyhedral  functions  will  not  be  carried  further  here  :  sufl&cient 
illustration  has  been  provided  as  an  introduction  to  the  theory  which,  in  its  various 
bearings,  is  expounded  in  Klein's  suggestive  treatise  already  quoted. 

JSa;.  1.  Shew  that  the  anharmonic  group  of  §  298  is  substantially  the  dihedral  group 
for  71=3  ;  and,  by  changing  the  axes,  complete  the  identification.  (Klein.) 

Ex.  2.  An  octahedron  is  referred  to  its  diagonals  as  axes  of  reference,  and  a  partition 
of  the  surface  of  the  sphere  is  made  with  reference  to  planes  of  symmetry  and  the  axes  of 
rotations  whereby  the  figure  is  made  to  coincide  with  itself. 

Shew  that  the  number  of  these  rotations  is  24,  that  the  sphere  is  divided  into  48 
triangles,  that  the  non-homogeneous  substitutions  which  transform  into  one  another  the 
partitions  of  the  plane  obtained  from  a  stereographic  projection  are 

'         s'  Z-1  2+1'  Z  +  t'  Z-%^ 

where  ^=0,  1,  2,  3 ;   and  that  the  corresponding  octahedral  function  is 

Z:  Z-\  :  1  =  (28 -I- 142*+!)^  :  (^^^ _ 3328 _ 33^+1)2  .  1082*  (s^ _ i )4.       (Klein.) 


303.]  ELLIPTIC   MODULAR-FUNCTIONS  767 

303.  We  now  pass  from  groups  that  are  finite  in  number  to  the 
consideration  of  functions  connected  with  groups  that  are  infinite  in 
number.  The  best  known  illustration  is  that  of  the  elliptic  modular- 
functions;  one  example  is  the  form  of  the  modulus  in  an  elliptic  integral 
as  a  function  of  the  ratio  of  the  periods  of  the  integral.  The  general 
definition  of  a  modular-function*  is  that  it  is  a  uniform  function  such  that 

an  algebraical  equation  subsists  between  i/r  i ^j  and  ■^{w),  where  a,  j3, 

7,  S  are  integers  subject  to  the  relation  aS-/37=l.  The  simplest  case  is 
that  in  which  the  two  functions  t^  are  equal. 

The  elliptic  quarter-periods  K  and  iK'  are  defined  by  the  integrals 

^K=\   [z(l-z){l-k-'z)]-^dz=Wz{l-z){l-cz)\-^'dz, 

.0  Jo 

2K'=  Wz{\-z){l-k''z)]-^dz=  W2{l-z){\-c'z)]-idz, 
Jo  Jo 

where  c  +  c'  =1.     The  ordinary  theory  of  elliptic  functions  gives  the  equation 

dc  dc  4cc" 

whatever  be  the  value  of  c.  To  consider  the  nature  of  these  quantities  as 
functions  of  c,  we  note  that  c  =  1  is  an  infinity  of  K  and  an  ordinary  point  of 
K',  and  that  similarly  c  =  0  is  an  infinity  of  K'  and  an  ordinary  point  of  K : 
and  these  are  all  the  singular  points  in  the  finite  part  of  the  plane.  The 
value  c  =  00  must  also  be  considered.  All  other  values  of  c  are  ordinary 
points  for  K  and  K'. 

For  values  of  c,  such  that  |  c  j  <  1,  we  have 

so  that,  in  the  vicinity  of  the  origin, 

d  fK'^ 


dc\K  4>K'cc' 


= i+o"*"  positive  integral  powers  of  cL 

Hence  in  the  vicinity  of  the  origin 

—  =--Iogc-HP(c), 

where  P (c)  is  a  uniform  series  converging  for  sufficiently  small  values  of  \c\: 
and  therefore,  still  in  the  vicinity  of  the  origin, 

^'  =  --logc  +  irP(c). 

TT 

*  This  is  the  definition  of  a  modular- function  which  is  adopted  by  Hermite,  Dedekind,  Klein, 
Weber,  and  others. 


768  ELLIPTIC  [303. 

Now  let  the  modulus  c  describe  a  contour  round  the  origin  and  return  to 
its  original  value.  Then  K  is  unchanged,  for  the  c-origin  is  not  a  singularity 
oi  K. 

The  new  value  of  K'  is  evidently 

--(27ri  +  \ogc)+KP(c), 

IT 

that  is,  iK'  changes  into  2K  +  iK'.  Hence,  ^uhen  c  describes  positively  a 
small  contour  round  the  origin,  the  quarter-periods  K  and  iK'  become  K  and 
2K  +  iK'  respectively. 

In  the  same  way  from  the  equation 

„,  dK     „  dK'  _       TT 

and  from  the  expansion  of  ^'  in  powers  of  c'  when  \c'\<  1,  we  infer  that 
when  c'  describes  positively  a  small  contour  round  its  origin,  that  is,  when  c 
describes  positively  a  small  contour  round  the  point  c  =  1,  then  iK'  is  unchanged 
and  K  changes  to  K—  2iK'. 

It  thus  appears  that  the  quantities  K  and  iK',  regarded  as  functions  of 
the  elliptic  modulus  c,  are  subject  to  the  linear  transformations 

U{K)  =  K  )  V{K)  =  K-2iK'] 

U  (iK')  =2K  +  iK']  '        V  (iK')  =  iK'\ 

without  change  of  the  quantity  c ;  and  the  application  of  either  substitution 
is  equivalent  to  making  c  describe  a  closed  circuit  round  one  or  other  of  the 
critical  points  in  the  finite  part  of  the  plane,  the  description  being  positive  if 
the  direct  substitution  be  applied  and  negative  if  the  inverse  be  applied. 

When  these  substitutions  are  applied  any  number  of  times — the  index 
being  the  same  and  composed  in  the  same  way  for  K  as  for  iK' — then, 
denoting  the  composite  substitution  by  P,  we  have  results  of  the  form 

PK=ZK  +  r^iK' 

PiK'  =  ^K  +  aiK' 

where  /3,  and  7  are  even  integers,  a  and  S  are  odd  integers  of  the  forms 
1  +  4<p,  1  +  4g,  say  =  1  (mod.  4),  and,  because  the  determinant  of  U  and  that 
of  V  are  both  unity,  we  have  aS  -  /37  =  1  by  §  282.  These  equations  give 
the  partially  indeterminate  form  of  the  values  of  the  quarter-periods  for  an 
assigned  value  of  the  modulus  c. 

iK' 

Conversely,  we  may  regard  c  as  a  function  of  lu  —  -^^ ,  the  quotient  of 

the  quarter-periods.  The  quotient  is  taken,  for  various  reasons  :  thus  it 
enables  us  to  remove  common  factors,  it  is  the  natural  form  in  the  passage 
to  g'-series,  and  so  on.     The    function   is   unaltered,  when   w  is  subjected 


303.]  MODULAR-FUNCTIONS  769 

to    the    infinite    group    of    substitutions    derived    from    the    fundamental 
substitutions 

JJlU  =W+   2,         VW  = rr—  . 

1  —  2w 
Denoting  the  function  c  by  (f){w),  we  have 

w 


c  =  0  (w)  =  (f)(iv  +  2)  =  ^ 


We  have  still  to  take  account  of  the  relation  of  iK'jK  to  c,  when  the  latter  has 
infinitely  large  values.     For  this  purpose,  we  compare  the  differential  expressions 

k  {x{\-x)  {l-Bx)r-^dx,     {y  (l-y)  {\-l'^y)Y-~dy, 

which  are  equal  to  one  another  if  k^x=y  and  kl=\.     As,  x  moves  from  0  to  1,  3/  moves 
from  0  to  k'^,  that  is,  from  0  to  1/^^ ;   integrating  between  these  limits,  we  have 

kK=\  +  i!s!, 

where  A  and  A'  are  quarter-periods  with  modulus  1=1  jk.     As  y  moves  from  0  to   1, 
X  moves  from  0  to  Ijk^;   integrating  between  these  limits,  we  have 

k{K+iK')  =  A, 

so  that  kili'  =  - ^A'. 

In  order  to  obtain  the  effect  on  K  and  iK'  of  an  infinitely  large  circuit  described 
positively  by  c,  we  make  I  describe  a  very  small  circuit  round  its  origin  negatively.  By 
what  has  been  proved,  the  effect  of  the  latter  is  to  change  A  and  iA'  into  A  and  ^A'  —  2A 
respectively.     Hence  the  new  value  of  kiK'  is 

-iA'  +  2A  =  k{3iK'  +  2K); 
and  the  new  value  of  kK  is 

A  -t-  iA'  -2A=-k  {2iK'  +  K). 

Hence  if  lo'  denote  the  new  value  of  w,  consequent  on  the  description  of  the  infinitely 

large  circuit  by  c,  we  have 

3W-1-2      „_,  „_, 
2w  +  \ 

No  new  fundamental  substitution  is  thus  obtained ;  and  therefore  U,  V  are  the  only 
fundamental  substitutions  of  the  group  for  c,  regarded  as  a  modular-function. 

Again,  c  is  a  rational  function  of  c  and  is  therefore  a  modular-function  : 
consequently  also  cc'  is  a  modular-function.  Being  a  rational  function  of 
c,  it  is  subject  to  the  two  substitutions  U  and  V,  which  are  characteristically 
fundamental  for  0  (w).  Now  cc'  is  unchanged  when  we  interchange  c  and  c\ 
that  is,  when  we  interchange  K  and  K' ;  so  that,  if  K^  and  iK^  be  new 
quarter-periods . for  a  modulus  cc,  we  have 

and  therefore  Wi  = . 

w 

Thus  cc'  as  a  modular-function  must  be  subject  to  the  substitution 

1 


w 


F.  F. 


49 


770  MODULAR- FUNCTIONS  [303. 

But  TUTw  =  -  j^  =  -^~r-=T^  =  ^'^> 

UTw        2  +  Tw     l-2w 

so  that  V  is  compounded  of  T  and  U.  Hence  the  substitutions  for  cc', 
regarded  as  a  modular-function,  are  the  infinite  group  which  is  derived  from 
the  fundamental  substitutions 

Uw=w  +  2,         Tw= . 

w 

Denoting  the  modular-function  cc'  by  ^  {w),  we  have 


CC'  =  %  (w)  =  %  (W  +  2)  =  ;^;  (  -  -  ) 


V    wj 

To  obtain  the  change  in  w  caused  by  changing  c  into  c/c,  we  use  the 
differential  expression 

When  the  variable  is  transformed  by  the  equation*  (1  —  y)(l  —  k^x)  =  1—  x, 
where  kfH^  =  —  k^,  the  expression  becomes 

k'  [x{l-x){\-  k''x)]~^  dx. 
When    y   describes   the    straight   line    from    0    to    1    continuously,   x   also 
describes  the  straight  line  from  0  to  1  continuously.     Integrating  between 
these  limits,  we  have 

A  =  k'K, 

where  A  is  a  quarter-period.  When  y  describes  the  straight  line  from  0 
to  1/^  continuously,  x  describes  the  straight  line  from  0  to  oo  continuously, 
or,  say,  the  line  from  0  to  Ijk"^  and  the  line  from  Ijk^  to  oo  continuously. 
Integrating  between  these  limits,  we  have 

A  -h  ^A'  =  k'  (K  +  iK')  +  \k'  T  [x{l- x){l-  k-'x)]'^  dx 

=  k'{K  +  iK')  +  k'K, 
on  using  the  transformation  k^xu  =  1  and  taking  account  of  the  path  described 
by  the  variable  u  :  and  therefore 

iA' =  k' {K  +  iK'). 

Hence  the  change  of  modulus  from  k  to  ikfk',  which  changes  c  to  —  cjc,  gives 
the  changes  of  quarter-periods  in  the  form 

A  =  k'K,    iM  =  k'{K+  iK') ; 
and  therefore  the  new  value  of  w,  say  Wi,  is 

Wi  =  w  +  1  =  8w. 
It  therefore  follows  that,  when  —  cjc'  is  regarded  as  a  modular-function 
of  the  quotient  lu  of  the  quarter-periods  K  and  iK',  it  must  be  subject  to 
the  substitutions 

S(w)=^w+l,       U{w)  =  iu+2,       V{tv)=^  _^^^. 
*  This  is  the  e(j[aation  expressing  elHptic  functions  of  k'u  in  terms  of  elliptic  functions  of  u. 


303.]  AUTOMORPHIC   FUNCTIONS  771 

Evidently  S^  =  U,  and  U  may  therefore  be  omitted ;  V  and  8  are  the 
fundamental  substitutions  of  the  infinite  group  of  transformations  of  w, 
the  argument  of  the  modular-function  c/c'. 

As  a  last  example,  we  consider  the  function 


/  = 


(c'^c  +  iy 


It  is  a  rational  function  of  cc,  and  therefore  is  a  modular-function  having  the 

substitutions  Tw  and   Uw.     By  §  298,  it  is  unaltered  when  we  substitute 

c 
— -y  for  c.     It  has  just  been  proved  that  this  change  causes  a  change  of  w 

into  w-f-1,  and  therefore  /,  as  a  modular-function,  must  be  suT^ject  to  the 
substitution 

Sw^w  -\- 1. 

Evidently  S^w  =  w  +  2=  Uw,  so  that  U  is  no  longer  a  fundamental  substitution 
when  aS^  is  retained.  Hence  we  have  the  result  that  J  is  unaltered,  when  w  is 
subjected  to  the  infinite  group  of  substitutions  derived  from  the  fundamental 
substitutions 

Sw  =  w  +  1,        Tw  =  — , 

w 

so  that  we  may  write 


J  = 


=  J{w)  =  J{w+l)  =  j{-^. 


This  is  the  group  of  substitutions  considered  in  §  284 :    they  are  of  the 
form 5^ ,  where  a,  /3,  7,  8  are  real  integers  subject  to  .the  single  relation 

These  illustrations,  in  connection  with  which  the  example  in  §  298  should  be  consulted, 
suffice  to  put  in  evidence  the  existence  of  modular-functions,  that  is,  functions  periodic 
for  infinite  groups  of  linear  substitutions,  the  coefficients  of  which  are  real  integers.  The 
theory  has  been  the  subject  of  many  investigations,  both  in  connection  with  the  modular 
equations  in  the  transformation  of  elliptic  functions  and  also  as  a  definite  set  of  functions. 
The  investigations  are  due  among  others  to  Hermite,  Fuchs,  Dedekind,  Hiu-witz,  and 
especially  to  Klein* ;  and  reference  must  be  made  to  their  memoirs,  or  to  Klein-Fricke's 
treatise  on  elliptic  modular-functions,  or  to  Weber's  ElUptische  Functionen,  for  an  exposi- 
tion of  the  theory. 

304.  The  method  just  adopted  for  infinite  groups  is  very  special,  being 
suited  only  to  particular  classes  of  functions :  in  passing  now  to  linear 
substitutions,  no  longer  limited  by  the  condition  that  their  coefficients  are 
real  integers,  we  shall  adopt  more  general  considerations.  The  chief 
purpose  of  the  investigation  will  be  to  obtain  expressions  of  functions 
characterised  by  the  property  of  reproduction  when  their  argument  is 
subjected  to  any  one  of  the  infinite  group  of  substitutions. 

*  Some  references  are  given  in  Enneper's  Elliptische  Functionen,  (2*«  Aufl.),  p.  482. 

49—2 


772  CONSTRUCTION  OF  '  [304. 

The  infinite  group  is  supposed  of  the  nature  of  that  in  §  290:  the 
members  of  it,  being  of  the  form 

(ni^:)'  -  <^'/'<^)>' 

are  such  that  a  circle,  called  the  fundamental  circle,  is  unaltered  by  any  of  the 
substitutions.  This  circle  is  supposed  to  have  its  centre  at  the  origin  and 
unity  for  its  radius. 

The  interior  of  the  circle  is  divided  into  an  infinite  number  of  curvilinear 
polygons,  congruent  by  the  substitutions  of  the  group :  each  polygon  contains 
one,  and  only  one,  of  the  points  in  the  interior  associated  by  the  substitutions 
with  a  given  point  not  on  the  boundary  of  the  polygon.  Hence  corresponding 
to  any  point  within  the  circle,  there  is  one  and  only  one  point  within  the 
fundamental  polygon,  as  there  is  only  one  such  point  in  each  of  the  polygons  : 
of  these  homologous  points  the  one,  which  lies  in  the  fundamental  polygon 
of  reference,  will  be  called  the  irreducible  point.  It  is  convenient  to  speak  of 
the  zero  of  a  function,  implying  thereby  the  irreducible  zero :  and  similarly 
for  the  singularities. 

The  part  of  the  plane,  exterior  to  the  fundamental  circle,  is  similarly 
divided :  and  the  division  can  be  obtained  from  that  of  the  -internal  area  by 
inversion  with  regard  to  the  circumference  and  the  centre  of  the  fundamental 
circle.  Hence  there  will  be  two  polygons  of  reference,  one  in  the  part  of  the 
plane  within  the  circle  and  the  other  in  the  part  without  the  circle :  and 
all  terms  used  for  the  one  can  evidently  be  used  for  the  other.  Thus  the 
irreducible  homologue  of  a  point  without  the  circle  is  in  the  outer  polygon 
of  reference  :  for  a"  substitution  transforms  a  point  within  an  internal  polygon 
to  a  point  within  another  internal  polygon,  and  a  point  within  an  external 
polygon  to  a  point  within  another  external  polygon. 

Take  a  point  z  in  the  interior  of  the  circle,  and  round  it  describe  a  small 
contour  (say  for  convenience  a  circle)  so  as  not  to  cross  the  boundary  of  the 
polygon  within  which  2  lies :  and  let  Zi  be  the  point  given  by  the  substitution 
fi{z).  Then  corresponding  to  this  contour  there  is,  in  each  of  the  internal 
polygons,  a  contour  which  does  not  cross  the  boundary  of  its  polygon :  and  as 
the  first  contour  (say  Cq)  does  not  occupy  the  whole  of  its  polygon  and  as  the 
congruent  contours  do  not  intersect,  the  sum  of  the  areas  of  all  the  contours 
Cj  is  less  than  the  sum  of  the  areas  of  all  the  polygons,  that  is,  the  sum  is 
less  than  the  area  of  the  circle  and  so  it  is  finite. 

If  fii  be  the  linear  magnification  at  Zi,  we  have 

dzi 


f^i 


dz 


and  therefore,  if  mj  be  the  least  value  of  the  magnification  for  points  lying 
within  Co,  we  have 

Ci>7n/G.. 


304] 


A   CONVERGING  SERIES 


773 


The    point is    the    homologue    of    z  =  <x>     by    the    substitution 

z,   ^ VI ,  and  therefore  —  84/7^  lies  without 

Ji2  +  OiJ 

the  circle :  though,  in  the  limit  of  i  infinite,  it 
may  approach  indefinitely  near  to  the  circum- 
ference*. 

Let  this  point  be  G :  and  through  G  and 
0,  the  centre  of  the  fundamental  circle,  draw 
straight  lines  passing  through  the  centre  of 
the  circular  contour.     Then  evidently 

,     ,  „    1 


^i=  |7i| 


GP'' 


and,  if  Mi  be  the  greatest  magnification,  then 


Fig.  122. 


1 

GQ^ 


so  that 


Mj^GP" 

mi~  GQ'' 


Now  G  is  certainly  not  inside  the  circle,  so  that  GQ  is  not  less  than  RA  : 

thus 

GP_        PQ^-,,A^     .      AB     RB 
GQ~    '^  GQ        ^  GQ^     '^  RA^RA' 

which  is  independent  of  the  point  G,  that  is,  of  the  particular  substitution 

fi  (z).     Denoting  (  ^^  j  by  K,  we  have 

M,      ^ 
<  K, 


or 

Evidently  fjb^  is  finite. 
Now 

and  therefore 

so  that 


mi 

Mi  <  Kmi. 


\yiZ  + 


-s-|2  =  /^i  <  Mi  <  Krrii 


1  K^ 

\'yiZ  +  bif  t>o 

S  \yiz  +  Si\-^  <  jT  S  C'i- 
4=0  ^0  r=o 


*  For,  in  §  284,  when  the  coefficients  are  real,  a  point  associated  with  a  given  point  may,  for 
i  =  cx> ,  approach  indefinitely  near  to  a  point  on  the  axis  of  x  :  and  then,  by  the  transformation  of 
§  290,  we  have  the  result  in  the  text. 


774 


A   CONVERGING   SERIES 


[304. 


It  has  been  seen  that   2  Ci  is  less  than  the  area  of  the  fundamental  circle  and 
is  therefore  finite :  hence  the  quantity 

00 

is  finite.     It  therefore  follows  that  2  /*/  is  an  absolutely  converging  series. 

i  =  0 

00 

Similarly,  it  follows  that    2    /uLi"^  is  an  absolutely  converging  series  for  all 

i=0 

integral  values  of  m  that  are  greater  than  unity*.     This  series  is  evidently 

i  =  0 

and  the  absolute  convergence  is  established  on  the  assumption  that  z  lies 
within  the  fundamental  circle. 

Next,  let  z  lie  without  the  fundamental  circle.     If  z  coincide  with  some 
one  of  the  points  —  Si/<yi,  then  the  corresponding  term  of 
the  series 

is  infinite. 

If  it    do    not  coincide   with   any  one   of  the   points 
—  Si/yi,  let  c  be  its  distance  from  the  nearest  of  them,  so 

that 

\yiZ  +  g^r^  <  \yi\-'^c-^^. 

Let  /  be  any  point  within  the  fundamental  circle :  then 

I  ji/  +  Si  \-^^  =  {  Gz')-"^  1 7,- 1-^"*. 

Now  Gz'  <1  +  0(t<1+— l,for  any  point  within  tlie  circle,  so  that 

It* 


Fig.  123. 


'.if 


Hence 


\yiZ  +  Si 


i7i/  +  Si|-^"* 

Only  a  limited  number  of  the.  points  —  8^/7^  can  be  at  infinity.  Each  of 
the  corresponding  substitutions  gives  the  point  at  infinity  as  the  homologue 
of  —  hij<yi ;  and  therefore,  inverting  with  regard  to  the  fundamental  circle,  we 
have  a  number  of  homologues  of  the  origin  coinciding  with  the  origin,  equal 
to  the  number  of  the  points  —  S^/7^  at  infinity.  The  origin  is  not  a  singularity 
of  the  group,  so  that  the  number  of  homologues  of  the  origin,  coincident  with 
it,  must  be  limited. 

*  A  completely  general  inference  as  to  the  convergence  of  the  series,  when  in=l,  cannot  be 
made :  the  convergence  depends  upon  the  form  of  the  division  of  the  plane  into  polygons,  and 
Burnside  (I.e.,  p.  754)  has  proved  that  there  is  certainly  one  case  in  which  S  jx^  is  an  absolutely 
converging  series. 


304.]  CONNECTED   WITH   INFINITE   GROUPS  775 

Omitting  the  corresponding  terms  from  the  series,  an  omission  which  does 
not  affect  its  convergence,  we  can  assign  a  superior  limit  to 
C-l.     Then 


7i 


let  it  be 


Thus  t\ryi2  +  Si\-^  <     -         S  |7i/  +  S^|-^'^ 

which  is  a  finite  quantity  by  the  preceding  investigation,  for  z'  is  a  point 
within  the  circle. 

Lastly,  let  z  lie  on  the  fundamental  circle.     If  it  coincide  with  one  of  the 
essential  singularities  of  the  group,  then  there  is  an  infinite  number  of  points 

—  ^i/yi  which  coincide  with  it :  and  so  there  will  be  an  infinite  number  of 
terms  in  the  series  infinite  in  value.  If  it  do  not  coincide  with  any  of  the 
essential  singularities  of  the  group,  then  there  is  a  finite  (it  may  be  small, 
but  it  is  not  infinitesimal)  limit  to  its  distance  from  the  nearest  of  the  points 

—  Si/ji :  the  preceding  analysis  is  applicable,  and  the  series  converges. 

Hence,  summing  up  our  results,  we  have  : — 


The  series  2  \yiZ  +  8i\~' 


2m 
=  0 


is  an  absolutely  converging  series  for  any  point  in  the  plane,  which  is  not 
coincident  with  any  one  of  the  points  —  Si/ji  {which  all  lie  without  the  funda- 
mental circle)  or  with  any  one  of  the  essential  singularities  of  the  assigned 
group  {which  all  lie  on  the  circumference  of  the  fundamental  circle)^. 

305.  Let  H{z)  denote  a  rational  function  of  z,  having  a  number  of 
accidental  singularities  aj,  ...,  a^,  no  one  of  which  lies  on  the  fundamental 
circle ;  and  let  it  have  no  other  singularities.     Consider  the  series 

%{z)=   i{y,z  +  8d--^H{''^^), 

the  group  being  the  same  as  above.  If  z  do  not  coincide  with  any  of  the 
points  «!,  ...,  ap,  or  with  any  of  the  points  homologous  with  a^,  ...,  ap  by  the 
substitutions  of  the  group,  there  is  a  maximum  value,  say  M,  for  the  modulus 

of  H  with  any  of  the  arguments  — -^ .     Then 

\@{z)\<Mt  IjiZ  +  Bil-'"^, 

4  =  0 

*  The  coefficients  a,  j8,  y,  d  of  the  substitutions  of  the  group  depend  upon  the  coefficients  of 
the  fundamental  substitutions,  which  may  be  regarded  as  parameters,  arbitrary  within  limits. 
The  series  is  proved  by  Poincare  to  be  a  continuous  function  of  these  parameters,  as  well  as  of 
the  variable  z :  this  proposition,  however,  belongs  to  the  development  of  the  theory  and  can  be 
omitted  here  as  we  do  not  prapose  to  establish  the  general  existence  of  all  the  functions. 


776  THETAFUCHSIAN    FUNCTIONS  [305. 

and  the  right-hand  side  is  finite,  if  in  addition  z  do  not  coincide  with  any  of 
the  points  -8^/7^  or  with  any  of  the  essential  singularities  of  the  group. 
Hence  @  {z)  is  an  absolutely  converging  series  for  any  value  of  z  in  the  plane 
which  does  not  coincide  with  (i)  an  accidental  singularity  of  E{z\  or  one  of 
the  points  homologous  with  these  singularities  by  the  substitutions  of  the 
group,  or  with  (ii)  any  of  the  points  -  8,:/7i,  which  are  the  various  points 
homologous  with  ^  =  00  by  the  substitutions  of  the  group,  or  with  (iii)  any  of 
the  essential  singularities  of  the  group,  which  are  points  lying  on  the  funda- 
mental circle. 

All  these  points  are  singularities  of  B  {z\ 

If  z  coincide  with  /^  (a),  and  if  fi  [fic  {z)]  =  z,  then  the  term  H  ['^      S) 

\'YiZ  -r  Ojv 

is  infinite,  the  point  being  an  accidental  singularity  of  H  (  J'j .     The 

rest  of  the  series  is  then  of  the  same  nature  as  @  (z)  in  the  more  general 
case,  and  therefore  converges.  Hence  the  point  is  an  accidental  singularity 
of  the  function  @  (z)  of  the  same  order  as  for  R,  that  is,  the  series  of  points, 
given  by  the  accidental  singularities  of  H  (z)  and  by  the  points  homologous 
with  them  through  the  substitutions  of  the  group,  are  accidental  singularities 
of  the  function  B  (z). 

In  the  same  way  it  is  easy  to  see  that  the  points  —  8^/7^  are  either 
ordinary  points  or  accidental  singularities  of  @  (z) ;  and  that  the  essential 
singularities  of  the  group  are  essential  singularities  of  @  (z).  Hence  we 
have  the  result : — 

The  series  B  (z)  -  i  (jiZ  +  Si)-^  H  (°^^^)  , 

where  the  summation  extends  over  the  infinite  number  of  members  of  an  assigned 
discontinuous  group,  is  a  function  of  z,  provided  the  integer  m  be>l  and  H{z) 
he  a  rational  function  of  z.     The  singularities  of  B  are : — 

(i),  the  accidental  singidarities  of  H  {z)  and  the  points  homologous 
with  them  by  the  substitutions  of  the  group  :  all  these  points  are 
accidental  singularities  of  B  {z) ; 

(ii),  the  points  —  S,:/7i,  which  are  the  'points  homologous  with  z  =  co  by 
the  substitutions  of  the  group :  all  these  points,  if  not  ordinary 
points  of  B  {z),  are  accidental  singularities ;  and 

(iii),  the  essential  singularities  of  the  group:  these  lie  on  the  fundamental 
circle  and  they  are  essential  singularities  of  B  {z). 

If  H{z)  had  any  essential  singularity,  then  that  point  and  all  points  homo- 
logous with  it  by  substitutions  of  the  group  would  be  essential  singularities 
of  B  {z).  The  function  B  {z),  thus  defined,  is  called*  Thetafuchsian  by 
Poincare. 

*  Acta  Math.,  i.  i,  (1882),  p.  210. 


305.]  PSEUDO-AUTOMOEPHIC   PROPERTY  777 

If  the  group  belong  to  the  first,  the  second,  or  the  sixth  family, 
it  is  known  that  the  circumference  of  the  fundamental  circle  enters  into 
the  division  of  the  interior  of  the  circle  (and  also  of  the  space  exterior  to 
the  circle)  only  in  so  far  as  it  contains  the  essential  singularities  of  the 
group.  But  if  the  group  belong  to  any  one  of  the  other  four  families, 
then  parts  of  the  circumference  enter  into  the  division  of  both  spaces. 

In  the  former  case,  when  the  group  belongs  to  the  set  of  families, 
made  up  of  the  first,  the  second,  and  the  sixth,  the  circumference  of  the 
fundamental  circle  is  a  line  over  which  the  series  cannot  be  continued :  it 
is  a  natural  limit  (§  81)  both  for  a  function  existing  in  the  interior  of  the 
circle  and  for  a  function  existing  in  the  exterior  of  the  circle  :  but  neither 
function  exists  for  points  on  the  circumference  of  the  fundamental  circle. 
The  series  represents  one  function  within  the  circle  and  another  function 
without  the  circle. 

It  has  been  proved  that  the  area  outside  the  fundamental  circle  can 
be  derived  from  the  area  inside  that  circle,  by  inversion  with  regard  to 
its  circumference.  Hence  a  function  of  z,  existing  only  outside  the  funda- 
mental circle,  can  be  transformed  into  a  ftmction  of  — ,  and  therefore  also 

of  - ,  existing  for  points  only  within  the  circle.     When,  therefore,  a  group 

belongs  to  the  first,  the  second,  or  the  sixth  family,  it  is  sufficient  to  consider 
only  the  function  defined  hy  the  series  for  points  within  the  fundamental 
circle :   it  will  be  called  the  function  @  {z). 

In  the  latter  case,  when  the  group  belongs  to  the  third,  the  fourth,  the 
fifth,  or  the  seventh  families,  then  parts  of  the  circumference  enter  into  the 
division  of  the  plane  both  without  and  within  the  circle.  Over  these  parts 
the  function  can  be  continued  :  and  then  the  series  represents  one  {and  only 
one)  function  in  the  two  parts  of  the  plane  :  it  will  be  called  the  function  0  {z). 

306.  The  importance  of  the  function  ©  {z)  lies  in  its  pseudo-automorphic 
character  for  the  substitutions  of  the  group,  as  defined  by  the  property  now 

to  be  proved  that,  if  ^  he  any  one  of  the  substitutions  of  the  group,  then 


<yz  -\- 

(ccz^+  /3 

KjZ 

az  +  ^ 


®  (°^i±^)  =  (^^  +  g)2m  (H)  (^). 
\ryz  -{■  0/ 


+  ^i 


J  ^  <yz  +  S  _  a/z  +  /3/ 

az  +  13  ^   ^       ryiZ+  6i 

'      ryZ  +  b 

which  is,  of  course,  another  substitution  of  the  infinite  group  :  then 

az  +  /3  ^   ^      y/z  +  Bi 

7i — o,  +  Oi  = — ^  . 

yz  +  6  yz+  0 


778  ZEROS   AND   SINGDXARITIES  [306. 

Hence  @  f  «^)  =  I  (^iU^"  ff  f  «il±f ) 

=  (7^  +  S)2^  0  (z), 
thus  establishing  the  pseudo-automorphic  character. 

This  function  can  evidently  be  made  subsidiary  to  the  construction  of 
functions,  which  are  automorphic  for  the  group  of  substitutions,  in  the  same 
manner  as  the  cr-function  in  Weierstrass's  theory  of  elliptic  functions  and 
the  so-called  Theta-functions  in  the  theory  of  Jacobian  and  of  Abelian 
transcendents.  But  before  we  consider  these  automorphic  functions,  it  is 
important  to  consider  the  zeros  and  the  accidental  singularities  of  a  pseudo- 
automorphic  function  such  as  (a)  (z). 

On  the  supposition  that  the  function  H,  which  enters  as  the  additive 
element  into  the  composition  of  ©,  has  only  accidental  singularities,  it  has 
been  proved  that  all  the  essential  singularities  of  @  lie  on  the  circumference 
of  the  fundamental  circle ;  and  that  the  accidental  singularities  of  0  are, 
(i)  the  points  homologous  with  the  accidental  singularities  of  H,  and 
(ii)  the  points  —  Si/ji,  which  all  lie  without  the  circle. 

When  the  function  H  (z)  has  one  or  more  accidental  singularities  within 
the  fundamental  circle,  then  there  is  an  irreducible  point  for  each  of  them, 
which  is  an  irreducible  accidental  singularity  of  0  (z).  Hence  in  the  case  of 
a  function  which  exists  only  within  the  circle,  the  number  of  irreducible 
accidental  singularities  is  the  same  as  the  number  of  {non-homologous)  accidental 
singularities  of  H  {z)  lying  within  the  fundamental  circle.  If,  then,  all  the 
infinities  of  the  additive  element  H  {z)  lie  without  the  fundamental  circle,  and 
if  the  function  ©  (z)  exist  only  within  the  circle,  then  0  (z)  has  no  irreducible 
accidental  singularities :  but,  in  particular  cases,  it  may  happen  that  0  (z)  is 
then  evanescent. 

When  the  function  JI(z)  has  one  or  more  accidental  singularities  without 
the  fundamental  circle,  then  there  is  an  irreducible  point  for  each  of  them, 
this  point  lying  in  the  fundamental  polygon  of  reference  in  the  space  outside 
the  circle :  and  this  point  is  an  irreducible  accidental  singularity  of  0  (z), 
when  0  (z)  exists  both  within  and  without  the  circle.  Further,  the  point 
—  Si/ji  is  an  infinity  of  order  2m :  there  is  a  homologous  irreducible  point 
within  the  polygon  of  reference  without  the  circle,  being,  in  fact,  the 
irreducible  point  which  is  homologous  with  z=  qo  .  Hence  taking  the  two 
fundamental  polygons  of  reference — one  within,  for  the  internal  division,  and 
one  without,  for  the  external  division, — it  follows  that  i??  the  case  of  a  function, 
which  exists  all  over  the  plane,  the  number  of  irreducible  accidental  singularities 


306.]  OF   A   PSEUDO-AUTOMORPHIC   FUNCTION    •  779 

is  equal  to  the  luhole  number  of  accidental  singularities  of  the  additive  element 
H(z),  increased  by  2m. 

307.  To  obtain  the  number  of  irreducible  zeros  we  use  the  result  of 
§  43,  Cor.  IV.,  combined  with  the  result  just  obtained  as  to  the  number  of 
irreducible  accidental  singularities.  A  convention,  similar  to  that  adopted 
in  the  case  of  the  doubly -periodic  functions  (§  115),  is  now  necessary:  for  if 
there  be  a  zero  on  one  side  of  the  fundamental  polygon,  then  the  homologous 
point  on  the  conjugate  side  of  the  polygon  is  also  a  zero  and  of  the  same 
degree :  in  that  case,  either  we  take  both  points  as  irreducible  zeros  and  of 
half  the  degree,  or  we  take  one  of  them  as  the  irreducible  zero  and  retain 
its  proper  degree.  Similarly,  if  a  corner  be  a  zero,  every  corner  of  the  cycle 
is  a  zero  :  so  that,  if  the  cycle  contain  \  points  and  the  sum  of  its  angles  be 

— ,  then  the  corner  is  common  to  X/u,  polygons ;  we  may  regard  each  of  the 

corners  of  the  fundamental  polygon  in  that  cycle  as  an  irreducible  zero,  of 
degree  equal  to  its  proper  degree  divided  by  X/x,  or  we  may  take  only  one  of 
them  and  count  its  degree  as  the  proper  degree  divided  by  //. — the  just 
distribution  of  zeros  common  to  contiguous  polygons  being  all  that  is 
necessary  for  the  convention— so  that  the  number  of  zeros  to  be  associated 
with  the  area  of  each  polygon  is  the  same,  while  no  zero  is  counted  in  more 
than  its  proper  degree.     A  similar  convention  applies  to  the  singularities. 

With  this  convention,  the  excess  of  the  number  of  irreducible  zeros 
over  the  number  of  irreducible  accidental  singularities,  each  in  its  proper 
degree,  is  the  value  of 

27^^■j@(^)      ' 
taken  positively  round  the  fundamental  polygon  within  the  circle  when  the 
function  @  (z)  exists  only  within  the  circle,  and  round  the  two  fundamental 
polygons,  within  and  without  the  circle  respectively,  when  the  function  ©  (z) 
exists  over  the  whole  plane. 

But  should  an  infinity  of  ^  ;  .^  lie  on  the  curve  along  which  integration 
•^        ®(z)  ^  ° 

extends,  (it  will  arise  through  either  a  zero  or  a  pole  of  B),  then,  in  order 
to  avoid  the  difficulty  in  the  integration  and  preserve  the  above  convention, 
methods  must  be  adopted  depending  upon  the  family  of  the  group. 

When  all  the  cycles  belong  to  the  first  sub-category  (§  292),  we  can 
proceed  as  follows :  the  general  result  can  be  proved  to  hold  in  every  case. 
If  an  infinity  occur  on  a  side,  another  will  occur  on  the  conjugate  side,  the 
two  being  homologous  by  a  fundamental  substitution.  A  small  semi-circle  is 
drawn  with  the  point  for  centre  and  lying  without  the  polygon,  so  that,  when 
the  element  of  the  side  is  replaced  by  the  semi-circumference,  the  point 
lies  within  the  polygon:  the  homologous  point  on  the  conjugate  side  is 
excluded  from   the    polygon  when  the    element   there    is   replaced    by  the 


780 


ZEROS   AND   SINGULARITIES 


[307. 


homologous  semi-circumference 
along  the  modified  sides. 

A  similar  process  is  adopted  when  a  corner  is  an  infinity  of 


The  subject  of  integration  is  then  finite 


small  circular  arc  is  drawn  so  as  to  have  the  point  included  in  the  polygon 
when  the  arc  replaces  the  elements  of  the  sides  at  the  point :  the  homologous 
circular  arcs  at  all  the  points  in  the  cycle  of  the  corner  will  exclude  all  those 
points,  also  poles,  when  they  replace  the  elements  of  the  sides  at  the  point. 
The  subject  of  integration  is  then  finite  everywhere  along  the  modified  path 
of  integration. 

First,  let  the  function  exist  only  within  the  circle, 
of  the  polygon,  A'B'  the  conjugate  side ; 
and  let 

^     yz  +  S 

be  the  corresponding  fundamental  substi- 
tution which  transforms  AB  into  A'B', 
so  that  ^  may  be  regarded  as  the  variable 
along  A'B'. 

Then  we  have 


Let  AB  be  any  side 


Q   B 


Fig.  124. 


and  therefore 


(H)  (^)  =  {r^z  -f  ly^  %  {Z\ 


dz. 


But  as  z  moves  from  J.  to  5,  ^  moves  from  A'  to  B'  (|  287) :  and  the  latter 
is  the  negative  direction  of  description.  Hence,  with  the  given  notation,  the 
sum  of  the  parts  of  the  integral,  which  arise  through  the  two  sides  AB 
and  B'A',  is 

•%'{z) 


%{z) 


dz,  for  ^5  +/{-  |y§  ^d  ,  for  B'A' 


2m 


z  + 


^  dz,  taken  along  AB; 


so  that,  if  E  denote  the  required  excess,  we  have 

„  m  [    dz 

E=- 


the  new  integral  being  taken  along  those  sides  of  the  polygon  which  are 
transformed  into  their  conjugates  by  the  fundamental  substitutions  of  the 
group. 

Consider  the  term  which  arises  through  the  integration  along  AB :  it  is 
evidently 

711  ' 

-— .    log (7^  +  8) 
m 


307.]  OF   THETAFUCHSIAN   FUNCTIONS  781 

Now  we  have 


dz      (7^  +  g)2' 

so  that,  if  M  be  the  magnification  in  transforming  from  A  to  A',  and  if  ^a  be 
the  angle  through  which  a  small  arc  is  turned,  we  have  at  A 

1 

Evidently  (pa  is  the  excess  of  the  inclination  of  A'F,  that  is,  of  A'C  to  the 
line  of  real  quantities  over  the  inclination  of  AP,  that  is,  of  J.  (7  to  that  line : 
and  therefore  at  A 

log  (7^  +  S)  =  -  ^  log  if  -  ii>«. 

Since  the  whole  integral  must  prove  to  be  a  real  quantity,  we  omit  the 
parts  —  -x — .  log  M  as  in  the  aggregate  constituting  an  evanescent  (imaginary) 

quantity :  hence  we  have 

m 

2^  (-  9a  +  96) 

as  the  part  corresponding  to  the  side  AB.  In  this  expression,  (p^  is  the  angle 
required  to  turn  AG  into  a  direction  parallel  to  A'C,  and  ^j  is  the  angle 
required  to  turn  QB,  that  is,  GB  into  a  direction  parallel  to  QB',  that  is, 
G'B',  both  rotations  being  taken  positively.     Thus 

(^a  =  inch  A'G'  —  incl.  AG, 

</>6  =  27r  -  incl.  BG  +  incl.  B'G' ; 
and  therefore 

<^„  _  (^j,  =  -  27r  +  incl.  A'G'  -  incl.  B'G'  +  incl.  BG  -  incl.  AG 

=  -27r  +  Ci'  +  Ci, 

where  Ci  and  c/  are  the  angles  AGB,  A' G'B'  respectively.  Hence,  if  we  take 
c  and  c  to  be  the  external  angles  AGB,  A' G'B'  as  in  the  figure,  we  have 

c  +  Ci  =  27r  =  c'  +  c/, 

and  therefore  (fib—  4>a  =  c  +  c'  —  27r. 

The  part  corresponding  to  the  arc  AB  in  the  above  integral  is  therefore 

(c  +  c'-27r). 
ztt 

There  are  no  sides  of  the  second  kind  in  the  path  of  integration,  because  the 
function  is  supposed  to  exist  only  within  the  circle.  Therefore  the  whole 
excess  is  given  by 

£;=^X(c  +  c'-27r), 

the  summation '  extending  over  those  sides  of  the  polygon,  being  in  number 
half  of  the  sides  of  the  first  kind,  which  are  transformed  into  their  conjugates 
by  the  fundamental  substitutions  of  the  group. 


782 


EXCESS   OF   NUMBER   OF   ZEROS 


[307. 


Draw  all  the  pairs  of  tangents  at  the  extremities  of  the  bounding  arcs 
of  the  fundamental  polygon  of  reference : 
then  the  angles,  such  as  c  and  c'  above, 
are  internal  angles  of  the  rectilinear 
polygon  formed  by  the  straight  lines. 
The  remaining  internal  angles  of  this 
new  polygon  are  the  angles  at  which 
the  arcs  cut,  which  are  the  angles  of 
the  curvilinear  polygon :  and  therefore 
their  sum  is  the  sum  of  the  angles  in 
the  cycles,  that  is,  the  sum  is  equal  to 

^27r 


iU-i 


2-; 


Fig.  125. 


where  ^^^  is  the  sum  of  the  angles  in 

one  of  the  cycles.  Now  let  2n  be  the  number  of  sides  of  the  first  kind  in 
the  curvilinear  polygon,  so  that  n  is  the  number  of  fundamental  substitutions 
in  the  group :  hence  the  number  of  terms  in  the  above  summation  for  E  is 
n,  and  therefore 

E  =  -mn  +  ^t(c  +  c'). 
Moreover,  the  rectilinear  polygon  has  4n  sides :  and  therefore  the  sum  of  the 


But  &is  sum  is  equal  to  S  (c  +  c')  +  2  — 


internal  angles  is  (4?i  -  2)  tt. 

where  the  first  summation  extends  to  the    different  conjugate   pairs  and 
the  second  to  the  different  cycles :    thus 


Therefore 


(4w  -  2)  TT  =  2  (c  +  c')  +  2'jrt 
E  =  —  mn  +  m  (2?i  —  1)  —  mX 


=  m{n  —  l  —  ^  — 


1^ 


where  the  summation  extends  over  all  the  different  cycles  in  the  fundamental 
polygon.  Hence  for  a  function,  which  is  constructed  from  the  additive 
element  H  {z)  and  exists  only  within  the  fundamental  circle  of  the  group,  the 
excess  of  the  number  of  its  irreducible  zeros  over  the  number  of  its  irreducible 
accidental  singularities  is 


m 


72-1-S- 


where  m  is  the  parametric  integer  of  the  function  constructed  in  series,  2n  is 


the  number  of  sides  of  the  first  kind  in  the  fundamental  polygon,  —  is  the  sum 


307.]  ^  OVEE   NUMBER   OF   SINGULARITIES  783 

of  the  angles  in  a  cycle  of  the  first  kind  of  corners,  and  the  summation  extends 
to  all  these  cycles. 

The  number  of  irreducible  accidental  singularities  has  already  been 
obtained ;    it  is  finite,  and  thus  the  number  of  irreducible  zeros  is  finite. 

Secondly,  let  the  function  exist  all  over  the  plane :  then  the  irreducible 
points  are  (i)  points  lying  within  (or  on)  the  boundary  of  the  fundamental 
polygon  of  reference  within  the  fundamental  circle  and  (ii)  points  lying 
within  (or  on)  the  boundary  of  the  fundamental  polygon  of  reference  without 
the  fundamental  circle,  the  outer  polygon  being  the  inverse  of  the  inner  poly- 
gon with  regard  to  the  centre.  For  such  a  function  the  excess  of  the  number 
of  irreducible  zeros  over  the  number  of  irreducible  accidental  singularities  is 
the  integral 

1     [(d'{z) 


27ri  J  S  (s)  ^^' 

taken  positively  round  the  boundaries  of  both  polygons.  We  shall  assume 
that  there  are  no  zeros  and  no  infinities  on  the  path  of  integration ;  the 
result  can,  however,  be  shewn  to  be  valid  in  the  contrary  case. 

For  the  sides  of  the  internal  polygon  that  are  of  the  first  kind  the  value 
of  the  integral  is,  as  before,  equal  to 

7n[n  —  l  —  z  — 

and  for  the  sides  of  the  external  polygon  that  are  of  the  first  kind,  the  value 
is  also 

1 


m  ( n  —  1  —  S 
V  /", 

Let  the  value  of  the  integral  along  the  sides  of  the  second  kind  in 
the  internal  polygon  be  /.  Those  lines  are  also  sides  of  the  second  kind 
in  the  external  polygon ;  but  they  are  described  in  the  sense  opposite  to 
that  for  the  internal  polygon,  the  integral  being  always  taken  positively: 
hence  the  value  of  the  integral  along  the  sides  of  the  second  kind  in  the 
external  polygon  is  —  /. 

Hence  the  excess  of  the  number  of  irreducible  zeros  over  the  number  of 
irreducible  accidental  singularities  of  a  function  %  {z),  which  is  constructed 
from  the  additive  element  E  {z)  and  exists  all  over  the  plane,  is 

2m  {n—l—'2  — 

where  the  summation  extends  over  all  the  cycles  of  the  first  category  of  either 
{but  not  both)  of  the  fundamental  polygons  of  reference. 

As  before,  the  number  of  irreducible  zeros  of  such  a  function  is  finite, 
because  the  number  of  irreducible  accidental  singularities  is  finite. 


784  FUCHSIAN   FUNCTIONS  [307. 

In  every  case,  this  excess  depends  only  upon 

(i)      the  parametric  integer  m,  used  in  the  construction  of  the  series : 
(ii)     the  number  of  sides,  2w,  of  the   first  kind  in   the   polygon  of 

reference : 
(iii)     the  sum  of  the  angles  in  the  cycles  of  the  first  category. 

Ex.     Prove  that  a  corner  belonging  to  a  cycle  of  the  first  category  is  in  general  a  zero ' 

of  order  ^,  such  that 

p=  -m  (mod.  fj), 

where  27r//i  is  the  sum  of  the  angles  in  the  cycle :  and  discuss  the  nature  of  the  corners 
which  belong  to  cycles  of  the  remaining  categories.  (Poincare.) 

308.  We  are  now  in  a  position  to  construct  automorphic  functions,  using 
as  subsidiary  elements  the  pseudo-automorphic  functions  which  have  just 
been  considered. 

For,  if  we  take  a  couple  of  these  functions,  @i  and  @2j  associated  with  a 
given  infinite  group,  characterised  by  the  same  integer  m,  and  arising  through 
different  additive  elements  H  {z),  then  we  have 


\'yz  +  h 


=  (7^  +  sr«@,(4 


where  ^  is  any  one  of  the  substitutions  of  the  group ;  and  therefore 


\^z  +  h)      @i(^) 


^  (az  +  /3\      @2  {2) 


&f)  =  ^»(^)' 


\^z  +  S  / 

that  is,  the  quotient  of  two  such  functions  is  automorphic.     Denoting  the 
quotient  by  Pn(z)*,  we  have 

^cizj\-  /3^ 

^yz 

the  automorphic  property  being  possessed  for  each  of  the  substitutions. 

It  thus  appears  that  such  functions  exist :  their  essential  property  is 
that  of  being  reproduced  when  the  independent  variable  is  subjected  to  any 
of  the  linear  substitutions  of  the  infinite  group. 

The  foregoing  is  of  course  the  simplest  case,  adduced  at  once  to  indicate 
the  existence  of  the  functions.  The  construction  can  evidently  be  general- 
ised: for,  if  we  have  any  number  of  functions  @i,  ...,  ©^,  ^i,  ...,  ^g  with 
characteristic  integers  wij,  ...,  mr,n^,  ...,  ng  and  all  associated  with  one  group 

*  Poincare  calls  such  functions  Fuchsian  functions:  as  already  indicated  (§  297),  I  have 
preferred  to  associate  the  general  name  automorphic  with  them.  But,  because  Poincare  himself 
has  constructed  one  class  of  such  functions  by  means  of  series  as  in  the  foregoing  manner,  his 
name,  if  any,  should  be  associated  with  this  class :   the  symbol  Pn  {z)  is  therefore  used. 


308.]  TWO   CLASSES   OF   AUTOMORPHIC   FUNCTIONS  785 

while  constructed  from  different  additive  elementary  functions  H  (z),  then, 
denoting 

^i(^) ^s{^) 

by  Pn  (2),  we  evidently  have 

SO  that,  provided  only  2  ti^  ^  2  Wg, 

3=1  9=1 

the  function  is  automorphic.  If  we  agree  to  call  m,  the  integer  characteristic 
of  a  pseudo-automorphic  function,  the  degree  of  that  function,  then  the  quotient 
of  two  products  of  pseudo-automorphic  functions  is  autovnorphic,  provided  the 
products  be  of  the  same  degree. 

There  are  evidently  two  classes  of  automorphic  functions:  those  which 
exist  all  over  the  plane,  and  those  which  exist  only  within  the  fundamental 
circle.  The  classes  are  discriminated  according  to  the  composition  of  the 
functions  from  the  subsidiary  pseudo-automorphic  functions. 

When  the  pseudo-automorphic  functions,  which  enter  into  the  composi- 
tion of  the  function,  exist  all  over  the  plane,  then  the  automorphic  function 
exists  all  over  the  plane.  But  when  the  pseudo-automorphic  functions,  which 
enter  into  the  composition  of  the  function,  exist  only  within  the  fundamental 
circle,  then  the  automorphic  function  exists  only  within  the  circle. 

Ex.     Consider  the  quantities  go,  and  ^3,  defined  in  §  123.     We  have 

o,  =  6022  -. TT , 

where  the  double  summation  extends  over  all  positive  and  negative  integer  values  of  TOi 
and  wi2,  simultaneous  zeros  alone  excepted.     Writing  w  =  a)]/(»2,  we  have 


and  therefore 


1 

Oo  (c<))  =  60a)2      22  7 ; TT  ; 


where 


yyod- 

Mi = mia + m^y,     M^ = m-^j^  +  m^^. 

Because  ab  —  ^y=\,  we  have 

m^  =  Mxb  -  M^y,     »i2  =  -  Mi^  +  M^a. 

If  then  h,  iS,  y,  8  be  integers,  subject  solely  to  the  condition  aS-/3y  =  l,  it  follows  that,  to 
every  pair  of  integer  values  of  mj  and  7^2,  there  corresponds  one  pair  (and  only  one  pair) 
of  integer  values  of  i/j  and  M^ ;  and  conversely.  Also,  simultaneous  zeros  for  the  one  set 
are  simultaneous  zeros  for  the  other ;  so  that 

1  1 

22.,.      .    ,,,,=22 


{Mia  +  M^y  {m^ca  +  mzY' 

50 


786  ESSENTIAL  SINGULAEITIES   OF   AUTOMORPHIC   FUNCTIONS  [308. 

Consequently, 


Similarly,  as 


we  have 


qr,  (a,)  =  14022  , rs 

=  140w2-<'22  -. ; r^, 


It  therefore  follows  that  g'^gi~^  is  automorphic ;   and  if  we  take 
J:J-\:\=gi:  27^3^  :  gi  -  ^^gi  ( =  A), 
then  J  is  an  automorphic  function,  such  that 

where  a,  /3,  y,  S  are  integers  subject  to  the  condition  ah-i^y=\.     Evidently  ^2  (<»)  ^-iid 
^3  (o))  are  pseudo-automorphic  for  the  group.     Taking 

ei=/i(l  +  c'),     e3=-/x(l+o),     e2=M(c-c')j 

where  c  +  c'  =  l,  we  have 

5r2=-12^2(cc'-l),      5f3=  -4/i3  (2 +  cc')(c-c'), 
A=16.272.,i6c2c'2^ 

so  that 

,    (1-C  +  C2)3 
^2^     C^(l-C)2    ' 

as  in  §  303,  where  other  examples  are  given. 

309.  It  is  evident  that  all  the  essential  singularities  of  an  automorphic 
function,  thus  constructed,  lie  on  the  fundamental  circle.  For  whether  the 
pseudo-automorphic  functions  exist  only  within  that  circle  or  over  the  whole 
plane,  all  their  essential  singularities  lie  on  the  circumference :  so  that, 
whatever  be  the  constitution  of  the  various  subsidiary  pseudo-automorphic 
functions,  all  the  essential  singularities  of  the  automorphic  function  lie  on 
the  fundamental  circle. 

Next,  the  number  of  irreducible  zeros  of  an  automorphic  function  is  equal 
to  the  number  of  its  irreducible  accidental  singularities.  For  an  irreducible 
zero  of  an  automorphic  function  is  either  (i)  an  irreducible  zero  of  a  factor 
in  the  numerator  or  (ii)  an  irreducible  accidental  singularity  of  a  factor  in 
the  denominator ;  and  similarly  with  the  irreducible  accidental  singularities 
of  the  function.  The  numerator  and  the  denominator  may  have  common 
zeros ;  this  will  not  affect  the  result. 

First,  let  the  automorphic  function  exist  only  within  the  circle :  then 
each  of  its  factors  exists  only  within  the  circle.  The  space  without  the  circle 
is  not  significant  for  any  of  the  factors  of  the  function,  because  they  do  not 


309.]  LEVEL   POINTS   OF   AUTOMORPHIC   FUNCTIONS  787 

there  exist.  Let  ei,  ...,  e^,  e/,  ...,  e/  be  the  excesses  of  zeros  over  accidental 
singularities  for  the  pseudo-automorphic  functions  within  the  fundamental 
circle :  then 

eg  =  7W„  ( n  —  1  -  2  — 
\  /til 

where  n  and  S  —  are  the  same  for  all  these  functions,  and 

Now  the  excess  of  zeros  over  poles  in  the  denominator  becomes,  after  the 
above  explanation,  an  excess  of  poles  over  zeros  for  the  automorphic 
function :  hence,  for  this  automorphic  function,  the  excess  of  zeros  over 
accidental  singularities  is 

r  s 

3=1  g=l 

=  f  /?,  —  1  —  X  —  )  (  S  rria—  X  ric 
=  0, 


1 

H^    \q  =  l  g  =  l 


by  the  condition   X  mq=  2  iiq.     Hence  the  number  of  irreducible  zeros  of 

9=1  q=l 

the  automorphic  function  is  equal  to  the  number  of  irreducible  accidental 
singularities. 

Secondly,  let  the  automorphic  function  exist  all  over  the  plane ;  then 
all  its  factors  exist  all  over  the  plane.  For  the  present  purpose,  the  sole 
analytical  difference  from  the  preceding  case  is  that  each  of  the  quantities  e 
now  has  double  its  former  value  :  and  therefore  the  excess  of  the  number  of 
zeros  over  the  number  of  poles  is 

2{n—l—2  —  ){  Sm^—  2  Ug 

\  fliJ    \q  =  l  3  =  1 

which,  as  before,  vanishes.  Hence  the  number  of  irreducible  zeros  of  the 
automorphic  function  is  equal  to  the  number  of  its  irreducible  accidental 
singularities. 

It  follows,  as  an  immediate  Corollary,  that  the  number  of  irreducible 
points  for  which  an  automorphic  function  assumes  a  given  value  is  equal  to 
the  number  of  its  irreducible  accidental  singularities.     For 

Pn{z)-A, 

where  J^  is  a  constant,  is  an  automorphic  function:  the  number  of  its 
irreducible  accidental  singularities  is  equal  to  the  number  of  its  irreducible 
zeros,  that  is,  it  is  equal  to  the  number  of  irreducible  points  for  which 
Pn  {z)  assumes  an  assigned  value. 

50—2 


788  DIFFERENT   FUNCTIONS    FOR   ONE   GROUP  [309. 

Moreover,  each  of  these  numbers  is  finite :  for  the  number  of  irreducible 
zeros  and  the  number  of  irreducible  accidental  singularities  of  each  of  the 
component  pseudo-automorphic  factors  is  finite,  and  there  is  only  a  finite 
number  of  these  factors  in  the  automorphic  function.  The  integer,  which 
represents  each  number,  will  evidently  be  as  characteristic  of  these  functions 
as  the  corresponding  integer  was  of  functions  with  linear  additive  periodicity. 

Note.  The  preceding  method,  due  to  Poincar6,  of  expressing  the  pseudo- 
automorphic  functions  as  converging  infinite  series  of  functions  of  the 
variable,  is  not  the  only  method  of  obtaining  such  functions.  It  was 
shewn  that  uniform  analytical  functions  can  be  represented  either  as 
converging  series  of  powers  or  as  converging  series  of  functions  or  as 
converging  products  of  primary  factors,  not  to  mention  the  (less  useful) 
forms  intermediate  between  series  and  products.  The  representation  of 
automorphic  functions  as  infinite  products  of  primary  factors  is  considered 
in  the  memoirs  of  Von  Mangoldt  and  Stahl,  already  referred  to  in  |  297. 

310.  Let  Pni{z),  Pn2{z),  say  Pj  and  P^,  be  two  automorphic  functions 
with  the  same  group,  constructed  with  the  most  general  additive  elements ; 
and  let  the  number  of  irreducible  zeros  of  the  former  be  k^,  and  of  the 
latter  be  k^,. 

Then  for  an  assigned  value  of  P^  there  are  /Ci  irreducible  points :  P^  has 
a  single  value  for  each  of  these  points,  and  therefore  it  has  Kj  values  alto- 
gether for  all  the  points,  that  is,  it  has  k^  values  for  each  value  of  Pj. 
Similarly,  Pj  has  k.2  values  for  each  value  of  Pg.  Hence  there  is  an  alge- 
braical relation  between  Pj  and  Pg  of  degree  k^  in  P^  and  of  degree  k^  in  Pg, 
which  may  be  expressed  in  the  form 

Let  Pn{z),  say  P,  be  any  other  uniform  automorphic  function,  having 
the  same  group  as  Pj  and  P^',  and  let  k  be  the  number  of  its  irreducible 
zeros.     Then  we  have  an  algebraical  equation 

F,iP,P,)  =  Q, 
which  is  of  degree  Ki  in  P  and  of  degree  /c  in  Pj ;  and  another  equation 

P,(P,P,)  =  0, 
which  is  of  degree  «2  in  P  and  of  degree  ic  in  Pg.     The  last  two  equations 
coexist,  in  virtue  of  the  relation 

P,,(A,P,)  =  o 

satisfied  by  P^  and  Pg.  Since  Pi  =  0  =  Po  coexist,  the  ordinary  theory  of 
elimination  leads  to  the  result  that  the  uniform  function  P  can  be  expressed 
rationally  in  terms  of  Pj  and  Pg,  so  that  we  have  the  theorem  that  evei'y 
automorphic  function  associated  with  a  given  group  can  be  expressed  rationally 
in  terms  of  two  general  a. utomorphic  functions  associated  with  that  gi^oup:  and 
between  these  two  functions  there  exists  an  irreducible  algebraical  relation. 


310.]  ALGEBRAICAL   RELATIONS  789 

The  genus  (§  178)  of  this  algebraical  relation  can  be  obtained  as  follows. 
Let  N  denote  the  genus  of  the  group,  determined  as  in  §  293  :  then  the 
fundamental  polygon  of  reference,  if  functions  exist  only  within  the  circle,  or 
the  two  fundamental  polygons  of  reference,  if  functions  exist  over  the  whole 
plane,  can  be  transformed  into  a  surface  of  multiple  connectivity  2iV+  1.  The 
automorphic  functions  are  uniform  functions  of  position  on  this  surface;  and 
hence,  as  in  Riemann's  theory  of  functions,  the  algebraical  relation  between 
two  general  uniform  functions  of  position,  that  is,  between  two  general  auto- 
morphic functions  is  of  genus  N,  where  N  is  the  genus  of  the  group*. 

It  is  now  evident  that  the  existence-theorem  and  the  whole  of  Riemann's 
theory  of  functions  can  be  applied  to  the  present  class  of  functions,  whether 
actually  automorphic  or  only  pseudo-automorphic.  There  will  be  functions 
of  the  same  kinds  as  on  a  Riemann's  surface :  the  periods  will  be  linear 
numerical  multiples  of  constant  quantities  acquired  by  a  function  when  its 
argument  moves  from  any  position  to  a  homologous  position  or  returns  to  its 
initial  position.  There  will  be  functions  everywhere  finite  on  the  surface, 
that  is,  finite  for  all  values  of  the  variable  z  except  those  which  coincide  with 
the  essential  singularities  of  the  group.  The  number  of  such  functions, 
linearly  independent  of  one  anot-her,  is  N ;  and  every  such  function,  finite 
for  all  values  of  z  except  at  the  essential  singularities,  can  be  expressed  as 
a  linear  function  of  these  N  functions  with  constant  coefficients  and  (possibly) 
an  additive  constant.     And  so  on,  for  other  classes  of  functions  f. 

311.     Because  Pn  {z)  is  an  automorphic  function,  we  have 

\'yz  +  h' 
and  therefore,  as  aS  —  ^^7  =  1, 

Hence,  if  @  {z)  be  a  pseudo-automorphic  function  with  m  for  its  characteristic 
integer,  so  that 

®(|Tf)  =  ^^^  +  ^)"®<^>- 

0iZ-\- 13\ 


@ 


r^z  +  hj  B  {z) 


p^./az  +  lBW^      {Pn(z)Y 


have 

J  P-n'     

\yz  +  B/ 

*  It  may  happen  that,  just  as  in  the  general  theory  of  algebraical  functions,  the  genus  of  the 
equation  between  two  particular  automorphic  functions  may  be  less  than  N :  thus  one  might  be 
expressed  rationally  in  terms  of  the  other.  The  theorems  are  true  for  functions  constructed  in 
the  most  general  manner  possible. 

t  The  memoirs  by  Burnside,  quoted  in  §  297,  develop  this  theory  in  full  detail  for  the  group 
which  has  its  (combined)  polygons  of  reference  bounded  by  2?i  circles  with  their  centres  on  the 
axis  of  real  quantities,  the  group  being  such  that  the  pseudo-automorphic  functions  exist  over 
the  whole  plane. 


790 


DERIVATIVES   OF   AUTOMORPHIC    FUNCTIONS 


[311. 


that  is,  ©  (2)  [Pn  {z)]~^  is  an  automorphic  function.  Such  a  function  can 
be  expressed  rationally  in  terms  of  Pn{z)  and  some  other  function,  say  of 
P  and  Q :  hence  the  general  type  of  a  pseudo-automorphic  function  with 
a  characteristic  integer  m  is 

(f  r^(^'  «■ 

where  /  is  a  rational  function. 

Corollary.     Two  automorphic  functions  P  and  Q,  belonging  to  the  same 
group,  are  connected  by  the  equation 

For  evidently  unity  is  the  characteristic  integer  of  the  first  derivative  of  an 
automorphic  function. 

This  equation  can  be  changed  to 

§=fiP.  Q). 

where  /  is  a  rational  function  :   moreover,  P  and  Q  are  connected  by  an 

equation 

FiP,Q)  =  0, 

which  is  an  algebraical  rational  equation,  and  can  evidently  be  regarded  as 
an  integral  of  the  above  differential  equation  of  the  first  order,  all  trace  of 
the  variable  2  having  disappeared.     Evidently  the  form  of/  is  given  by 

dF     j.,y~,  r\\^F     ^ 

Again,  denoting ^  by  ^,  and  Pn  (  — — -»  j  by  11  (^),  we  have 


say 
Then 


{yz  +  Sf 


27 


jyz  + 


P" 
I'^P' 


so  that 

n' 


rr 


=  (yz  +  8y 


=  2y{rYZ  +  8)  +  {jz  +  8y-^ 


P" 


rP' 


2Y  +  2y{yz  +  8)^+(yz  +  8Y(^~ 


P'2 


and  therefore         ^[7  ~  f  \  fF  [  ={yz  +  oy 


P'      MP' 


whence 


n'2 


P'2 


where  [P,  z]  is  the  Schwarzian  derivative.     It  thus  appears  that,  if  P  be  an 


311.]  DIFFERENTIAL  EQUATIONS  791 

automorphic  fiinction,  then  (P,  2]  P'~^  is  a  function  automorphic  for  the 
same  group. 

But  between  two  automorphic  functions  of  the  same  group,  there  subsists 
an  algebraical  equation :  hence  there  is  an  algebraical  equation  between 
P  and  [P,  2]  P'~^,  that  is,  P  {2),  an  automorphic  function  of  2,  satisfies 
a  differential  equation  of  the  third  order,  the  degree  of  which  is  the  integer 
representing  the  number  of  irreducible  2eros  of  P  and  the  coefficients  of  which, 
where  they  are  not  derivatives  of  P,  are  functions  of  P  only  and  not  of  the 
independent  variable. 

This  equation  can  be  differently  regarded.     Take 

2/,  =  P'i        y,  =  2P'^; 

then  it  is  easy  to  prove  that 

ld'y,_ld^y,_^{P,2] 
y^dP^     y^dP^     2     p/^    • 

The  last  fraction  has  just  been  proved  to  be  an  automorphic  function  of  2 ; 
and  therefore  it  is  rationally  expressible  in  terms  of  P  and  any  other  general 
function,  say  Q,  automorphic  for  the  group.  Then  y^  and  y^  are  independent 
integrals  of  the  equation 

%  =  y^{P,Q), 

where  Q  and  P  are  connected  by  the  algebraical  equation 

F{P,Q)  =  0. 
.  Conversely,  the  quotient  of  two  independent  integrals  of  the  equation 

^,  =  y<j>(P,Q), 

where  Q  and  P  are  connected  by  the  algebraical  equation 

P(P,  Q)  =  0, 

can  be  taken  as  an  argument  of  which  P  and  Q  are  automorphic  functions : 
the  genus  of  the  equation  P  =  0  is  the  genus  of  the  infinite  group  of  sub- 
stitutions for  which  P  and  Q  are  automorphic*. 

Ex.  One  of  the  simplest  set  of  examples  of  automorphic  functions  is  furnished  by 
the  class  of  homoperiodic  functions  (§  116).  Another  set  of  such  examples  arises  in  the 
triangular  functions,  discussed  in  §  275 ;  they  are  automorphic  for  an  infinite  group,  and 
the  triangles  have  a  circle  for  their  natural  limit.  A  third  set  is  furnished  by  the  polyhedral 
functions  (§§  276—279). 

As  a  last  set  of  examples,  we  may  consider  the  modular-functions  which  were  obtained 
by  a  special  method  in  §  303. 

*  Klein  remarks  (Math.  Ann.,  t.  xix,  p.  143,  note  4)  that  the  idea  of  uniform  automorphic 
functions  occurs  in  a  posthumous  fragment  by  Eiemann  (Ges.  Werke,  number  xxv,  pp.  413 — 416). 
It  may  also  be  pointed  out  that  the  association  of  such  functions  with  the  linear  differential 
equation  of  the  second  order  is  indicated  by  Riemann. 


792  MODULAR-FUNCTIONS  [311. 

First,  we  consider  them  in  illustration  of  the  algebraical  relations  between  functions 
automorphic  for  the  same  group.  It  follows,  from  the  construction  of  the  group  and  the 
relation  of  c  to  w,  that,  in  the  division  of  the  plane  by  the  group  with  Uw  and  Vw  for  its 

fundamental  substitutions,  where 

^        ^  w 

uw  =  w  +  2,        Vw=z — jr— , 
1— 2w 

there  is  only  a  single  point  in  each  of  the  regions  for  which  c  has  an  assigned  value ;  hence, 

regarding  c  as  an  automorphic  function  of  w,  the  number  k  (§  310)  is  unity.     If  there  be 

any  other  function  C  of  w,  automorphic  for  this  group,  then  between  C  and  c  there  is  an 

algebraical  relation  of  degree  in  C  equal  to  the  number  k.  for  c,  that  is,  of  the  first  degree 

in  G.     Hence  every  function  automorphic  for  the  group,  ivhose  fundamental  substitutions 

are  U  and  V,  where 

w 

Uw  =  W  +  '2,  Vw=- ;r— , 

'  1  -  2w 

is  a  rational  function  of  c. 

In  the  same  way,  it  can  be  inferred  that  every  function  automorphic  for  the  group, 
whose  fundamental  sicbstitutions  are 

Uw=w  +  2,        Tw= , 

is  a  rational  function  of  cc' ;   and  that  every  function  automorphic  for  the  group,  whose 
fundamental  substitutions  are 

Sw  =  iv-\-l,       Tw= , 

w 

that  is,  automorphic  for  all  substitutions  of  the  form  -^,  where  a,  b,  c,  d  are  real 

integers,  such  that  ad-bc=\,  is  a  rational  function  of  J—^-^j v^  • 

Secondly,  in  illustration  of  the  general  theorem  relating  to  the  differential  equation 
of  the  third  order  which  is  characteristic  of  an  automorphic  functioUj  we  consider  the 

iK' 
quantity  c  as  a  function  of  the  quotient  of  the  quarter-periods.     Let  z  denote  -^  :  then 

because  every  function,  automorphic  for  the  same  group  of  substitutions  as  c,  is  a  rational 
function  of  c,  we  have 

{c  z\  . 

^-^  =  I'ational  function  of  c: 

*  c^ 

and  therefore,  by  a  property  of  the  Schwarzian  derivative, 

{z,  c]=  —same  rational  function  of  e. 
By  known  formulae  of  elliptic  functions,  it  is  easy  to  shew  that 

thus  verifying  the  general  result. 

Similarly,  it  follows  that  j-^  >  ^j-,  where  O  —  cc',  is  a  rational  function  of  cc',  the  actual 

value  being  given  by 

UK'     J       1  -  5(9 -h  16^2 


K  '     \       2^2(1  _ 4^)2  ' 
and  that  j-^j  J\  is  a  rational  function  of  J,  the  actual  value  being  given  by 

\iK'      1  _  16J^-mJ-330 

In  this  connection  a  memoir  by  Hurwitz*  may  be  consulted. 
*  Math.  Ann.,  t.  xxxiii,  (1889),  pp.  345—352. 


311.]  CONCLUSION  793 

The  preceding  application  to  differential  equations  is  only  one  instance 
in  the  general  theory  which  connects  automorphic  functions  with  linear 
differential  equations  having  algebraic  coefficients.  This  development 
belongs  to  the  theory  of  differential  equations  rather  than  to  the  general 
theory  of  functions :  its  exposition  must  be  reserved  for  another  place. 


Here  my  present  task  comes  to  an  end.  The  range  of  the  theory  of 
functions  is  vast,  its  ramifications  are  many,  its  development  seems  illimit- 
able :  an  idea  of  its  freshness  and  its  magnitude  can  be  acquired  by  noting 
the  results,  and  appreciating  the  suggestions,  contained  in  the  memoirs  of 
the  mathematicians  who  are  quoted  in  the  preceding  pages. 


MISCELLANEOUS   EXAMPLES. 
I. 

1 .  Find  the  curves  which  cut  orthogonally  the  family  of  curves  r^  r^  r^ = a,  where  ri ,  r2 ,  r^ 
are  the  distances  of  a  variable  point  from  three  fixed  points. 

2.  P  and  Q  are  conjugate  functions,  so  that  P  +  ^■$  is  a  function  f{z)  of  a  compl^;^ 
variable  z=x-\-iy  and 

P-§=(cos.r  +  sin^-e-'')/(2cos.r-e!'-e-!'). 

Find  /(?),  subject  to  the  condition  /(i?r)  =  0. 

3.  Evaluate  the  integral 

j  _„l-2a;cos^  +  ^2    ■*' 

where  a,  5  are  real,  obtaining  the  necessary  limitation  of  the  values  of  a,  for  which  the 
integral  is  finite. 

4.  Evaluate  rigorously  the  integrals 

n  sinao?.^ 

^  '     j  _i  1  —  2«cosa  +  .»2' 

(2)  dx, 

Jo       ^ 

in  which  a  may  have  any  real  value. 

State  in  each  case  the  value  or  values  of  a  for  which  the  integrals  are  discontinuous 
functions  of  a. 

5.  Prove  that,  if  a  and  b  are  real,  then 

f  °°   sin (x—a)  , 

I         ^ -dx=ir, 

J  _oo      X  —  a 

/■*    sin  {x  -  a)  sin  (^  -b)  j  _     sin  (a-b) 
J -oc        {x-a){x-h)  _  {a-b)     ' 

6.  Prove  that,  if  a  is  positive, 

jo  2Va/ 

Justify  differentiating  the  integral  under  the  sign  of  integration  any  number  of  times 
with  respect  to  a,  and  so  obtain  the  formula 

/""       ,  ,    ,  1.3 (2%-l)    f      TT       1^ 

Finally  justify  the  deduction  of  the  formula 

1        6* 

I     e-aa;'^cos26^'c?^=-i— i    e    «, 
by  expansion  of  the  cosine  as  a  power  series  and  integration  term  by  term. 


MISCELLANEOUS   EXAMPLES  796 

7.     Prove  that,  if  0  <  w  <  1, 


/; 


smo;  , 


0     ^"  2r(w)sin(i7r?i) 

8.     Prove  that 


X  cos  ax  ,         TT^e-aT 
ax—  ■ 


0    sinh  a;  (1  +  e-ajr)2 ' 

-=loar2. 


0  {l-hx'^)  cosh  ^TT^ 

9.  Obtain  expansions  of  the  cosine  and  sine  integrals  defined  by 

...      f^cosx  J         ...       f'^sinx  ^ 
ci[x)=  j     — —ax,     si{x)=  I     dx, 

J  X       ^  J  0       ^  ^ 

appropriate  for  calculation  (i)  when  x  is  large,  and  (ii)  when  x  is  small. 

/■"    g—iex 

Express    I     ,„       „  dk  in  terms  of  the  cosine  integral  and  sine  integral. 

10.  Shew  that,  if  m  is  real  and  positive, 
sin  mx 


/: 


11.  Prove  that,  when  a  is  real  and  positive, 
°°    cos  a?     ,       7re~«, 

12.  By  contour  integration  (or  any  method)  prove  that 
*  sin  a?  ,       TT  ,  .      , 
_.^(^2-2.r+2)^^  =  2^^^-"°'^  +  '^^y)' 

1  —  cos  X 


/; 


/; 

j  _oo  « {x"^  -  "ix  +  i)  """~2e 
the  angle  y  being  the  unit  of  circular  measure. 

13.  Prove  that,  when  n  is  even, 

,  r      sinhM^     "I      '•='*,        ,  r         2rw      .  ,     1 

tan-M — T-^ 7^^r=    2  tan-Msec;r^ — vsmha?}-. 

(cosh(n  +  l)^J       ,.=1  (       2»+l  j 

14.  Prove  that,  if  a  >  0,  the  value  of  the  integral 


/ 


taken  round  three  sides  of  a  rectangle  from 

z  =  R  to  R  +  ni,     R  +  ni  to  —R  +  ni,     —R  +  ni  to  —R, 
will  tend  to  zero  as  R  and  n  tend  to  oc  (n  being  an  even  integer). 
Deduce,  or  otherwise  prove,  that 


/; 


l+x' 
15.     Prove  that,  if  a  <  1, 


tanh  (^  ttx) 5  dx  =  a  cosh  a  —  sinh  a  log  (2  sinh  a). 


sinrx  ,       tt  (    ,,  1 

a.r  =  -  i  coth  Trr  - 


0   e=^-l  2   (.  Trr 

1         f^    -x'^-e^  _    1       /■" 


=lH 1 v. 


796  MISCELLANEOUS   EXAMPLES 


16.     Prove  that,  if  a  is  positive, 
X  —  sin  X 


/: 


(^^=|7ra-*(^a2-a  +  l-e-«) 


0  x^{cfi  +  x^) 
If  ■ 

.■?;  =  a+^sin  ^, 

obtain  the  expansion  of  in  powers  of  t,  in  the  form 


x  —  a 


cos  a    .  =°  f^         o?«'  +  i(sin"a) 

-    2  ^  - 


x  —  a      t  sin  a     sin  a     n=\  (?i  + 1) !  *i        c?a' 
18.     By  evaluating  the  integral  (where  c  is  real) 


/ 


—  cosh  {v/(c2  -  z^)}  &, 


taken  round  a  path  which  consists  of  the  real  axis  and  a  large  semicircle,  prove  that 

/ 

Prove  similarly  that 


cosh  {x/(c2  -  x'^)}  dx  =  \'n  (cosh  c  -  ^). 


r  cos  iax)  ''°Mf "'  .f^^  dx  =  Uh  {c  v/(l  -«^)}, 

if  0<a<l,  where  /^(^)  =  i +|!  +  _^^  +  ...  ; 

and  shew  that  the  integral  is  equal  to  Jtt  if  a  =  l,  and  is  zero  if  a>  1. 

19.     Criticise  the  following  argument : —  ~ 

"Putting  2=rcos^  =  rja;  r^^x^^-y^-^-z^,  ar  =  rsin^,  the   functions  (where  %  is   a 
positive  integer) 


2^i 


2» 

(2  +  ^ar  cos  (^)"  0?^', 


277 

(2  +  m  COS  0)  -  ("^  + 1)  C?^ 
0 


are  finite  over  a  unit  sphere,  and  for  ar  =  0  they  reduce  to  r"  and  r~^^'^^^  respectively. 
They  must  therefore  be  equivalent  to  r»P„  {]i)  and  r-(''  +  i)P^(ju.)  respectively." 
Shew  that  the  conclusion  is  invalid  with  regard  to  the  second  integral  when  z  is 

negative :   and  give  the  correct  result. 

Prove  that,  if  the  coefficients  a^  are  such  that  the  series  2  |a„  — a^+i  |  is  convergent, 

then  the  series  2a„  P„  (/x)  converges  uniformly  in  any  interval  for  /x  which  falls  within  the 

interval  (  —  1,  +1). 

Determine  an  asymptotic  formula  for  the  sum  of  2 P^  {fj),  as  r  tends  to  1  and  fi 

tends  to  1. 

20.     Prove  that,  when  the  real  part  of  ?i  is  greater  than  —  1, 


l       fc  +  cci    ,     .,      ... 


where  J^  (x)  is  Bessel's  Function  of  order  n  satisfying  the  equation 


Establish  the  relations 


/; 
/ 


dx^     X  dx      \       x^J  " 


bn  '' 


e~ax"'Jn{bx)x^'-^dx  =  j^^^^e   4«,     ?i>-l. 


1  (26')" 

e-axJn(bx)x'^dx=-=  T(n  +  h) — ^ — - — rr?     «>-i- 


MISCELLANEOUS   EXAMPLES  797 

21.     Prove  that  e«»'cose  may  be  expanded  in  either  of  the  two  forms 
JoW  +  2  2  ^■»J„(r)cos?^<9, 


2  {2n  +  l)  i^Jn^i  (r)  P„  (cos  6). 
By  employing  the  first  expression,  or  otherwise,  prove  the  theorem 

Jo{(?-H/2-2rr'eos(9)4}  =  Jo(0«4(''')  +  2  2  J"„  (r) /„  (/)  cos  to^. 
22.     Prove  that 


^m  /•"&       ri 

'^™^^)  =  2;;?j./^Pl2 
where  the  path  of  integration  encloses  the  negative  real  axis  in  the  w-plane. 

Prove,  (i),  that 


/: 


J^,{qx)e-^^^-x"^^^dx=~'^—    -  ^^ 


(ii),  that,  with  In{x)  —  i~'^J^{ix\ 

1    r** 

-— .         ey'h{z)z'^dz 

_     8m{n  +  m)TT        T{n+m  +  l)  /n  +  ??i  +  l     n  +  7n  +  2  ^A 

-  ~  2»r(?i+l)2/™+'"  +  i      V       2        '         2        ''*  +  ^'.y     y' 

where  |y  |  >  1,  and  the  real  parts  of  ay,  6y,  are  negative  ;  and 

(iii),  that  j  ^  J^{px)J„{qx)a;r--^^dx=^  2--^-4{n-my 

where  q>  p,  n'>m>  -1. 

23.     Shew  that 

eXcose  =  2«-in(«-l)^r(n  +  X-)  C,»(cos^)%^\ 
fc=o  '>■  * 

where  Cfc"  (cos  5)  is  the  coefl&cient  of  k''  in  the  expansion  of  ( 1  —  2A  cos  ^  4-  A^)  ""■  in  powers  of  h. 
Prove  further  that 


27^^  j  (._ooi  \ 


^(x6\  (if 

TyT' 

where  c  is  any  real  positive  quantity,  the  real  part  of  n  is  greater  than  - 1  and  the  real 
parts  of  (a  ±  b)^  are  positive,  and  hence  deduce  the  addition  theorem  for 

J„J{a^  +  b^-2abGosd)  , 

{J(a^^b^'-2ab  cos  6)}'' ' 

/■"  e^f*"^*^)    ,    .  1  ,         27r         >_i      ■ 

24.     If  X,  g,  be  positive  numbers,  prove  that  /  .      dz  is  equal  to  jY?x^)  ^^ 

to  0,  according  as  c  is  positive  or  negative.     Explain  how  this  integral  may  be  employed 
in  the  transformation  of  a  multiple  integral  taken  through  a  limited  domain. 

Hence  or  otherwise,  prove  that 

where  the  double  integration  is  taken  for  all  values  of  ^,  ?;,  such  that  1  — 2~p^^^^  ^^^ 
6q  is  the  positive  root  of  the  equation  1  — ^-y-^  -  tAxa  -  -^  =  0. 


798  MISCELLANEOUS   EXAMPLES 

25.  If  the  contour  of  the  integral  separates  positive  and  negative  sequences  of  poles 
of  the  subject  of  integration  and  at  a  large  distance  from  the  origin  is  parallel  to  the 
imaginary  axis,  shew  that,  for  general 'complex  values  of  a^,  Cg,  ^i  and  /32,  the  integral 


^-.  / r (ai  +  s)  r {a2+s)  r (/3i -s)  r  (/^2-«)  <^« 


27 

is  convergent  and  equal  to 

r  (ai  +  ^i)  r  (g^  +  132)  r  (ai  +  jgg)  r  (a2  +  ^l)  ' 
r(ai  +  a2+^i+^2) 
Deduce,  or  obtain  independently,  Riemann's  transformation 

F{2a,2^;  y;  x}  =  F{a,^;  y,  4x{l-a;)}, 
when  y  =  a  +  ^  +  ^,  R{x)<^,  |^|<1,  |  4.*  (1 -a;)  |  <  1. 

26.  Give  the  definition  of  V  (z)  by  means  of  an  infinite  product  and  prove  that  it 
agrees  with  the  equation 

r{z)    J 

where  the  integral  is  taken  along  a  path  which  encloses  the  negative  real  axis  and  inter- 
sects the  positive  real  axis. 

Prove  that 

2Tri        fc+ix  

~=  T{t)z-*dt,         (c>0), 

^  J    C—icc 

where  the  phase  of  ^  is  between  —  ^tt  and  -t-^Tr. 

27.  The  generalised  Riernann  ^-function  is  defined,  when  ^  (  -  j  >  0,  by  the  relation 

,     2r(l-s)   r    ,        ,,_!     e-«^      , 
t(s,a,a))  =  — I -I    (-^)i    dx, 

the  contour  of  the  integral  embracing  a  straight  axis  L  from  the  origin  through  the  point 
-  to  infinity,  and  ( —  ^)1~^  having  a  cross  cut  along  the  axis  L  and  being  real  when  a;  is 
real  and  negative. 

Obtain  an  expression  for  the  continuation  of  the  function  for  all  finite  values  of  \a\. 

Shew  further  that,  except  for  particular  values  of  a/co  or  s,  the  equality 

™-i     1        >/     V     '^s'n(a)rd''  x^-n 

^=0  («  +  '*<»)  "=o    nl     \_aaf'   l-sJx=m<o 

exists,  and  is  asymptotic  in  Poincare's  sense  when  m  is  large. 

28.  Prove  by  contour  integration  that 

I     cos -f^lossnu  du= -^KtAnhi  ~  -=] . 
j  0  ^  -  \2  KJ 

(It  may  be  assumed  that  K  and  K'  are  real  and  positive.) 

11. 

29.  Find  the  most  general  values,  for  any  path  of  integration,  of 

p  ^  n      dz 

30.  Investigate  the  periods  of  the  integrals 


MISCELLANEOUS   EXAMPLES  799 

Let  f{z)  denote  a  function  that  is  regular  over  the  whole  of  any  finite  region  of  the 
plane  of  z,  including  the  finite  part  of  the  range  of  the  axis  of  real  quantities.  Prove  that 
the  function 


where  the  integration  extends  along  that  axis,  is  regular  for  all  places  within  the  region 
off  the  axis  ;  prove  also  that,  for  two  adjacent  places  on  opposite  sides  of  the  point  k  on 
the  real  axis  between  0  and  1,  the  difference  of  values  of  the  function  is  2iTif{k). 

31.  A  variable  u  is  defined  as  the  integral  of  a  rational  function  of  z  by  the  equation 

du  _l+z^  _ 

dz~l+z^'' 
shew  that  s  is  a  periodic  (but  not  uniform)  function  of  u,  and  determine  its  independent 
periods. 

32.  Within  a  closed  path  of  integration  C  in  the  plane  of  the  complex  variable  t,  the 
function  <^  {t)  has  no  singularity, .  while  /  {t)  has  poles  only  ;   express  the  value  of  the 

integral  I     -:ft4  4>  (0  ^^  ^^  terms  of  the  values  of  cf)  (t)  at  the  zeros  and  poles  of  f{t)  which 

J  c  J{t) 
fall  within  C. 

It  f{t)  =  fig  {t)  —  7/  where  ^(0)  =  1  and  g{t)  has  no  singularity  within  a  small  circuit 
round  t  =  0,  prove  that/(^)  has  two  zeros  (say  t  =  Xi,  x-^  within  the  circuit,  when  \y\  is 
Sufficiently  small.     Shew  further  that 

(Pix,)  +  cf>  {X2)  =  %4>  (0)  +   i  A^y\ 

n=l  / 

where  nAf^  is  the  coefficient  of  ^2»-i  in  the  expansion  of  j    /\(m  in  powers  of  t. 

33.  Extend  Cauchy's  Theorem  of  Residues,  so  as  to  establish  the  following  result : — 
If  T'  be  a  simple  contour  in  the  plane  of  a  variable  t,  and  Uhe  a,  simple  contour  in  the 

plane  of  a  variable  u,  etc.,  and  if  n  analytic  functions  f{t,u,...z),  (f){t,u,...z)...,  of  the 
n  variables  t,u, ...  z,  have  no  singularities  or  zeros  for  values  of  t,  ti,  ...  z,  on  these  contours, 
then 

(27r^•)«,^j^^•••jz/(«,^*,...^)0(^,«,...2)^(^,«,...^) 
is  an  integer. 

34.  Prove  that  the  integral  I      dt  is  an  analytic  function  of  x  in  any  region  from 

J  0   x-\-t 

which  the  negative  real  axis  is  excluded. 

35.  Shew  that  the  integral 

f{t,z)dt 


will  converge  uniformly  for  a  given  domain  of  values  of  s,  provided  a  function  <^  {t)  exists 
independent  of  z  and  such  that    I      \<p{t)\dt  converges,  and      .        r     is  always  greater 

'  J  a  .      _  I  /  (^)  ^J  I 

than  some  fi:sed  positive  quantity,  for  values  of  z  within  the  domain  and  for  values  of  t  on 
the  range  of  integration. 

Investigate  a  condition  which  depends  on  the  theory  of  uniform  convergence  and 
subject  to  which  the  integral 

rf(t,z)dt  :  . 

J  a, 

can  be  differentiated  under  the  integral  sign. 

Evaluate  the  integral   I     e"'^  sin  tz  —  in  terms  of  the  error- function  I     e~*'  dt. 
7  0  t  J  ^ 


800  MISCELLANEOUS   EXAMPLES 

36.  If  one  development  of  a  monogenic  analytical  fmiction  be  the  convergent  series 

y  =  x-\-a'i,x^-\-azO(?'-V ...-, 
the  values  of  y  to  be  considered,  say  y=f{x\  are  those  arrived  at  by  continuation  of  this 
series ;  prove  that  if  ^  be  considered  as  a  function  of  y  one  convergent  development  of 
X  reducing  to  zero  when  2/  =  0  is  obtained  by  reversing  this  series  ;  and  if  (^(3/)  be  the 
function  arising  by  continuation  of  this  single  series,  discuss  the  question  whether  <^  (y) 
takes  all  the  values  which  satisfy  the  equation  y=f{x). 

37.  Prove  that,  if  ai  +  a2  +  «3+---  is  convergent,  then  the  series 

1«      2»      3«      ••• 
represents  a  function  of  the  complex  variable  s  =  o-4-^^;  which  is  analytic  throughout  any 
domain  B  lying  entirely  to  the  right  of  the  axis  of  t. 

Prove,  by  taking  a„  =  (-l)"~i,  that  Riemann's  ^-function  is  analytic  for  o->l  except 
for  a  simple  pole  at  s=  1. 

38.  By  considering  the  expression 

„=o    n\     1  +  2™ .3;' 
or  otherwise,  shew  that  if  x~c  be  a  singular  point  of  a  function  f{x)  represented  in  a 
certain  region  by  an  expression  Fix)  it  may  happen,  not  only  that  F{x)  and  all  its 
diflFerential  coefficients  are  finite  at  that  point,  but  also  that  the  series 

I    ^F(-){c).{x-cY 

converges  for  all  finite  values  of  x. 

39.  Prove  that  the  series 

i     (l+?i2 ^-2) -1(1+^2  +  ^2)-! 

TO  =  1 

converges  for  all  real  values  of  x,  but  represents  a  function  which  cannot  be  expanded  in 
positive  powers  of  x. 

Shew  further  that  when  x  lies  between  1  and  ^^2  it  can  be  expanded  in  powers  of  x 
and  x~^. 

40.  If  the  triangle  formed  by  z^,  Z2,  S3  does  not  enclose  the  origin  the  series 

00  00  00  1 

2  2  2    

1  1    1    imiZi+7n2Z2  +  '>n3ZsY 

is  absolutely  convergent  when  p>3. 

41.  By  considering  the  behaviour  of  both  sides  of  the  identity 

\_Z-Zi        Z-Zz  Z  —  ZnJ 

where /(s)  =  {z-  Zi)  {z  —  z,^...{z  —  z^),  on  passing  round  an  oval  path  on  the  Argand  diagram 
(or  otherwise),  prove  that  an  oval  path,  enclosing  all  the  roots  of  f{z)  =  0,  necessarily 
encloses  all  the  roots  of /'(«)  =  0. 

42.  Shew  that  the  equations 

e*  =  ax,       eF  =  ax'^,       (F  =  ax^, 
where  a  >  e,  have  respectively  (i)  one  positive  root,  (ii)  one  positive  and  one  negative  root, 
and  (iii)  one  positive  and  two  complex  roots  within  the  circle  |  ^  |  =  1. 


MISCELLANEOUS   EXAMPLES  801 

43.  li  f{z)  denote  a  single-valued  monogenic  function,  and  Rifizj]  be  a  rational 
function  of  f{z),  shew  that  a  pole  oi  f{z)  is  equally  a  pole  of  i?  [/(«)],  unless  it  is  an 
ordinary  point. 

What  statement  can  be  made  for  an  essential  singularity  of  f{z)  1 

44.  -  Find  for  what  values  of  z  the  series 

00     g  —  im 

represents  a  monogenic  function  of  z,  and  for  what  values  its  diflferential  coefficient  is 
represented  by  the  series  of  the  diflferential  coeflficients  of  its  terms. 

46.     Rearrange  the  series 

in  powers  of  {z-^i),  and  find  the  value  of  z  of  greatest  modulus  on  the  circle  of  con- 
vergence of  the  new  series,  and  the  sum  of  the  series  for  this  value  of  z. 

46.  Investigate  the  convergence  of  the  series 

00  ^™  +  2 

^^  n=i  (a»-t-l)K  +  2)' 
where  a  is  a  positive  integer  >  1 ;  also  the  region  of  existence  and  the  singularities  of  the 
function  represented  by  it. 

47.  Prove  that,  when  n  is  an  integer, 

{z  +  ix  cos  a  +  iy  sin  a)" 

n  n 

=  Ao+    2    ^^{(^■^-f-y)™^-(^>-«/)™}  cos  ma -j-^■   2   A^  {{ix  +  2/)"" -{ix-y)^}  sin  ma, 
m=l  m=l 

where  ^™  =  —  ^ ,-. =-^ ^—, 

'"     2»  {n  +  m)\         dz^^'^        ' 

r^  being  equal  to  x^-{-y^-\-z^  and  being  treated  as  a  constant  in  the  differentiations. 

Deduce  the  expressions  for  the  2n  +  1  complete  spherical  harmonics  which  are  of 
order  n. 

48.  If  f{z),  (j)  (2),  and  -r-p.  ^^  ^^^^  regular  functions  of  the  complex  variable  z  in  the 
vicinity  of  the  origin,  shew  that 

for  sufficiently  small  values  of  \z\. 
If  i  z  I  <  >/2  —  1,  shew  that 

Ana  ^^^'=    ;    (-)""'  ^•4- (2^-2)  fjz_\^n 

\^l-z)      nil      nl      S.5...{2n-l)\l-zy     ' 

1  '^  A 

49.  If  • — -  is  expressed  in  the  form    2 ,   n   being   a 

nL!;+tau2^  '^  '^-ftan2— — - 

r=i\  2n+lJ  2n+l 

positive  integef,  shew  that 

Ar=  -^ ^  Sm2 COS^"     '^  rr -^  . 

271  +  1  2n  +  l  2n  +  l 

50.  Shew  that  a  straight  line  can  be  drawn  in  the  plane  of  the  complex  variable^/ 
which  divides  the  plane  into  two  regions  in  one  of  which  the  function 

.ziz-l)      0(0-1)  (.-2) 
■^"2!       ~  3!  "*"••■ 

is  everywhere  zero,  and  in  the  other  of  which  it  is  everywhere  infinite. 

P.  F.  51 


802  MISCELLANEOUS  EXAMPLES 

51.  A  regular  function  f{z)  has  zeros  ai,  ...,  a,„  of  orders  ai,  ...,  a^  and  it  has  poles 
Ci,  ...,  Cm  of  orders  71,  ...,7n  within  a  given  simply-connected  region,  which  contains  no 
other  singularities  of  the  function  ;  also  f  {£)  is  regular  and  has  no  zero  along  S,  the 
boundary  of  the  region  ;  and  R  (z)  is  any  function  of  z  which  is  regular  within  and  along  S. 
Prove  that 

m  n  1       f  f  (z) 

^2  a,.  R  (a,.)  -  ^^y,R{c,)=^.\R  (z)  ''j^  dz, 

the  integral  being  taken  along  the  contour. 

Apply  this  result  to  prove  (or  otherwise  obtain)  the  theorem  that,  if /(s,  C)  be  a  regiolar 
function  of  its  arguments  within  finite  domains  round  the  respective  origins -and  ii  f{z,  0) 
has  the  origin  for  a  zero  of  order  n,  then  the  equation 

/(2,  0  =  0, 
regarded  as  an  equation  in  2,  has  n  roots  which  vanish  with  {"  and  which  are  the  roots  of 
an  equation 

2»  +  C]2'^~'  +  ---  +  t«  =  0, 

where  ^1,  ...,  ^„  are  regular  functions  of  f  all  of  which  vanish  when  ^=0. 

52.  By  consideration  of  the  expression  2  —^ . ^ ,  or  in  any  other  way,  prove  that 

n=\  "^^       I  —  Z 

the  function  defined  by  the  series  2  a^z^,  where  a„  denotes  the  sum  of  the  reciprocals  of 

n=l 

the  squares  of  all  the  divisors  of  n,  exists  only  within  the  unit  circle  whose  centre  is  the 
origin. 

53.  Let  (f)  (x)  be  an  analytic  function  defined  by  a  series  of  powers  of  x,  of  unit  radius 
of  convergence.     The  points  x  =  a  and  .v=b  are  singularities  of 

and  <i    

1-x/  ^  \   X 

respectively.  Shew  that  the  real  parts  of  a  and  b  are  respectively  not  less  than,  and  not 
greater  than,  -j. 

CO 

li  f{x)  is  the  function  defined  by  the  series  2  anX^^  of  radius  of  convergence  greater 
than  unity,  and  the  functions 

00         /  X    \^^  ^        /IX"' 

are  expanded  in  series  of  powers  of  x,  shew  that  the  necessary  and  sufficient  condition 
that  all  the  singularities  of  /  {x)  are  of  the  form  -^  +  ki  is  that  the  radii  of  convergence  of 
these  two  latter  series  should  both  be  unity. 

54.  Shew  that,  if  the  series  ^a^x''^  has  positive  (or  zero)  coefficients  and  unit  radius  of 
convergence,  then  .^  =  1  is  a  singular  point  of  the  function  represented  by  the  series. 

Shew  that  every  point  of  the  unit  circle  is  a  singular  point  of  the  function 

/(^)  =  2^. 

55.  Prove  that  the  region  of  existence  of  the  function  2  ^x'"",  where  c„  =  l,  2,  3,  ...,  «.,  ' 

is  bounded  by  a  line  of  essential  singularities  everywhere  dense  upon  this  line. 

Investigate  Runge's  theorem  that  a  function,  expressible  as  a  series  of  rational 
functions,  can  be  constructed  whose  region  of  existence  is  any  arbitrarily  given  connected 
finite  area. 


MISCELLANEOUS   EXAMPLES  803 

CO         2  . 

56.  Shew  that  2  3™    has  the  circumference  of  its  circle  of  convergence  as  a  line  of 

re=l 

essential  singularity. 

Discuss  the  statement  "  si  I'ou  donne  au  hasard  une  serie  de  Taylor  dont  le  cercle  de 
convergence  ait  un  rayon  fini,  en  geniral  la  fonction  qu'elle  represente  ne  pourra  gtre 
prolongee  au  delk  de  ce  cercle."     (Hadamard.) 

57.  Shew  that  the  function  represented  by  the  power  series  2a„2"  has  at  least  one 
singular  point  on  the  circle  of  convergence. 

If  the  terms  of  the  series  2a„  are  all  positive  after  a  certain  stage,  shew  that  Sa^s"  has 
a  singularity  at  the  point  of  intersection,  of  the  circle  of  convergence  with  the  positive 
real  axis. 

Deduce  that  the  function 

has  the  unit  circle  as  its  natural  boundary,  although/ (2)  and  all  its  differential  coefl&cients 
converge  absolutely  at  every  point  of  the  circle. 

58.  If  2a„  have  a  finite  sum,  prove  that  the  sum  of  Sa^s'*  for  |  z  |  <  1  is  continuous  as 
z  suitably  approaches  z=l. 

If  2a„  is  a  real  series  which  tends  to  infinity,  prove  that 

Lim  2(X,j 2*^=00 , 
when  z  tends  to  1  along  the  radius  from  the  origin. 

59.  If  the  radii   of  convergence  of  the  power   series   2  a„^",  2  6„;t'"  be  p  and  o- 

ra=0  m=0 

respectively,  shew  that  the  radius  of  convergence  of  the  conjoined  series 

CO 

2  aj)^x^  is  $;pcr. 

Shew  fui'ther  that  the  singularities  of  the  conjoined  series  are  all  included  in  the  points 
obtained  by  conjunction  of  the  singularities  of  the  two  original  series. 

Consider  as  an  example  the  case  when 

_         1.3...(2..-1) 

60.  Shew  that  the  nature  of  the  singularity  at  x=\  of  the  function  defined,  when 

I  ;y  I  <  1,  by  the  series 

71=0'^  +  '*' 

is  given  by 

,      ,,    ,    ,      ,  ,,,      ,,  ,  -^   a-xY  {a-V)...{a~n)  (\      X  \\ 

-i°g^i--)+^^i)-^^«)+"\i-^^ — ^-^ — ^(i+2+-»)- 

Determine  the  behaviour  near  x=\  0I  the  more  general  function  defined  similarly  by 

°°         X"" 

2 r-  where  s  is  any  complex  quantity. 

61.  Prove  that,  if  ^  be  small,  and  if  x  (assumed  to  lie  between  0  and  ^)  be  not  small 

compared  with  p,  the  sum 

sinh  mru, 

2„  -l^r^-^ — >^  -    ^  cos  ?2-7rA', 

n''  (smh  2?i7rp  -  zmrfj,) 

where  ?i  =  l,  3,  5,  ...,  is  approximately  equal  to 

^      {l-ex^+4x^)-^il-2x); 


128fi^^  '     320 

and  obtain  an  expression  for  the  difference  when  -  is  large,  in  terms  of  the  roots  of  the 

equation 

sinh  2=2. 

51—2 


804  MISCELLANEOUS  EXAMPLES 

III. 

62.  Shew  that,  when  m  and  n  become  infinite  together  in  a  finite  ratio, 

s 

™     /        2  \  _  /»*V  sin  z 

63.  Discuss  the  convergence  of  the  infinite  product 

w=i  \     c„y 

where  c^,  C2,  C3,  ...  have  the  single  limiting  point  2=co .       ■ 

Form  an  integral  function  whose  zeros  are  the  points  log  2,  log  3,  log  4,  .... 

64.  Assuming  2  |  a„  |  is  convergent,  deduce  the  form  of  the  product 
Prove  that,  if  |  5- 1  <  1, 

n (1  -y2«)  n {(1 + j2«-i^)  (1  +  ^2n-i/^)}  =  i +2  (x-^ +— j  q^. 

65.  If  ?  and  m  become  infinite  in  such  a  way  that  —  =  jo,  where  jo  is  a  finite  quantity, 
shew  that 

2(«2+t;2)  n'  i  22         [  =  (cosh2y-cos2M)p'^  , 

where  the  accent  in  n'  denotes  that  the  term  corresponding  to  7i=0  is  to  be  omitted. 

66.  Discuss  the  problem  of  finding,  where  possible,  the  most  simple  type  of  integral 
function  whose  zeros  are  given  by 

n 
r=l 

where  m^  can  take  all  positive  integral  values  (zero  included)  and  the  a's  are  general 
complex  quantities. 

67.  If  0(.)=^_n{(l-?).^}. 

and  if  the  values  of  (f)  (x)  and  its  differential  coefficient  for  x=^  he  denoted  by  a  and  b, 

prove  that 

bx 


J.  {0-■;^i)'"i=^"°  *<-+*>■ 


68.  Discuss  the  problem  of  constructing  an  integral  function  whose  zeros  are  known. 
What  is  meant  by  the  class  of  a  function  ?  Is  it  possible  to  determine  it  without  knowing 
its  zeros  ? 

Prove  that,  if  F  (2)  be  an  integral  function  of  class  p,  and  have  all  its  zeros  real,  then 
F'{z)  has  at  most  p  imaginary  zeros,  if  p  be  even,  and  {p  -  1),  if  p  be  odd. 


MISCELLANEOUS   EXAMPLES  805 

69.     Shew  that  the  function 

r(g)r(a) 

T{z+a)  ' 
which,  when  a  is  positive,  can  be  expressed  in  the  form 

°°      R 

where  ^,(-l)-(.-l)^.-2)...(a-.)^    ^_^_ 

can,  when  a  is  negative,  be  expressed  by  the  series 

,,=0  V  +  ^i         "^^j' 
where  (r„(2)  is  a  polynomial  of  degree  {v-l)\az,  defined  by  the  equation 

whUe  0{z)  =  (\^~\U  +  ^-^...(\  + 


aj\      a+lj       V       a  +  v-ly 
and  where  v  is  the  integer  next  greater  than  —  a. 

70.  If  ai,  a2,  as . . . ,  he  a.n  assigned  simple  sequence  of  zeros  whose  moduli  ultimately 
increase  without  limit;    if  the  real  part  of  a„,  for  n=l,  2,  3..,,  is  positive;    and  if 

2  1/1  a„  I**  is  convergent,  shew  that  the  product 

is  convergent  if  o-^2p  >  2,  where 

71.  Construct  a  function  f(x)  which  has  a  pole  at  each  of  the  points  «=1,  x=2, ... 
and  no  others,  and  such  that/(.'«7)  — .r  cot  7r^-*-0,  at  each  of  these  points. 

72.  Discuss  the  continuity  of  the  function 

and  prove  that,  whatever  be  the  character  oif{z)  and  g{z),  the  function 

^  {zg  (?)  -f{z)}  +  {/(.)  -  g  {z)}  4>  (z) 

represents  the  function  f{z)  within  the  circle  |  2  |  =1  and  the  function  g  (z)  without  that 
circle. 

73.  A  finite  region  of  the  plane  being  given  which  does  not, include  the  part  of  the 
real  axis  between  z  =  l  and  z=+cc,  obtain  a  series  of  polynomials  which  converges  uni- 
formly over  this  region  and  represents  (1  —  2^)"*  therein. 

74.  For  the  region  between  the  two  circles 

(x  -a){x-b)+f= 0,       (x  -  a')  (x  -  h')  +y'^  =  0, 

where  a,  b,  a',  b'  are  real  and  positive,  and  a<a'<b'<b,  ab  =  a'b',  obtain  a  function  of  z, 
single-valued  and  developable  save  at  2/  =  0,  x=^(a  +  a'),  where  it  has  a  pole  of  the  second 
order,  whose  values  are  equal  at  any  two  corresponding  points  of  the  circumferences 
where  they  are  intersected  by  any  circle  orthogonal  to  both. 


806  MISCELLANEOUS   EXAMPLES 

75.     Let  it  be  assumed 

(1)  that  f{z)  is  a  regular  function  of  z  in  an  annular  region  4,  bounded  by  an  outer 
curve  S  and  an  inner  curve  s ; 

(2)  that  6  {z)  is  a  regular  function  at  all  points  in  the  interior  of  S,  and  has  a  single 
.zero  a  within  this  contour ; 

(3)  that  .r  is  a  point  within  A  ; 

(4)  that  for  all  points  z  of  the  contour  S  the  inequality 

\6{x)\<\e{z)\ 

is  satisfied,  and  for  all  points  z  of  the  contour  "s  the  inequality 

I  ^  (^)  I  >  I  ^  (2)  I 
is  satisfied. 

Shew  that  f{x)  can  be  expanded  in  powers  of  6  {x)  in  the  form 

00  00      7? 

/(^)=   2  Ar,e-{x)+   2    -^  , 
w=o  51=1  ^    K^J 

where  ^   -    '     '   f^'^''^'^'' 


Bn  =  ^^f{.^)e^'-'{z)6'{z)dz; 


and  shew  how  the  coefl&cients  A^  and  5„  can  be  evaluated  when  the  function  f{z)  has  no 
singularities  in  the  interior  of  the  curve  s  except  poles. 

76.  Obtain  the  theorem  known  as  Mittag-LefHer's,  for  the  expression  of  a  single 
valued  monogenic  function  whose  singularities  have  only  the  infinite  point  as  their  point 
of  condensation,  as  a  series  of  functions  each  with  only  one  finite  singularity. 

Prove  the  ordinary  formula  for  cot  irz  as  an  infinite  series  of  rational  functions  each 
with  one  pole,  pointing  out  the  properties  of  the  trigonometrical  functions  which  you 
assume.  Prove  the  corresponding  formula  for  the  logarithmic  differential  coefficient  of 
the  Gamma  function  r(l  +  2). 

77.  Explain  what  is  meant  by  a  branch  and  a  branch -point  of  an  analytic  many 
valued  function.     Illustrate  your  remarks  by  consideration  of  the  following  functions  : 

1or(1— ^)  /riog(l  — ^)) 

log(l-^),      J{\og{\-x)},      -^V~'       Vl       \        }• 

In  each  case  enumerate  all  the  branches  of  the  function  and  the  singularities  attaching 
to  each  branch,  and  shew  how  to  pass  from  any  branch  to  any  other  branch  by  the 
description  of  a  suitably  chosen  contour. 

78.  Defining  the  principal  value  of  log  2  as  that  value  whose  amplitude  lies  between 
-  TT  and  TT,  prove  that  the  coefficient  of  i  in  the  expression 

log(l+^■^)-log(l-^■^), 

where  r<\  and  each  logarithm  has  its  principal  value,  is  that  value  of 

/2?-cos(9'^ 

arc  tan    — ^ -, 

\  1-r 

which  lies  between  — -jtt  and  -f^rr. 


MISCELLANEOUS   EXAMPLES  807 


IV. 

79.  By  the  relation 

with  the  customary  significance  for  Z  and  z,  shew  that  a  family  of  circles  in  the  Z-plane, 
having  the  origin  for  a  common  centre,  and  a  family  of  straight  lines,  concurrent  in  the 
origin,  are  transformed  into  a  double  family  of  confocal  conies  in  the  s-plaue.  Draw  the 
family  of  curves  which  are  the  representation,  in  the  .Z'-plane,  of  the  circles 

I  2  —  1 1  =  constant. 

80.  The  interior  of  the  circle  j  s  j  =  1  is  to  be  conformally  represented  on  part  of  the 
w-plane  by  the  relation 

log(l-ag) 
''-log(l-.)' 

where  a  is  a  real  constant  lying  between  0  and  1 ;  indicate  the  part  of  the  ■z<;-plane 
required  for  the  purpose. 

81.  Find  all  systems  of  values  of  m,  y,  for  which  sn2(w+w)  is  real,  where  u  and  v  are 
real  and  0<P<1. 

82.  Determine  completely  an  area  on  the  .r-plane  of  which  the  conformal  repre- 
sentation on  the  half  ^plane  is  given  by  a  function  t  satisfying  the  equation 

where  h  is  real  and  less  than  unity. 

83.  The  coordinates  of  two  points  are  connected  by  the  equation 

X+i  Y=  en  {x + iy)l  {1  -J-  sn  (.^'  +  iy)]^ 

the  modulus  k  being  as  usual  positive  and  less  than  unity.  Shew  that,  as  {x,  y)  describes 
the  boundary  of  the  rectangle  formed  by  the  lines  ^  =  0,  x  =  K,  y  =  0,  y  =  K',  the  point 
(Z,  T)  describes  the  complete  boundary  of  a  quadrant  of  a  circle  of  imit  radius. 

84.  Prove  that,  \i  X -\-iY= f  {x ^-iy)^  the  curves  X=constant,  I''=constant  cut  ortho- 
gonally, provided  that  at  the  point  of  intersection  /  has  a  finite  differential  coefficient 
which  is  not  zero. 

Find  what  these  curves  are,  if /(^^-^?/)  =  tan-l(.r-|-^»  ;  and  prove  further  that  if  the 
point  JT,  r  travels  along  the  axis  of  A"  from  X=0  to  Z=7r/4,  along  the  hne  X=7r/4  in  the 
positive  direction  to  infinity,  and  back  along  the  axis  of  F,  the  point  {x,  y)  describes  the 
complete  boundary  of  a  quadrant  of  a  circle  of  unit  radius. 

85.  Prove  that  the  relation 

^  =  7+7 

transforms  the  part  of  the  axis  of  x,  between  the  points  z  =  \  and  z=-l,  into  a  semi- 
circle passing  through  the  points  z'=\  and  z'=-l.  Find  all  the  figures  that,  by  suc- 
cessive applications  of  the  relation,  can  be  obtained  from  the  originally  selected  part  of 
the  axis  of  x. 


808  MISCELLANEOUS   EXAMPLES 

86.     If  a  be  real  and  greater  than  unity,  if 

shew  that  in  that  sheet  of  the  Eiemann  Surface  which  represents  u  on  the  z-plane,  for 
which  ulz  tends  to  the  limit  + 1  as  2  tends  to  infinity,  u  will  have  the  value  —  1  when  z 
is  zero,  and  will  be  real  and  positive  for  that  part  of  the  real  axis  for  which  z>a. 

Taking  x=-  (a  +  - 

2  \       a 

so  that  vP'=-z^-'±xz-k-\., 

dz 


1      (zH 
2117  J     u 


shew  that  „ .      , 

2^7^  J     u 

is  a  solution  of  Legendre's  Equation 

fJ2  p  dp 

(l-^2)^.-2^^'  +  »(v^  +  l)P„  =  0, 

the  integral  being  taken  round  a  closed  contour  including  the  points  z  =  a,  z=- ,  but 
excluding  the  origin. 

Assume  (after  verification)  that 

dz  \^  u^  ^       "      II  )       {  u^  u^  It] 

87.  Writing  .^=Jr+zF,  where  X and  Fare  real,  and  taking 

Z=  sin  z, 

determine  a  simply-connected  region  of  the  plane  of  z  which  is  transformed  conformally 
into  the  half-plane  Y>  0. 

88.  Prove  that  the  relation 

2  =  aCOs(|3log.Z'), 

a^  +  lfi 
where  a^  =  a^  —  h'^,     cosh/37r  =  -5 — j^,, 

a^  —  0^ 

gives  the  conformal  representation  of  the  interior  of  the  region' boimded  by  the  ellipse 

-2  +  ^2  =  1?  aiid  two  lines  joining  the  foci  to  the  extremities  of  the  major  axis,  upon  the 

interior  of  an  annular  region  in  the  .^-plane  bounded  by  two  concentric  semi-circles  and 
two  segments  of  a  diameter. 

89.  Find  the  area  on  the  z-plane  of  which  the  upper  half  of  the  w-plane  is  the 

{z  —  c\^ 
conformal  representation,  when  w  and  z  are  connected  by  the  relation  w^l \  . 

If  ?;•=  -  ic  cot  ^z,  shew  that  the  infinite  rectangle  bounded  by  .^"=0,  a'=  tt,  ^=0,  y  =  oo  on 
the  5-plane  is  conformally  represented  on  a  quarter  of  the  i'j-plane. 

90.  Shew  that  the  most  general  representation  of  the  interior  of  a  circle  of  unit  radius 
upon  itself  is  expressed  by  the  formula 

z'  =  ize^''  +  ixe'^)/{zixe-'^  +  e-'''), 

where  fi  is  real  and  positive  but  less  than  unity.     Obtain  the  most  general  conformal 
representation  of  the  interior  of  a  square  upon  itself 


MISCELLANEOUS   EXAMPLES  809 

91.  Prove  that  the  surface  of  revolution  engendered  by  the  revolution  of  the  tractrix 
about  its  base  can  be  conformally  represented  upon  the  plane  of  2,  in  virtue  of  the  fact 
that  the  square  of  its  arc-element  can  be  conformally  represented  in  the  form 

where  x  ranges  from  0  to  27r,  and  y  ranges  from  1  to  +  00  . 

Discuss  its  representation  upon  the  s'-plane,  when  z  and  z'  are  connected  by  the  relation 

/  =  sin2{i(s-^•)}, 
indicating  what  part  of  the  s'-plane  is  covered  by  the  conformal  representation. 

92.  Develope  the  Schwarz-ChristofFel  formula  for  conformal  representation  in  detail 

in  the  case  when  the  polygon  is  a  triangle  whose  angles  are  — ,  — ,  — ,  discussing  the 

Ji      S      Xi 
character  of  each  variable  as  a  function  of  the  other. 

93.  Obtain  the  conformal  representation  of  the  interior  of  the  equilateral  triangle 
whose  vertices  are  2  =  1,  z  —  i  J^,  z=  —  I,  upon  the  upper  half  of  the  plane  of  ^,  expressing 
each  of  the  variables  z  and  f  explicitly  in  terms  of  the  other. 

94.  State  briefly  in  precise  analytical  terms  what  you  understand  by  a  closed  oval 
curve  everywhere  convex  and  with  a  definite  tangent.  For  such  a  curve  drawn  in  the 
plane  of  the  complex  variable  z,  an  analytic  function  exists,  single-valued  and  finite, 
and  having  its  imaginary  part  positive  within  the  oval,  which  is  real  and  has  one  pole  of 
the  first  order  on  its  perimeter.     Explain  how  this  is  to  be  proved. 

Find  the  function  for  the  conformal  representation  of  the  part  of  an  infinite  plane 
which  is  exterior  to  two  intersecting  circles  upon  the  upper  half  of  another  plane,  verifying 
that  it  has  the  properties  desired. 


V. 

/T' 

95.  Assuming  that  -=.  is  real  and  positive,  establish  the  formula 

9^  ^^  ~^  "    ^  1   _„2»-l  ' 

determining  the  values  of  x  for  which  it  holds ;  and  find  the  corresponding  formula  for 
any  finite  value  of  x. 

96.  Prove  that 

2Kx      .        -r/l  +  o2™-i\2  /     l-2o2»cos2^-f-o*»     \"1 
^^^r  =  '^^^TLVTT?^j   U-2g^»-cos2^^-fj^»-2.jJ' 

a     :Anx)n\(^-^"^~y       (l-2g'>sin.^  +  y^-)^      I 
(i-sm^;ii|^^  j_l_^2»  j   (i_2^2«-ico8  2^-i-?*«-2)J' 

1  /I       \^ 

97.  Prove  that,  if  ^'  =  -  ( -  -  a )   ,  «.  being  positive  and  less  than  unity, 


and  that 

2Kx 

1  —  sn 


snH^=  ^'"^ 


■(l-f-a2)(l+2a-a2)' 
and  prove  that  the  value  of  sn^  |^  is  obtained  by  writing  -  -  for  a  in  this  expression. 


810  MISCELLANEOUS   EXAMPLES 

98.  Discuss  the  functional  relations  connecting  the  pairs  of  the  three  quantities 
(i)  the  cross  ratio  of  the  roots  of  the  quartic  polynomial  f{z\  (ii)  the  absolute  invariant  J 
of  this  quartic,  (iii)  the  ratio  r  of  two  fundamental  periods  of  the  elliptic  integral 

99.  Obtain  a  formula  of  reduction  for  Jsn»  udu ;  and  thence  shew  how  to  perform  the 
integration  when  n  is  an  even  positive  integer. 

Establish  the  formula 

sn^  u  du 


(i+F)t-/;,-^ 


=1. 


[-\-CQ.u)diV?U 

100.  Prove  that 

?5_^=-y&'-l{f(^i-^)-C(tt-^-2lX')-f(2^X')}, 
en  w 

the  f -functions  being  formed  with  the  periods  2^,  AiK'. 

101.  Establish  the  identity  of  the  infinite  series 

1  +2  2  ( — )"  q"^  cos  'inx 
1 
with  the  infinite  product 

(?n(l-2j2«-icos2A'  +  g^™-2), 
1 

where  G  =  li.{\-q^'^). 

1 

Either  of  these  expressions  being  denoted  (as  in  Jacobi's  lectures)  by  ^  (^),  prove  that,  if 

Ig-I^  <|e*^|  <  l^pi, 

2Z#5(0)  ,  „^  „ 

•  —     t^  ,       =  tto  +  2  2  (X„  cos  2%^, 

ir      ^  {x)  1 

where  a«  =  2  i  ( -  1)™  j(»^+i)(2»+m+i). 

0 

102.  If  Mi  +  M2  +  W3  +  ^4=2^,  shew  that  the  anharmonie  ratios  of  the  four  quantities 
sn  2fi,  ...,  sn  Ui  are  equal  to  those  of  the  quantities  sn  (?<i  +  ^),  ...,  sn  {ui  +  E). 

103.  Prove  that  one  of  the  values  of 

~fdn^  +  cnM"|2       ("dn  M  -  en  m]  ^~[  r  [      l-sn^t      p       f     l-i-sn'j<^     "j  ^~j 
_\    1  +  cn  M   J        \   1  -  en  M    J    J  L  V^  ic  —  k'  sn  u]         \dn  u  +  k'  sn  u]  J 

is  2(1+^')-  

104.  Establish  the  formula 
%Kx      .         "    f/l  +  g^^-^V      l-2q^''cos2x+q* 


sn 
Prove  also  that 


»   j-/i  +  ^2n-i\2      l-2^2»eos2.r+^*''     "1 
= sm X  n^  |(^  i  +  ^2»  J    i_2j2«-icos2^+2*™-4  ' 


dn k  sn  — 


MISCELLANEOUS   EXAMPLES  811 

105.  If  ^i(it)  =  25'o?isin7rM  n  {l-g-^^eaTrwi  {i_g,2ne-27rm}^ 

n=l 

where  qo=   U  {1  -  q^''),     §'=e'^»^,  /(r)  >  0,  and  | /(m)  j</(r), 

m=l 

/  (r)  being  the  imaginary  part  of  r,  shew  that 

^i(m  +  t)=  -e-2'=-m^-15^(^t); 

and  express  Si  (i«)  as  a  series  of  sines  of  multiples  of  ttu.  '■ 

Shew  further  that,  if  |/(m)  |  and  1/(2^)1  are  both  less  than  /(r), 

^n    ;   X  n    /         =C0t7rit  +  C0t7r'i;+4  2      2  0-2™"  sin  (27r«% +  277^1;). 

TT      5i(m)5i(v)  „=,i^=i^ 

106.  Shew  that,  if 

then  H(u)  =  C  sin  ^  n  A  -  2a2»  cos  —  +  g*™  ^ 

where  g'=e«'rw7"  and  C  is  a  constant. 

Further,  taking  for  periods  2(o  =  2K,  2a)'=2iK',  shew  that 

1  irru 

and  deduce  that 

^(M)  =  2jism^-2gTsm^^  +  2g*  sm  ^  +  ...  admf. 

How  are  the  fimctions  sn  m,  en  m,  dn  z<,  connected  with  5  (^i)  and  the  allied  functions  ? 

107.  If  5'  =  e     -^  ,  x  =  Tvu\2K  and  the  real  part  of  K'\K  be  positive,  shew  that 

1  +  2g'2«  cos  2a; +  g*" 


-\/|^J* 


cos.^'  n 


j=i  1  -  222»-i  cos  2.r  +  g*™-2  ■ 
Shew  further  that,  if  the  modulus  of  the  imaginary  part  of  ujK'  is  less  than  unity, 

GO  qm 

log  en  («)  =  log  cos  X  -  ^2^  ^|i^(_^)^|  (2  sin  mx)\ 

108.  Assuming  the  addition  formulee  for  the  Jacobian  elliptic  functions  sn  u^  en  %, 

dn  w,  shew  that 

-1  —  dn  ^.u      h'^  sn^  m  cn^  ii 
l  +  dnazi"        dn^tt 

If  P  {u)  =  |\+  ^"  ^H  '  ,  shew  that 
^  ^     ll+dnwj    ' 

P(^/.)  +  P(M  +  2^■Z■')_     sn  2t^  en  u 

P{u)-P{u  +  2iK')  ~~     en  2u  sn  %  * 
Determine  the  zeros  and  poles  of  the  function  P  {u),  and  the  first  term  of  the^expansion 
of  the  function  in  the  vicinity  of  each  of  these  places. 

109.  By  considering  the  intersections  of  the  curves 

y  =  l  +  mx  +  nx'^, 
y'^=x{\-x)(l-k'^x\ 
or  otherwise,  shew  that,  if  %,  ...  %  are  such  that  «i  +  ^<2  +  %  +  ^4=0, 

^(Ml)+...+^(M4)  =  ^{S]2+...  +  S4^  +  2ClC2C3C4-2SlS2S3«4-2)■^ 

where  Si  =  sn  Mi  ,  Ci  =  en  Ui ,  etc. 


812  MISCELLANEOUS   EXAMPLES 

110.     "With  the  usual  notation  for  the  Jacobian  elliptic  functions,  prove  that 

sn{u+v)     _      Gn{u+v)      _       dn{u+v) 1 

SiC2(i2+S2Ci«^i      C1C2  — SiS^did2     did^  -  k"^  SiS^CxCi      1—kh-^s^^ 

where  Si,Ci,  d^  denote  the  elliptic  functions  of  u,  and  §2,  ^2,  d^  denote  the  elliptic  functions 
of  V. 


Prove  that 


Sk'h-i^s-^CiC^did^ 


(1-PsiV)^     * 


sn2(M  +  y),  ^n{u  +  v)^n{u  —  v),  sn^  (%  —  ?;) 
cn^  (%  +  -?;),  cu  (w  +  y)  en  (%  — v),  g\\^{u-v) 
dn2(w  +  v),     din.{ii  +  v)dn{u—v),     dn^{u-v) 

111.  Shew  that,  if  Mi  +  M.2  +  %  +  t64=0,  the  expression 

(sn  Ui  sn  U2  —  sin  M3  sn  u^  sn  (?ti  +  u^ 
is  unchanged  by  any  permutation  of  the  suffixes  ;  and  that,  \i  u  +  v+w  =  Q, 
1  -  dn^M  — dn^z;-  dn2w  +  2dn  ?f  dn  v  dn  to=k^sn^u  sn^v  sn^w. 

112.  Shew  that,  iix+iy  =  sn^  (m  +  iv)  and  x  —  iy  —  sn^  (?f  —  i'?'), 

{(.r-  l)2+3/2}i  ={^2+y2}i(in  2i(  +  cn  2%. 

113.  Shew  that  w=cn  z  dn  0  satisfies  the  equation 

^2y,'4  _  ( 1  +  ^2)  (^'4  +  4^2^2)  M,'2  +  ( 1  _  ,j,2)  (^.'4  +  4^2^2)2  =  Q. 

114.  Denoting  the  second  Jacobian  elliptic  integral  by  E{%i\  prove  that 

E  (Su)  -  ZE  (u)  =  ^  _  g^2,4  +4  (^2  +  ^)  s6  _  3/5:2^8 '        - 
where  «,  c,  c^,  are  the  elliptic  functions  of  ^l. 

115.  Prove  that,  if  the  imaginary  part  of  r  be  positive,  the  series  - 

7l=— 00 

is  absolutely  and  uniformly  convergent  for  all  finite  values  of  u. 

Starting  with 

.|_^io(Q)  ,,    _   1    ^nW. 

5oo(0)'  4^X^oiW 

deduce  the  equation 

1         /"*  c?a; 


TT^oo  (0)  7  0  \/4^(l-a,')(l-X.r) ' 

vo  complex  numoers,  sucn   tnat;   tne  quotieni, 
imaginary  part  positive,  and 


116.     If  0)1  and  0)3  be  two  complex  numbers,  such   that   the  quotient   —    has    its 

0)1 


TTlCUo 

x  =  —  ,     z  =  e'*,      q  =  e  '^\ 

0)1 


find  the  relations  between  6-^  (w  +  ioai),  6x  (^  +  20)3)  and  ^i(if),  where 

61  {u)  =  'iq^&ixi\x-  2g''^sinf.r  +  2g'~^sinfa^-,... 
Writing  ^ 

F{z)  =  A6x{u)l{dx{u-a)6x{a)},     A  =  2iqi -3qi  i-dq''-^- ...),     b  =  e<^  , 
prove  that  bF{qh)  =  F{z). 


MISCELLANEOUS   EXAMPLES  818 

Shew  further  that  F{z)-i  ^  _^  has  a  Laurent-expansion  valid  within  the  ring 

\q\''<\zlb\<\llq\\ 


and  that  F{z)=-2i^'  i_5  2n>  if  |  g|^<  |  g/^)  |  <1, 


0 

1 

h 

< 

? 

117.  Prove  that,  if  |  5- 1  <  1,  the  series 

^1  (^)  =  2  2  (  -  1)"  qin^W  sin  (2%  + 1)  .*', 
0 
and  the  product 

sin  A-  n  ( 1  -  22'2»  cos  2.^  +  ^*"), 
differ  from  one  another  only  by  a  constant  factor. 
Prove  also  that,  \i  q  =  e     K  ^ 

m  being  any  integer. 

118.  Defining  the  Jacobian  theta-functions  by  the  equations 

- »  -00 

i(9i(A')=    2    (-l)'»^(m+i)2e(2m  +  l)a;i^         ^2  (^')  =    2   ^(»i +*)''' e(2m  +  l)a;t^ 

00  — QO 

prove  that  6^^  62,  O3  are  even  functions  of  .r,  while  di  is  an  odd  function  of  x ;  that  ^q  and 
^3  are  periodic  with  tt  as  period,  while  B^  and  ^2  are  periodic  with  Stt  as  period ;  and  that 
all  four  functions  are  pseudo-periodic  with  i  log  g  as  a  period,  two  of  them  with  -  e^^t  as 

1       .  ' 

a  factor,  and  the  other  two  with  —  e^m  ^s  a  factor. 

Prove  that 

^3-^  (0)  6s  {X + i/)  Os  {X  -I/)  ==  6i  (x)  ds'  (y) + e;^  {x)  e,^  {y) 

=^e,^x)e,\y)  +  6i{x)e.i{y); 
and  deduce  from  these  equations  similar  exjiressions  for 

ei{Q)6,{x+y)e,{x-y) 


for  r  =  0,  1,  2. 

119.     With  the  notation 


^{x)=   2   (_l)»j»2g2ma;i^ 

^2  {x)  =  2  ^i  (2-1  - 1?  e(2«  - 1)  a;i, 

establish  the  periodic  and  pseudo-periodic  properties  of  the  ^-functions. 

Prove  also  that  the  expression 

4  4 

n  ^2  (■«?«)  +  n  .^3  {x^) 
1  1 

is  unaltered  when  jti,  X2,  Xs,  x^  are  replaced  by 

^{Xi  +  X2  +  X3  +  Xi),       |(.^i-t-.^2-*'3-'*4)5       |  G^'l  -  .^2  +  ^3  " -^'4),       i  i^l' •'^2  — ^3  + ^i)  ', 

respectively. 


814  MISCELLANEOUS   EXAMPLES 

VI. 

120.  Examine  under  what  circumstances  p  {ku)  is  a  rational  function  of  p  (u),  k  being 
a  suitable  constant  multiplier  ;  and  if  ?7,i  be  a  function  of  u  defined  by  the  equations 

^1  =  1,     U^=-p'{u\     U^^,  =  ^{p{u)-p{'>iu)}, 

prove  that  f2m  +  i  '^^^  C^2m/P'  ^^^  rational  integral  functions  of  p  (u),  ^g^  and  g^,  with 
integral  coefficients,  and  that 

If  IT  —  IJ  TT  7^7    2  _   TJ  TT  TJ  2 

^  n  —  in'^  n-\-in       ^  n+X'^  n—\^  m         '^m  +  l  ^  m—l  "-^ »  > 

where  m  and  n  are  positive  integers. 

Prove  that,  if  p  (w),  g^-,  and  ^3  are  rational  numbers, 

p{nu)<e'^'\ 
where  X  is  independent  of  the  positive  integer  n. 

121.  Prove  that,  if  pu^,  Pu2,  Pu^  are  distinct  solutions  of  the  equation 

ap{u)  +  b  =  p'{u), 
then  u-^  +  U2  +  Us  =  ma  +  nco',  where  co  and  m'  are  the  periods,  and  m  and  n  are  integers. 
Prove  that 

122.  If  PiM  denote  the  function  with  2(Bi,  2<oi  as  primitive  periods,  r=—  ,  ti  =  — , 

CO  6)1 

the  imaginary  parts  of  t,  ti  being  positive,  and 

A      I    7? 
prove  that  t^  =  y^~ — ^  ,  where  A,  B,  C,  D,  are  integers  such  that  AD-  BC=  1  and  such  that 

B  and  C  are  even. 

123.  Shew  that,  if  u  and  z  are  connected  by  a  relation 


1? 

(2*  +  6a2  z''  +  4a3  2  +  ^4)  ^ 


then  ^^5^.7   w       ' 

where  the  invariants  of  the  elliptic  functions  are  given  by 

g2  =  ai  +  Sa2^,        g^i^^a^a^  —  a^^-a^^. 
Express  in  terms  of  functions  of  ib  and  Mq  the  value  of  the  integral 


/: 


-^dz, 


where  u  and  2  are  connected  as  above,  and  «o  is  related  to  z^  in  the  same  way  as  u  to  z. 
124.     Shew  that  the  determinant 

1,    ^K),    ^'K),    ^"(«i), -,    ^(»-^)K) 
1,     ^(%),     ^'(»3),     ^"(%),  ...,     ^("-2)(^*3)  ; 


1,    P(«j,    ^'K),    P"K),-,    ^<"-')K) 

^  ^  0-(Mi+M2  +  — +Wj  H;^,  ^i  O-  (M;^  -  M^) 
{(T  (Ml)  O-  (Ms)  O-  (Ms)  • .  •  O-  (m„)}" 


-     MISCELLANEOUS   EXAMPLES  815 

where  X  </x ;  X,  /x=l,  2,  3,  ...,  %  ;  and  C  is  a  numerical  constant.  Determine  the  value 
of  C. 

Prove  that  the  four  roots  of  the  equation 

are  the  values  of  ^  (a),  where  3a  is  any  period  of  the  function  ^  {u),  a  not  being  itself  a 
period. 

Prove  also  that  the  value  of  the  product 

{g3(«)-P(a)}{g?0.  +  a)-^(a)}{^(l6  +  2a)-^(a)} 
is  independent  of  u. 

125.  If  'p  {u)  denote  Weierstrass's  doubly-periodic  function,  and  X  be  a  constant  not 
an  aliquot  part  of  any  period  of  this  function,  pi'ove  that  the  function 

considered  in  the  infinite  plane  of  x  with  the  exclusion  only  of  the  points  x=0,  ^'=qo,  is 
capable  of  assuming  for  every  given  value  of  x,  values  as  near  as  may  be  desired  to  any 
assigned  value. 

126.  The  symbol  <p  being  used  to  denote  Weierstrass's  elliptic  function,  prove  that  if 
the  relation 

is  satisfied  by  io=Ui  and  by  u  =  Uo,,  it  is  also  satisfied  by  m=  -  u^  -  u^-  Deduce  the  addition 
formula  for  the  function. 

With  the  customary  notation  for  ej,  e^,  e^  {e^  >ei>  63),  prove  that 


{^(M+i-)- 63}!^  («-.') -63}' 


{^(«)-g2}  {^W-e2}-eie3-  262' 

LIP  W- 63}  Wky)~e^]-ex^i 


-2622-12 
-2.32  J 


< 

iu 

{PW 

-p(a)}2 

:-[' 

dx 

127.  Prove  that  a  doubly-periodic  meromorphic  function  of  %i  can  be  expressed 
linearly  in  terms  of  functions  f(?«  — a),  f'('w  — a),  ...,  where  f  (?<)  =  — -j-^- 

Evaluate  / 

J 

128.  Shew  that  the  equation 

■   -r  — ^ 

J  xo  Jf{x) 

where /(a")  is  any  quartic  function  of  x,  and  Xq  is  a  root  oi  f{x),  is  equivalent  to  the 
equation 

_  /'  (.To) 

^-^■«  +  4{^(.)-Jj/"K)}' 
where  the  doubly -periodic  function  ^  {z)  is  formed  with  the  invariants  g^  and  g^  of  f{x). 

If  Xq  be  not  a  root  of  /(^),  shew  that  the  last  equation  must  be  replaced  by 

^^^     ,    /^  (^0)  F  (^)  +  i/  (-^0)  W  {^)—hf'  (^0)}+ 2V/(^0)/"  (^0) 

2  {^(2)_  Jj./"(^o)F--^L/(^o)/"(^o) 

129.  Shew  that,  if  n  is  an  odd  prime,  the  value  of  the  elliptic  function  of  the  wth 
part  of  a  primitive  period,  e.g.  ^  ( —  J ,  may  be  obtained  by  solving  an  equation  of  degree 
71-1-1,  and  ?i-|-l  equations  of  degree  n  —  \. 


816  MISCELLANEOUS   EXAMPLES 

1 30.  Starting  from  the  definition  of  o-  (z)  as  an  infinite  product,  prove  the  formula " 

a  {b  +  c)  cr{b  —  c)  a{a  +  d)  a- {a  —  d)  +  (r{c+a)  a-  {c  —  a)  a  {h  -\- d)  a-  {h  -  d) 

+  (T  {a-\-h)  <T  {a  -h)  (T  (c+d)  a  {c-d)  =  0. 
Shew  that,  if  (bcu)  be  written  for 

P(b),     &{c),     P{^c) 
P'ib),     P'{c),     f{u) 
1,  1,  1 

then  the  relation  between  the  cr-functions  resulting  from  the  identity 

(bcu)  (adu)  +  {cau)  {bdu)  +  {abu)  {cdu)  =  0 
is  substantially  the  same  as  the  above. 

131.  Prove  that  \i  A,  B,  C,  D  are  any  constants, 

A(Ti{u)  +  B(T2{u)  +  C(Tz{u)  +  D(r  {u) 
can  be  factorized  in  the  form 

„    fu -a\     (u-  /3\     (u -y\     (u- h\ 

where  a+/3+7  +  S=0. 

Deduce,  or  prove  otherwise  that  2a-  ia)  a  (b)  or  (c)  or  {d)  is  equal  to 

0-1  (W)  0-1  {V)  0-1  JW)  0-1  (0        0-2  JU)  0-2  {V)  q-2  (w)  (Tg  {t) 

(ei  -  62)  (ei  -  63)  (^2  -  ^i)  (62  -  63) 

where  2«  =  'M+'y4-'2<'  — ^,         2b  =  -u  +  v  +  w+t, 

2c  =  u  —  v  +  w  +  t,         2d=u  +  v  —  w  +  t. 

132.  Prove  that 

Prove  also  that  o-  (^iw)  {o-  («)}  ~  "■'^  is  a  uniform  doubly-periodic  function  of  u. 

133.  From  the  definition  of  the  a--function 

l-|jen    202^ 

where  O  =  2?«.a)  +  2?H/a)',  and  the  product  refers  to  every  pair  of  integers  m,  m',  not  both 
zero,  deduce  directly  the  value  of  a-{u  +  2(o)l(T{u).     And  proceed  thence  to  deduce  the 

series  for  the  function  o-  {u)  in  integral  powers  of  e  "  ,  it  being  assumed  that  the  imaginary 
part  of  the  ratio  ajca  is  positive. 


134.  Express  x  as  a  single- valued  function  of  u,  when 

_  /■«  ■    dt 

135.  If 


[X          dx 
'=/    i  = 


•{\+x^-2x^Y 

express  .27  as  a  single- valued  function  of  u,  by  the  help  either  of  Jacobi's  or  Weierstrass's 
functions. 


MISCELLANEOUS   EXAMPLES  817 

1 36.  Shew  that,  ii  Zi+Z2+Zs=0, 

[  {P  ih)  -  «2}  W  (^2)  -  ^3}  W  (^s)  -  es}]*  -  [  W  ih)  -  es}  {&  (22)  -  e,}  {P  (S3)  -  62}]* 

137.  Using  the  customary  notation,  obtain  the  formulae 

/A/   N     //I/  \  o-(?<  +  v)a-(?<-^) 

P(«)-^(^)= ^2(^),2(,)  > 

,.,, , ,.   {g>(^)^(.y)-ig^2}  {g>(^-)+^(y)}-i^'(^)F(y)-^^3 

Prove  also  that 

0'3 


3C(3«)-9a«)=^._.^^p_^^^_^^^^2; 

and  shew  that  n^  (nu)  -  n^ ^  (it)  can  be  expressed  in  terms  of  p  and  ^'  only,  n  being  an 
integer. 

138.  Prove  that 

2p'  (2«)  f  (u)  =  {^  (?^)  -  p  {u  +  o))}  {p  (u)  -p{u  +  0)')}  {^  (u)  -Piu  +  o)")}. 

r=4  1 

139.  Express  (62  -  63)  n  {p  (t<,.)  —  ^il   +two  similar  terms,  by  means  of  Jacobi's  elliptic 

r=l 

4 

functions;  and  hence  (or  otherwise)  shew  that,  when  2  ^^,.  =  0,  the  expression  is  equal  to 

r=l 

-  (61-62)  (^2- 63)  (63-61  )• 

140.  Shew  that 

2o 

.•2i7(m+(o)/)h 


'(4Kt)] 

where  2a>  is  a  period  of  p  {u)  and  m  is  an  integer,  is  the  mth  root  of  a  rational  integral 
function  of  p  (u)  and  p'  (u). 

141.     Obtain  a  general  formula  for  p  (nu)  in  terms  of  p  (ii). 

The  periods  of  p  (u)  being  2co,  2a)',  and  a,  b,  c,  d,  denoting  the  quantities 


prove  that  a^  +  eb^  +  e^c^  =  0,       e'^b^  -  ec^  +  d^  =0, 

where  e  is  an  imaginary  cube  root  of  unity. 

142.  Specify  briefly  the  descriptive   properties  characterising  the  elliptic   function 
P  (•?«),  and  prove  that  the  function  is  determined  thereby. 

Shew  that  the  function  (p  (u)  =  {p  (u)  —  e,}"^  is  a   single-valued   function;    obtain   its 
periods,  and  the  expression  of  cf)  {u+v)  in  terms  of  (f)  (u),  0'  (m),  0  (v),  0'  (v). 

Find  the  ^-function  having  the  periods  of  <p  (u),  and  the  relation  connecting  this  new 
^-function  with  cf)  (u). 

143.  Prove  that 

P'  {%)  -  p'  {v{)  _  p'{u)-p'{v.^  ^  2,T{u)<T{u+v^  +  v^)<r{v^-v^) 
Piu)-p{vi)        P{u)-p{v2)       cr{u  +  Vi)a{u  +  V2)a-(v{)(r(v2)' 
Deduce  that  the  cross-ratio  of  the  pencil  of  tangents  from  any  point  on  a  non-singular 
cubic  to  the  curve  is  constant. 

F.  F.  52 


818 


MISCELLANEOUS    EXAMPLES 


144.  Shew  how  to  build  an  integral  function  of  two  variables  whose  second  logarithmic 
differential  coefficients  possess  fourfold  periodicity,  obtaining  the  necessary  relation  between 
the  periods. 

If,  with  the  usual  notation,  o-  (u)  be  the  odd  elliptic  theta-function  of  one  variable  and 
m,  a,  b,  n  be  constants,  obtain  the  simultaneous  periods  of  the  second  logarithmic 
derivatives  of  the  function  of  u  and  v 

gma  ^gmv  a-(u-a)  +  e-^'>(r{u-  6)], 
and  the  algebraic  relation  connecting  these  derivatives.  ■ 

145.  Prove  that  every  elliptic  function  can  be  expressed  linearly  in  terms  of  ^(m  -  aj), 
^(w  — 02),  ...  and  their  differential  coefficients,  where  C(^)  =  o"'  {u)/(r{u). 

Prove  that 

1  yw+FW  _  &'Xv)+&'H] 

146.  Establish  the  formula  : 


=  -C{'^-u)  +  C{w-v)-i-C{v)-C{u). 


1,    ^W,    F(«) 

I,     Piw),     p'{w) 
147.     Express 


=  -2a-  {u-v)cr{v  -to)cr  (w-u)  cr  {u  +  v  +  w)j a^  (u)  a^  (v)  a^  (vj). 


1,     p(a;),     p(^),     p'{a;) 

1,    ^(3/),    PH^),   P'{y) 

1,     ^(^),     P(2),     F(2) 
1,     p{u\     P(«),     p'{u) 
as  a  fraction  whose  numerator  and  denominator  are  products  of  a--functions.     Deduce 
that,  if  a  =  p  {x\  /3  =  p  (3/),  7  =  ^  {z),  8  =  p  {u),  where  x+y  +  z  +  u  =  0, 

(62  -  63)  {(a  -  ei)  O  -  ei)  (7  -  ei)  (S  -  ei  )}*  +  (63  -  ^i)  {(a  -  eg)  (/3  -  63)  (y  -  e^)  (8  -  e,)}^ 

+  (ei  -  62)  {(«  -  63)  (3  -  63)  (7  -  63)  (S  -  63)}*  =  (62  -  %)  (63  -  61)  (^1  -  62). 
.  148.     Denoting  the  roots  of  4^^  — ^2^-5'3  =  0  by  ei,  62,  63,  prove  that 

where  I,  m,  n=l,  2,  3,  and  the  summation  extends  over  the  three  corresponding  terms. 

149.  Prove  that,  for  any  three  arguments  u-^,  u^,  u^, 

C(%)  +  CC«2)  +  t(%)-CK  +  W2  +  %) 

_2 W  (^1)  -  P  (^2)}  W  K)  -  <P  (%)}  {P  {Us)  -  Pin,)}      . 

&'  (%)  {^  («2)  -  &  (%)}  +  ^'  (^*2)  {P  («3)  -  ^  («l)}  +  &'  («3)  {^  («l)  -  P  (U2)}  ' 

150.  Express  the  function  {p  (s)  — ej}^  as  the  sum  of  an  integral  function  of  z,  having 
no  singularities  in  the  finite  part  of  the  2-j)lane,  and  of  a  series  of  rational  functions  of 
each  of  which  has  only  one  pole. 


151.     Prove  the  formula 


P{u-coj)-ei 


P{u)-ei      {ei-e„,){ei-en)' 
where  I,  m,  n  are  the  numbers  1,  2,  3  in  any  order. 
152.     Prove  that 

P%^)  +  P{k<^  +  c^')^2ey, 
P(ia>)-^(ia,  +  a,')  =  2  {(61-62)  (61-63)}*, 


^'  (i»)  =  -  2  {(61  -  62)  (61  -  63)}^  {(61  -  62)^  +  (61  -  63)^}. 


153.     Shew  that 


r(.)=6 


MISCELLANEOUS   EXAMPLES  819 

a- (z  +  a)  a- (z  —  a)  a- (z-hc)  a  (z  —  c) 


0-*  (z)  0-2  (a)  0-2  (c) 
where  P  {a)  =  {^92)^-,      ^  («)= -(tV5'2)*- 

154.     State  the  properties  of  the  eUiptic  function  p  {u)  which  prove  that  there  is  a 
single-vakied  function  a  (u),  such  that  a^{ii)  =  p  (^)  — ^i,  and  ua{u)  =  l  when  u  =  0. 

Defining  similarly  b  (w)  =  (p  (u)  -  e^^ ,  c  {u)  =  (p  (w)  —  gg)^ ,  prove  that 

a  (u)  b  {v)  c  (v)  —  a{v)b  (u)  c  (u) 


Shew  also  that 


a{tL  +  v)=- 

a^  {v)  —  a^  (u) 

a{u  +  a>)a  (?<)  =  a'  {a)  =  —  or'^  ( i^  ^j,), 

2a  (m)  6  (m)  c{u)  a  (2u)  =  a^  (u)  —  (X*  (^  co), 

'<^  fl  1 

-^ a{u)\-  dzo  =  log  [^ M  {6  («)  +  c  («)}], 

da{u) 

where  a  iu)  =  — j^  . 

du 

155.     From  the  theory  of  doubly-periodic  functions,  or  otherwise,  obtain  the  formula 

(.) 

Shew  how  to  express  the  points  of  a  plane  quartic  curve,  which  has  two  double  points, 
by  means  of  elliptic  functions. 


^(»)-^<»-)=-.s{frgl^ 


VII. 

156.  Obtain  Euler's  relation  C+  F=E+2  connecting  the  numbers  of  corners,  faces 
and  edges  of  an  oi'dinary  convex  solid  bounded  by  plane  faces ;  and  extend  this  result  to 
the  case  of  a  solid  for  which  the  surface  is  not  simply  connected. 

A  closed  rectangular  box  has  a  partition  lying  midway  between  two  opposite  faces,  and 
this  partition  is  pierced  with  two  holes.     Discuss  the  connectivity  of  the  inner  surface. 

157.  Prove  that  on  any  closed  sui-face  in  space  (or  on  a  Riemann's  surface)  a  system 
of  closed  curves  can  be  drawn,  so  that  (i)  it  shall  be  possible  to  pass  from  any  one  point  of 
the  surface  to  any  other  by  a  continuous  path  which  does  not  cut  the  closed  curves,  and 
(ii)  any  two  paths  so  drawn  between  two  given  points  shall  be  continuously  deformable 
the  one  into  the  other  without  crossing  the  closed  curves. 

Construct  such  a  system  of  closed  curves  for  a  three-sheeted  Riemann's  surface  with 
two  branch  lines  at  each  of  which  the  three  sheets  are  connected  cyclically. 

158.  A  table  consists  of  a  rectangular  parallelepiped  resting  on  two  other  rectangular 
parallelepipeds,  placed  vertically,  and  each  pierced  with  a  hole.  Upon  the  table  is  laid  a 
book,  and  upon  this  a  much  smaller  book.  Estimate  the  connectivity  of  the  surface  of  the 
whole  I'esulting  solid,  each  book  being  regarded  as  a  rectangular  parallelepiped. 

159.  There  are  n  rings  placed  in  order,  each  connected  to  those  on  either  side  by  a 
cylinder,  whilst  the  two  rings  at  the  end  have  each  a  point  boundary.  Find  the  con- 
nectivity of  the  surface  so  formed. 

There  are  n  rings  placed  in  order,  each  connected  to  those  on  either  side  by  a  cylinder, 
whilst  the  first  and  last  rings  are  connected  to  each  other  by  a  cylinder.  To  the  surface 
so  formed,  a  point  boundary  is  supplied.     Find  the  connectivity  of  the  surface. 

52—2 


820  MISCELLANEOUS   EXAMPLES 

160.  Prove  that,  if  an  anchor  ring  be  hollowed  out  and  the  exterior  and  interior 
surfaces  connected  by  a  hollow  cylinder,  and  if  a  point  boundary  be  supplied  to  the  surface 
so  formed,  then  its  connectivity  is  5. 

161.  A  surface  of  connectivity  n  has  h  boundary  lines.  Shew  that  it  is  impossible, 
without  dividing  the  surface,  to  make  more  than  ^{n  —  h)  cross-cuts,  each  of  which  passes 
from  a  point  of  an  original  boundary  to  another  point  of  the  same  boundary. 

162.  A  variable  u  is  defined  by  means  of  the  relation 

/"«  dx 

where  y^  =  Ax^- g^x- g^, 

and  the  inversion  of  this  relation  is  expressed  in  a  form 

apply  Abel's  Theorem  to  shew  how  an  expression  can  be  obtained  for  Q{u  +  v)  in  terms  of 
Q  (w),  Q'  {u),  Q  (v),  Q'  (v).     Find  also  the  periods  of  the  function  Q  (u). 

Construct  the  integrals  that  remain  finite  on  the  Riemann's  surface  associated  with 

i/i  =  4x^-  g2X-  gs. 

163.  The  equation  f{w,  z)  =  0  is  algebraic  in  w  and  z  ;  and  it  is  satisfied  hj  w  =  a,z=a. 
Shew  that,  when  z=a  +  z',  where  \z'  \  is  small,  then  values  of  w  are  given  by  a  +  v/,  where 
I  ti^  I  is  small,  and  where  w',  if  not  a  uniform  function  of  /,  belongs  to  a  set  of  values  the 
members  of  which  interchange  cyclically  when  z  describes  a  small  circle  round  a. 

Obtain  explicitly  the  branches  of  the  function  w,  as  defined  by  the  equation 

w^  +  3wz  +  z^  =  l 
for  values  of  z,  (i)  near  the  origin  z  =  0,  (ii)  near  the  point  z=-l. 


J  (■ 


164.  Shew  that  the  integral 
{{x  -  ai)  {x  -  a^)  {x  -  as)  {x  -  ai)}    ^  dx 

is  transformed  to  the  integral 

by  the  relations 

(a2-ai)(^-a4)'  («3-«i)(«2-a4)  ' 

and  obtain  an  expression  for  the  general  value  of  the  former  integral. 

165.  Prove  that  every  integral  of  the  first  class  associated  with  the  equation 

iv^  +  z'^  =  l 
is  of  the  form 

\aw  +  Bz^C)'^^^, 

where  A,  B,  C  are  arbitrary  constants ;  and  construct  integrals  of  the  second  and  the  third 
classes,  associated  with  the  same  equation. 


/< 


166.     Obtain  the  integrals  of  the  first  kind  connected  with  the  equation 

vfi  —  z{z—l){z~a)  (z  —  b), 

where  |  a  |  >  |  6  |  >  1  ;  and  shew  how  to  obtain  the  addition -theorem  for  the  functions  that 
arise  in  the  inversion  of  the  integrals. 


MISCELLANEOUS   EXAMPLES  821 

167.  For  the  equation 

y^  —  byx^  +  4^  =  0, 

find  the  form  of  the  everywhere  finite  integrals. 

168.  Determine  the  genus  of  the  equation 

y'^  =  x  {\  —  xy. 

Find  a  system  of  integrals  of  the  first  kind,  and  also  an  elementary  integral  of  the 
third  kind  with  its  infinities  at  the  values  x  =  0,  x  =  l. 

169.  Construct  and  dissect  the  Riemann's  surfaces  associated  with  the  equations 

(i)      w^=z^{z'^-\\ 

(ii)     v^  =  {z-aif{z-a2f{z-a^f, 

(iii)     -?«;*= (0-ai)2(0-a2)3  (2 -a3)3. 

Give  two  dissections  for  the  last  surface,  one  of  which  does  not  involve  any  cut  in  one 
of  the  sheets. 

170.  Construct  a  Riemann's  surface  on  which  the  function  w  defined  by  the  equation 

2<^-(22  +  3)w2  +  l=0 
can  be  exhibited  as  a  uniform  function. 

171.  Explain  in  general  terms  the  principles  of  the  theory  of  the  dissection  of  a 
Riemann's  surface,  to  render  the  surface  simply  connected,  and  the  part  which  the 
number  of  everywhere  finite  integrals  upon  the  surface  plays  in  the  number  of  necessary 
dissections. 

Shew  how  to  find  the  number  of  everywhere  finite  integrals  associated  with  an  equation 

y»»  =  (^-aj)»i(^-a2)"2 .... 
Find  these  integrals  in  particular  for  the  equation 

{x  —  a){x  —  h) 


r= 


and  dissect  the  surface. 


{x-c)(x-d)' 


172.     Describe  the  character  and  position  of  the  infinities  of  the  integral 

'y^  +  ax'^  +  Ax  +  By  +  C 


/^ 


{2y^-x^-l)y 

where  w  is  an  imaginary  cube  root  of  unity,  and  x,  y  are  connected  by 

x^+x'^y'^+y'^  =  x^-\-y^  +  ^; 

and  find  the  sum  of  its  values  extended  from  the  points  where  Px-\-Qy  +  R=Q  to  the 
points  where  P'x+Q'y  +  R'  =  0. 

173.  Construct  a  Riemann's  surface  to  represent  the  equation 

{x-l)y^  =  a^{x  +  lf; 

draw  cuts  reducing  it  to  a  simply  connected  surface  ;  and  construct  an  Abelian  integral 
of  the  first  kind  associated  with  the  equation. 

174.  Discuss  the  general  pharaoter  of  the  Riemann's  surface  which  represents  the 
equation 

{w  —  z)  (w^  —  z'^)  (w^  —  2^)  =  1 ; 

and  prove  that  its  genus  is  equal  to  4. 


822  MISCELLANEOUS   EXAMPLES 

175.  lu  the  whole  of  a  Riemann's  surface  of  n  sheets,  the  total  number  of  branch 
points  is  s ;  and  the  numbers  of  the  branches  of  the  represented  algebraic  function, 
which  interchange  at  the  branch-points,  are  m^^  m^,  ...,  mg  respectively.  Prove  that  the 
connectivity  of  the  surface  is 

(    2  9n,.^-(2w  +  s-3), 

and  that  its  genus  is 

Prove  that  the  genus  of  the  Riemann's  surface  associated  with  the  equation 

w^^=A{z-af{z-hf{z-c)\^ 

where  a,  b,  c  are  unequal  constants,  is  7 ;  and  indicate  the  relations  between  the  sheets 
of  the  surface  at  each  of  the  branch-points. 

1 76.  Indicate  various  classes  of  functions  of  position  on  a  Riemann's  surface  of  genus 
p,  explaining  specially  the  characteristic  properties  of  the  functions  which  usually  are 
called  of  the  first  kind,  the  second  kind,  and  the  third  kind,  respectively,  as  well  as  of 
adjoint  polynomials  j  and  prove  that  an  adjoint  polynomial  possesses  2p  — 2  zeros  on  the 
surface. 

At  a  set  of  m  among  these  2p—2  places,  q  other  adjoint  polynomials  vanish  together : 
at  the  remaining  2p  — 2  — m  places,  q'  other  adjoint  polynomials  similarly  vanish  together. 
Prove  that,  on  a  general  Riemann's  surface, 

q'  —  q  =  m—p  +  \. 

177.  Find  the  genus  p  of  the  relation 

Construct  a  rational  function  associated  therewith  with  p+\  arbitrary  poles,  and 
obtain  the  forms  of  p  everywhere  finite  integrals. 

178.  For  the  Riemann's  surface  associated  with  the  equation 

^y  -f  Ms  +  ^4  =  0, 

where  %,  %  are  homogeneous  polynomials  in  x,  y  respectively  of  orders  3  and  4  with 
general  coefficients,  construct  a  set  of  linearly  independent  integrals  of  the  first  kind 
and  an  elementary  integral  of  the  third  kind  whose  infinities  are  at  ^=0,  ?/  =  0. 

Find,  save  for  an  additive  constant,  the  sum  of  the  values  of  the  integral 


/ 


^-dx 

X 


at  the  intersections  with  the  given  curve  of  the  line  Ax-\-By-^C=-'d^  expressed  in  terms 
of  -4,  5,  C,  and  the  coefficients  of  the  curve. 

179.  Construct  a  Riemann's  surface  suitable  for  the  representation  of  a  function  y 
given  by  the  equation 

Make  a  series  of  cuts  which  will  render  the  surface  simply  connected. 

180.  Shew  that  the  genus  of  the  equation 

y'^-\-y{,x,y)''-\{x,y)^  =  0  . 
is  unity ;  and  construct  an  integral  of  the  first  kind  associated  with  the  equation. 
Similarly  for  the  equation 

/=(l-^2)(l_^2^2)_ 


MISCELLANEOUS   EXAMPLES  823 

181.  Obtain  the  position  and  character  of  the  branch  places  of  the  Riemann's  surface 
representing  the  equation 

and  the  forms  of  the  everywhere  finite  integrals. 

182.  For  the  fundamental  equation 

y*  =  .K^  +  ^  +  1, 
describe  the  behaviour  of  the  integral 


/-[|--5^.]^ 


and  find  the  sum  of  its  four  values  integrated  from  the  four  points  where  P^^x  +  §oy  +  -^0  =  0, 
to  the  four  points  where  Px  +  Qi/  +  R  =  0. 

183.     Prove  that  the  equation  of  a  curve  of  order  n,  having  ^n{7i  —  3)-l  double  points, 
can  be  transformed  birationally  into  the  hyperelliptic  form 

where/  is  a  polynomial  of  degree  5  or  6.     Transform  in  this  way  the  equation 

f  =  x^'{x^--3x+2). 


184.     Prove  that  the  equation 


fix,  y)  dx 


wherein  y^=^-  g%oc-g^^  defines  x^  y  as  single-valued  meromorphic  functions  of  %. 

If  y'^=a?'X^-\-4tbx^-\-Qcx"-\-'^dx-\-e,  and  R{x,y)  denote  a  rational  function  of  x  and  y, 
find  three  integrals 

R  {x,  y)  dx 


/^ 


respectively  (i)  everywhere  finite,   (ii)  algebraically  infinite  to  the  first  order  bvit  not 
logarithmically  infinite,  at  .■v  =  |,  y  =  r),  (iii)  logarithmically  but  not  algebraically  infinite 

s,t  x—^,  y=rj,  andat.r  =  co,  y  =  ax'^;  and  shew  that  every  integral  jR(x,y)dx  is  expres- 
sible, save  for  rational  functions  of  x  and  y,  in  terms  of  integrals  of  these  three  forms. 

185.     If  JO  and  n  be  positive  integers,  yP  =  l  +  aj^x  +  a2X^  +  ...+anX";  the  right-hand 
side  having  no  repeated  roots,  prove  that 


v=  I    dx/yP    ^ 
J  0 


is  an  elliptic  integral,  provided  that  p  and  n  have  the  greatest  values  consistent  with 
the  inequalities 

-+  ->:!  and   -+  ->1.  . 

p      n  n     p 

Determine  the  possible  values  of  p  and  b. 

Shew  that  p  (v)  can  be  expressed  in  the  form 

where  t>=0  corresponds  to  ,^;=0,  y  =  l. 

Completely  determine  the  invariants  ^2  ^^'^  ffs  "^  *^®  cases  ji9  =  2,  n  =  4,  and  p  —  3,  n—3. 


824  MISCELLANEOUS   EXAMPLES 

VIII. 

186.     If  fix)  be  a  quintic  polynomial  in  x,  and 

investigate  the  character  and  general  form  of  Xi  +  x^a.^,  fi,  function  of  Uy  and  U2 . 


187.     If  3/^  be  a  quintic  polynomial  in  .r,. vanishing  when  x=ai  and  when  .2; =0^2 5  and 
discuss  the  character  of  XyX^  and  of  {(«!  — ^1)  («i  —  *'2)}*  as  functions  of  %  and  'M2' 


r^i  0?.^;       /'■^s  0?^  /"^i  xdx       f' 

J  a^     y        J  a^    y  J  a^      y         Jo 


X  ~~~  Xq 

■(^0  Ex  +  F  ,        f^^'^'>  Ex+F 


dx  + 

y 


188.  If  A,  B,  C,  D,  E,  F  be  constants,  f{x)  a  quintic  polynomial,  (^o^o)  a  variable 
pair  satisfying  y'^=f{x),  and  {xi),  (x^),  ...,  be  the  zeros  of  the  rational  function  in  {x,  y) 

A  {x-\-x^)^-Bxx^-\-Cy;^^^D, 

d   Y  f^^^^  Ex  +  F  ,        f 

evaluate  -^—       |        ax+  I 

"■^^o  L ./  y  J 

189.  Quantities  u  a,ud  v  are  defined  by  the  relations 

2u=r^:^dx+ri^^d,, 

where  X={X'~  ao)  (x  —  aj)  {x  -  a^  (x  —  a^)  (x  —  a^), 

T  is  the  same  function  of  j/  as  X  is  of  x,  and  the  constants  ao,  «!,  a2,  «3,  a^  are  real, 
unequal,  and  (so  arranged)  are  in  descending  order  of  magnitude.  Prove  that  any 
rational  symmetric  function  of  x  and  y  is  an  even  quadruply-periodic  function  of  u  and  v, 
and  that,  in  particular, 

{Ur  -  x)  (a,.  -  y) 

(for  '/■=0,  1,  2,  3,  4)  is  the  perfect  square  of  a  quadruply-periodic  function  of  u  and  v,  which 
is  even  for  even  values  of  r  and  is  odd  for  odd  values. 

Writing  p^?  =  {a^  -  x)  (a,.  —  y), 

(«™  -  «.)  Pmn  -Pm  {j^i   +  J^  )  '  Pn[  ^^    +    ^^  )  , 

obtain  the  types  of  quadratic  relations  connecting  the  fifteen  functions  ;  and  prove  that 

Pi  ,      Pm  ^      Pn      I  ) 
Plrj      pmri      Pnr 
Pis  5      Pms )      Pns     1 

where  I,  m,  n,  r,  s  are  any  arrangement  of  0,  1,  2,  3,  4,  is  constant. 


MISCELLANEOUS   EXAMPLES  825 


/: 


190.  If  the  curve  F  {z,  m)  =  0  be  of  degree  m,  shew  that  every  integral  of  the  first 
kind  is  of  the  form 

Fu  (^,  m)     ' 
where  Q  (s,  u)  is  a  polynomial  in  z  and  ii  of  degree  (m-  3)  at  most. 

If  the  curve  F  {z,  u)  =  Q  have  no  multiple  points  except  such  as  have  distinct  tangents, 
shew  that  any  multiple  point  of  order  q  on  the  curve  i''  is  a  multiple  point  of  order  q—1 
on  the  curve  Q. 

Find  the  integrals  of  the  first  kind  for  the  curve 

z'^-u'^  +  ahu^O  ; 

and  discuss  the  transformation  of  this  curve  to  the  hyperelliptic  form. 

191.  Shew  that  every  uniform  function  of  position  on  a  Riemann's  surface,  connected 
with  an  algebraic  equation  f{w,  z)  =  0  of  degree  m  in  io,  the  function  having  infinities 
only  of  finite  order,  can  be  expressed  in  a  form 

dw 

where  C^  is  a  polynomial  in  w  of  degree  ^m  —  2  having  rational  functions  of  z  for  its 
coefficients,  and  ff  (2)  is  a  rational  function  of  z. 

Prove  that  there  are  integrals  of  rational  functions  of  w  and  z,  which  do  not  acquire 
an  infinite  value  upon  the  surface. 

Construct  these  integrals,  when  the  algebraic  equation  is 

w3  =  s  (l-z)  (1  -az)  (1  -  bz)  (l-cz), 

where  the  constants  a,  b,  c  are  unequal  to  one  another  and  no  one  of  them  is  either  zero 
or  unity. 

192.  If  an  analytical  correspondence  be  set  up  between  two  variable  points  x  and  y 
of  a  non-singular  Riemann's  smface,  of  such  a  nature  that  to  every  point  x  there  cor- 
respond m  variable  points  3/1,3/2)  ■•■■>ymi  distinct  in  general  from  x,  prove  that 

m 

2  %(,yj)-fy%(.^)  =  Ji,     (-^=1,  2,  ...,/•), 
i=l 

where  y  is  a  certain  positive  or  negative  integer,  Jj  is  a  quantity  independent  of  x,  and 
%i,  U2,  ...,  Wp  are  p  independent  and  everywhere  finite  integrals  on  the  surface. 

Prove  further  that,  if  Cj,  C2, ...,  Cp  be  suitably  chosen  constants,  and  if 

denote  the  ^-function  belonging  to  the  surface,  then 

6{ui{y)-Ui{x)-c,] «=™ e{Ui{y)-Ui{y^)-G,) 


6  K  {.y)  -  Wi  i^)  -  Ci}  0  bh  if)  -  Mi  {^)  -  Ci}  n=i  B  {Mi  ( /)  -  y'i  iyn)  -  Ci)  e  k  {y)  -  Ui  (y„o)  -  c,] 

is  an  algebraic  function  of  x  and  y  ;  x^,  j/",  denoting  fixed  points  and  x,  y  a.  variable  point 
of  the  surface,  z/i",  3/2**,  ...,  ym^  the  points  corresponding  to  x^. 

Deduce  that  such  a  correspondence  can  always  be  represented  by  an  algebraic  function 
properly  interpreted. 


826  MISCELLANEOUS   EXAMPLES 

193.  If  Wi,  Wo,  ...,  u>p  be  the  p  normal  integrals  of  the  first  kind  on  a  Riemann's 
surface,  and  5;,(*)  be  the  modulus  of  periodicity  of  the  normal  integral  of  the  second 
species  to  tc,^{z)  with  respect  to  the  cut  b^,  shew  that 

Hence  or  otherwise  shew  that  the  necessary  and  sufficient  conditions,  that  q  distinct 
points  Ci,  C2,  ...,  Cq{q^p)  should  be  such  that  at  a  certain  number  of  them  a  regular 
function  assumes  the  same  value,  are 


0  = 


w/(Ci),    ...,Wi'{Cq) 
W2'{Ci),   ...,  W2'(Cg) 


Wp'{Cl),      ...,'Wp'{Cg) 

Shew  also  that  if,  for  every  integral  of  the  first  kind  W  (z), 

W  (ci)  dci  +  W  (cg)  dc2+...+  W  (Cg)  dcg  =  0, 
then  Ci,  Cg,  ...',  Cg  are  points  at  which  a  regular  function  assumes  the  same  value. 


IX. 

194.  Give  an  outline  of  a  proof  that  a  potential  function  ti  exists,  subject  to  the 
conditions, 

(i)  at  all  points  within  the  area  of  a  circle  of  radius  unity,  the  quantities 
M,  ^  ,  ^  ,  —  ,  ^-"2  are  regular  functions  of  x  and  y  such  that  ^^  +  ^-2  =  0  ; 

(ii)  the  quantity  u  acquires  assigned  values  along  the  circumference,  which  are 
regular  functions  of  the  position  on  the  circumference  ;  and  obtain  the  function  in 
the  form 

2     r  27r  \ _^2  _  y2 

""^^nj  ^^^'^\x-  COS  ^/.)2  +  Cy  -  sin  y\rf  '^'^' 
where  /  (■v//')  represents  the  values  along  the  circumference.     Prove  also  that  /  (yj^)  can  have 
a  finite  discontinuity  at  a  limited  number  of  points. 

Apply  the  transformation  x+iy  —  -TjT. ^^ — ^  to  the  above  integral  so  as  to  prove 

that  a  potential  function  u,  which  exists  over  the  whole  plane  and  is  such  that  its  value 
is  unity  between  —  1  and  + 1  on  the  real  axis  and  elsewhere  is  zero  on  the  real  axis,  is  given 

by  -  ^  where  6  is  the  angle  subtended  at  the  point  in  the  plane  by  the  part  of  the  real  axis 

lying  between  —  1  and  +  1. 

195.  Deduce,  from  Cauchy's  integral  formula,  Poisson's  expression  for  the  value  of  a 
developable  potential  function  at  any  point  interior  to  a  given  circle,  in  terms  of  the 
values  of  the  function  on  the  circumference. 

Shew  ab  initio  in  the  case  when  u  (t)  is  finite  and  continuoiis  for  all  real  values  of  t, 
the  values  ^(  +  00)  being  the  same,  that  the  integral 

1   /■+"    u{t)ydt 


represents  a  developable  potential  function   of  {x,  ?/),  for  y>0,  reducing  when  (x,  y) 
approaches  to  {xq,  0)  to  the  value  t(,{xQ). 

Evaluate  the  integral  when  u  (t)  =  -x — -. 


MISCELLANEOUS   EXAMPLES  827 

196.  Taking  as  an  area  the  whole  of  the  plane  of  z  with  the  exception  of  the  finite 
straight  line  joining  z=  —  \  to  ^  =  1,  find  a  function  of  a,  which  is  single-valued  in  the  area, 
is  real  on  the  boundary,  and  is  discontinuous  within  or  upon  the  boundary  of  the  area 
only  like  {z  -  i)~^. 

197.  Find  a  function  which  shall  be  regular  within  the  circle  1^1  =  1  and  shall  have  on 
the  circumference  the  value 

a^  -  2a  cos  d  +  cos  26  +  i  {2a  sin  6  -  sin  26) 
{a^-2acos6  +  lf  ' 

where  |  «  |  >  1, 

198.  Prove  that  the  most  general  form  for  a  function  which  (i)  is  to  be  single-valued 
and  analytic  in  a  rectanglis  in  the  plane  of  the  complex  variable  u  whose  corners  are  u  =  0, 
m  =  2q),  u  =  iH,  u=2a>  +  iH,  where  co,  iTare  real;  and  (ii)  is  to  be  further  such  as  to  assume 
equal  values  at  opposite  points,  ?i  =  t^,  ^t  =  ^■A-f2«,  of  one  pair  of  sides;  is  a  series  of 

integral  powers  of  exp  (  —  m  j . 

If  a  <  6  be  real  and  positive,  and  the  function  p  {u)  be  consti-ucted  with  the  real 
period  2a)  and  the  period  2co'  given  by 

,     im ,      b 
&)  =  —  log  -  , 

IT     ^  a' 

find  the  region  in  the  plane  of  z  given  by  the  formula 

when  (  varies  in  the  annulus  lying  between  two  circles  with  centre  at  ^=0  of  respective 
radii  a  and  h. 

Shew  that  there  is,  in  this  annulus,  only  one  value  of  (  corresponding  to  any  point  in 
the  region  obtained. 

199.  Shew  how  to  define  an  integral  function  of  the  two  variables  u,  v,  which  shall 
satisfy  the  equations 

0(W+1,  v)  =  (f,(u,  v)  =  (t){u,  v+l), 

(})  (2i  +  p,v  +  a-)  =  )u/''  (f)  (u,  v),         (p  (m  +  p',v  +  a-')  =  /x'e^'"  ^  {u,  v), 
obtaining  any  necessary  conditions  for  the  constants  p,  cr,  p',  a',  X,  X',  p.,  p.'. 


200.     If  the  substitutions  of  an  infinite  discontinuous  group  be 


t^J^         (^=1.2,  ...CO), 


shew  that  the  series 

2   (7^3  + Si)-''" 
i=l 

is  absolutely  convergent  when  m  is  a  greater  integer  than  unity,  except  for  special  points  z. 

Construct  a  group  of  substitutions  of  genus  p,  for  which  the  fundamental  polygon  is 
the  space  outside  2p  circles,  and  the  fundamental  substitutions  make  these  circles  corre- 
spond in  pairs  ;  and  shew  that,  subject  to  certain  inequalities,  the  series 

^{yiZ+K)-^ 

i 

is  absolutely  convergent  for  groups  of  this  character. 


828  MISCELLANEOUS   EXAMPLES 

201.  Give  an  example  of  a  rational  function  R  (x)  which  is  unaltered  by  a  group  of 
transformations  of  the  form  x'  =  {ax  +  h)l{cx  +  d),  finding,  for  your  example,  a  region  in 
the  plane  of  the  complex  variable  x  in  which  the  function  assumes  every  value  just  once. 

Explain  some  general  method  of  expressing  automorphic  functions  when  the  group  of 
linear  transformations  is  assigned. 

202.  In  the  equation 

which  is  satisfied  by  the  quarter  periods  K  and  iK'  of  the  Jacobian  elliptic  functions 
formed  with  the  modulus  k=\/z,  shew  that  0  is  a  uniform  function  of  the  quotient 

t  =  -^  of  two  solutions  of  the  equation,  being  automorphic  for  the  group  generated  by  the 

substitutions 


(M  +  2)and(^,^). 


Shew  how,  by  using  this  as  an  auxiliary  differential  equation,  any  linear  differential 
equation  with  uniform  coefficients  and  three  singular  points  can  be  solved  in  terms  of 
uniform  functions. 


GLOSSARY  QF  TECHNICAL  TERMS. 


{The  numbers  refer  to  the  pages,  ivhere  the  term  occurs  for  the 
first  time  in  the  book  or  is  defined.) 


Abbildung,  conforme,  11. 

Absolute  convergence  of  series,  21 ;  of  pro- 
ducts, 91. 

Absoluter  Betrag,  3. 

Absolute  value,  3. 

Accidental  singularity  (pole),  17,  61. 

Addition-theorem,  algebraic,  344. 

Adelphic  order,  864. 

Adjoint  curves,  445. 

Adjoint  polynomials,  445. 

Algebraic  addition-theorem,  344. 

Algebraic  function,  determined  by  an  equation, 
190. 

Amplitude,  3. 

Analytical  curve,  458,'  478,  658. 

Analytic  function,  10;    monogenic,  67. 

Anharmonic  group,  754. 

Argument,  3. 

Argument  and  parameter,  interchange  of,  513. 

Arithmetic  mean,  method  of  the,  458. 

Ausserwesentliche  singuldre  Stelle,  61. 

Automorphic  functions,  715,  753. 

Betrag,  absoluter,   3. 
Bicursal,  555. 
Bien  defini,   190. 
Bifacial  surface,  372. 
Birational  transformation,  537. 
Boundary,  369. 
Branch,  16. 
Branch-line,  385. 
Branch-point,  17,  183. 
Branch-section,  385. 

Canonical  resolution  of  surface,  402. 
Categories  of  corners,  cycles,  725,  729. 
Circle,  discriminating,  133. 


Circle  of  convergence,  22. 

Circuit,  374. 

Class,  or  genus,  (of  connected  surface),  371. 

Class  of  doubly-periodic  function  of  second 
order,  263. 

Class  of  equation,  395. 

Class  of  group,  742. 

Class  of  singularity,  177. 

Class  of  tertiary-periodic  function,  335. 

Class  of  transcendental  integral  functions,  109. 

Class-moduli,  545. 

Combination  of  areas,  480. 

Compound  circuit,  374. 

Conditional  convergence  of  series,  21;  of  pro- 
ducts, 91. 

Conditional  equation  in  Abel's  theorem,  581. 

Conformal  representation,  11. 

Conforme  Abbildung,  11. 

Congruent  figures,  631,  724. 

Conjugate  edges,  725. 

Connected  surface,  359. 

Connection,  order  of,  364. 

Connectivity,  364. 

Constant  modulus  for  cross-cut,  427. 

Contiguous  regions,  724. 

Continuation,  67. 

Continuity,  region  of,  67. 

Continuous  substitution,  717. 

Convergence  of  series,  22;    of  products,  91. 

Convexity  of  normal  polygon,  727. 

Corner  of  region,  724. 

Coupure,  165,  220. 

Critical  point,  17. 

Cross-cut,  361. 

Cross-line,  385. 

Cycles  of  branches  of  algebraic  function,  570. 

Cycles  of  corners,  726. 


830 


GLOSSAEY   OF   TECHNICAL   TERMS 


Deficiency,  403. 

Deformation  of  loop,  407. 

Deformation  of  surface,  379. 

Degree  of  cycle,  570. 

Degree  of  pseudo-automorphic  function,  785. 

Degree    of    rational    function    on    Biemann's 

surface,  420. 
Derivative,   Schwarzian,  657. 
•Dihedral  group,  757. 
Diramazione,  punto  di,   17. 
Dirichlet's  principle,  458. 
Discontinuity,  polar,  17. 
Discontinuous  groups,  717. 
Discontinuous  substitution,  717. 
Discrete  substitution,  717. 
Discriminating  circle,  133. 
Divergence  of  series,   22 ;    of  products,  91. 
Domain,  60. 
Domaine,  60. 
Dominant  function,  39. 
Double  (or  fixed)  circle  of  elliptic  substitution, 

746. 
Doubly-periodic  function  of  first,  second,  third, 

kind,  320,  321. 

Edge  of  region,  724. 

Edges  of  cross-cut,  positive  and  negative,  424. 

Eindndrig,  16. 

Eindeutig,  16. 

Einfach  zusammenhcmgend,  360. 

Element,  67. 

Element  of  doubly-periodic  function  of  third 

kind,  338,  340. 
Elementary  integral  of  the  second  kind,  third 

kind,  446,  452. 
Elliptic  substitution,  631. 
Equivalent  homoperiodic  functions,  260. 
Essential  singularity,  19,  61. 
Exceptional  value,  66. 
Existence-theorem,  416,  455, 

Factor,  primary,  101. 

Factorial  functions,  531. 

Families  of  groups,  740. 

Finite  groups,  719. 

First   kind,    doubly-periodic   function   of  the, 

321. 
First  kind  of  Abelian  integrals,  444. 
Fixed  (or  double)  points  of  substitution,  628. 
Forlsetzung,  67. 

Fractional  factor  for  potential  function,  476. 
Fractional   part   of  doubly-periodic   function, 

259. 
Fuchsian  functions,  753. 


Fuchsian  groups,  740. 
Fundamental  circle  for  group,  737. 
Fundamental  loops,  407. 
Fundamental  parallelogram,  237,  244. 
Fundamental    polyhedron    (of    reference    for 

space),  748. 
Fundamental  region  (of  reference   for  plane), 

724. 
Fundamental  substitutions,  716. 

Gattung  (kind  of  integral),  444. 

Genere,  109. 

Genere  (genus  of  connected  surface),  371. 

Genre  (applied  to  singularity),  177. 

Genre     (applied     to     transcendental     integral 

functions),  109. 
Genre  (genus  of  connected  surface),  371. 
Genus  (of  connected  surface),  371. 
Genus  (of  equation),  395. 
Genus  (of  group),  742. 
Geschlecht,  395,  403. 
Giramento,  punto  di,  17. 
Gleichverzweigt,  419. 
Grenze,  natilrliche,  153. 
Grenzkreis,  133. 
Group  of  substitutions,  715. 
Grmidzahl,  364. 

Harmonic  functions,  9. 

Hauptkreis,  737. 

Holomorphic,  17. 

Homogeneous  substitutions,  756. 

nomographic  transformation,  or  substitution, 

625. 
Homologous  (points),  237. 
Homoperiodic,  263. 
Hyperbolic  substitution,  631., 
Hyperellipti^  curves  or  equations,  565. 

Improperly  discontinuous  groups,  718. 

Index  of  substitution,  717. 

Infinitesimal  substitution,  636,   717. 

Infinity,  17. 

Integrals  of  the  first  kind,  second  kind,  third 

kind,  Abelian,  444,  446,  452. 
Interchange  of  argument  and  parameter,  513. 
Invariants  of  elliptic  functions,  295. 
Inversion-problem,  517. 
Irreducible  circuit,  374. 
Irreducible  (point),  236,  237. 
Isothermal,  707. 

Kleinian  functions,  753. 
Kleinian  group,  743. 


GLOSSAEY  OF  TECHNICAL  TERMS 


831 


Lacet,  182. 

Lacunary  functions,   166. 
Level  values,  269. 
Ligne  de  passage,  385. 
Limit,  natural,  153. 
Limitrophe,  724. 
Linear  cycles,  570. 
Linear  substitution,  625. 
Loop,  182. 
Loop-cut,  362. 
Loxodromic  substitution, 


631. 


Majorante,  39. 

Mehrdeutig,  16. 

Mehrfach  zusainmenhdngend,  361. 

Meromorphic,  17. 

Modular-function,  767. 

Modular  group,  721. 

Modulus,  3. 

Modulus  for  cross-cut,  constant,  427. 

Modulus  of  periodicity  (cross-cut),  427. 

MonadeliDbic,  360. 

Monodromic,  16. 

Monogenic,  15. 

Monogenic  analytic  function,  67. 

Monotropic,   16. 

Multiform,  16. 

Multiple  circuit,  374. 

Multiple  connection,  362. 

Multiplicateurs,  fonctions  a,  531. 

Multiplier  of  substitution,  628. 

Natural  limit,  153. 
Natiirliche  Grenze,  153. 
Negative  edge  of  cross-cut,  424. 
Non-essential  singularity,  61. 
Normal  (connected)  surface,  381. 
Normal  form  of  linear  substitution,  715. 
Normal  form  of  transformable  equations,  567. 
Normal   function  of   first  kind,   second  kind, 

third  kind,  508,  510,  511. 
Normal  polygon  for  substitutions,  728. 

Order  of  a  doubly-periodic  function,  25?. 

Order,  of  connection,  adelphic,  364. 

Order    of    rational    function     on    Eiemann's 

surface,  420. 
Ordinary  point,  60. 
Origin  of  cycle,  570. 
Orthomorphosis,  11. 
Oscillating  series,  21. 

Parabolic  substitution,  631. 
Parallelogram,  fundamental  or  primitive,  237, 
244. 


Path  of  integration,  21. 
Period,  235. 

Periodicity  for  cross-cut,  modulus  of,  427. 
Permanent  equation  in  Abel's  theorem,  581. 
Polar  discontinuity,   17. 
Pole,  17,  61. 
Polyadelphic,  361. 
Polyhedral  functions,  706. 
Poly  tropic,  16. 

Positive  edge  of  cross-cut,  424. 
Potential  function,  457. 
Primary  factor,  101. 
Primfunction,  101. 
Primitive  parallelogram,  244. 
Products,  convergence  of,  91. 
Properly  discontinuous  groups,  718. 
Pseudo-periodicity,  301,  304,  320,  321. 
Punto    di    diramazione,    punto    di    giramento, 
17. 

Querschnitt,  361. 

Eamification  (of  Eiemann's  surface),  395. 

Ramification,  point  de,  17. 

Eational  function,  84. 

Eeal  substitutions,  631. 

Eeconcileable  circuits,  374. 

Eeducible  circuit,  374. 

Eeducible  (point),  236,  237. 

Eegion  of  continuity,  67. 

Eegular,  17,  60. 

Eegular  singularities,  192. 

Mepresentation  conforme,   11. 

Eesidue,  48. 

Eesolution  of  surface,  canonical,  402. 

Retrosection,  362. 

Eiemann's  surface,  382. 

Eoot,  17. 

RUckkehrschnitt,   862. 

Schleife,  182. 

Schwarzian  derivative,  657. 

Second  kind,  doubly-periodic  function  of  the, 

821. 
Second  kind  of  Abelian  integrals,  446. 
Secondary-periodic  functfcns,  322. 
Section,  69,  165,  220. 
Section  (cross-cut),   361. 
Series,  convergence  of,  22. 
Sheet,  382. 

Simple  branch-points,  208. 
Simple  circuit,  374. 
Simple  connection,  360. 
Simple  curve,  24. 
Simple  cycle  of  loops,  408. 


882 


GLOSSARY  OF  TECHNICAL  TERMS 


Simple  element  for  tertiary-periodic  function, 

338,  340. 
Singular  point,  17. 
Singularity,  accidental,   17,  61. 
Singularity,  essential,  19,  61. 
Special  function  on  Riemann's  surface,  526. 
Species  of  singularity,  177. 
Sub-categories  of  cycles,  740. 
Sub-rational  representation  of  variables,  551. 
Substitution,  homogeneous,  756. 
Substitution,  linear  or  homographic,  625. 
Syneetic,  17. 


Umgebung,   60. 

Unconditional  convergence  of  series,  22 ;  of 
products,  91. 

Unicursal,   548. 

Unifacial  surface,  372. 

Uniform  convergence  of  series,  22;  of  pro- 
ducts, 91. 

Uniform  function,  16. 


Verzweigungschnitt,  385. 
Verzweigungspunkt,  17. 


Taglio  trasversale,  361. 
Tertiary-periodic  functions,  322. 
Tetrahedral  group,  759. 
Thetafuchsian  function,  776. 
Third  kind,  doubly-periodic  function  of  the,  321. 
Third  kind  of  Abelian  integral,  452. 
Transcendental  function,  84. 
Transformation,  birational,  537. 
Trasversale,  361. 


Wesentliche  singulare  Stelle,  61. 
Winding-point,  392. 
Winding-surface,  392. 
Windungspunkt,  17. 

Zero,  17. 

Zusammenhangend,    einfach,    mehrfach,    360, 
361. 


INDEX. 


(The  numbers  refer  to  the  pages. 


Abel,  269,  518,  580. 

Abel's    formula    for    sum    of    transcendental 
integrals,  585 :    examples  of,  587-590  ;    ap- 
plied   to    integrals    of    first   kind,   590 ;     of 
second  kind,  594;    of  third  kind,  597. 
Abel's    Theorem    on    integrals,   quoted,    519 ; 

proved,  579-601  :    the  main  result,   585. 
Abelian  transcendental   functions,   arising  by 
inversion  of  functions  of  the  first  kind  on 
a  Eiemann's  surface,  517; 

Weierstrass's  form  of,  518. 
Absolute  convergence,   of  series,   21 ;    of  pro- 
ducts, 91. 
Accidental  singularities,  17,  61,  78 ; 

must  be  possessed  by  uniform  function, 

78; 
form  of  function  in  vicinity  of,  78; 
are  isolated  points,  78; 
number  of,  in  an  area,  82,  86; 
if    at    infinity   and   there   be   no   other 
singularity,  the  function  is  polynomial, 
83; 
if  there  be  a  finite  number  of,  and  no 
essential     singularity,     the     uniform 
function  is  rational  and  meromorphic, 
■      85. 
Addition-theorem,  for  uneven  doubly-periodic 
function  of  second  order  and  second  class, 
290; 

for  Weierstrass's  ^-function,  307; 
quasi-form   of,   for   the   (7-function  and 

tbe  f-function,  307; 
definition  of  algebraical,  344; 
algebraical,  is  possessed  by  algebraical 
functions,  344; 

by  simply- periodic  functions,  345; 

by  doubly -periodic  functions,  846; 

function  which  possesses  an  algebraical,  is 

either     (i)  algebraical,  347; 

or  (ii)  simply -periodic,  350,  352; 

or         (iii)  doubly-periodic,  354; 

F.  F. 


satisfies  a  differential  equation  be- 
tween itself  and  its  first  derivative, 
355; 
condition  that  algebraical  equation  be- 
tween three  variables  should  express, 
357; 
form    of,    when    function    is    uniform, 
358. 
Adjoint  curves,  445. 

Adjoint    polynomial    on    Eiemann's    surface, 
quotient    of    one   by   another,    is   a   special 
function,  527. 
Adjoint  polynomials,  445. 
Algebraic    equation   between    three    variables 
should  express  an  addition-theorem,  condi- 
tion that,  357; 
Algebraic  equation,  defining  algebraic  multi- 
form functions,  190  (see  algebraic  function); 
genus  of,  395 ; 

for  any  uniform  function  of  position  on 
a  Eiemann's  surface,  417. 
Algebraic  equation  defines  functions  that  are 

analytic,  207. 
Algebraic  equation  has  roots,  88. 
Algebraic  function,  cycle  of  branches  of,  570: 

birationally  transformed,  571-577. 
Algebraic  function  is  analytic,  207. 
Algebraic    (multiform)    functions    defined    by 
algebraical  equation,   190; 
branch-points  of,  191 ; 
infinities   of,   are    singularities    of   the 
coefficients,  192 ; 

graphical  method  for  determination 
of  order  of,  194 ; 
branch-points  of,  197; 
cyclical  arrangements  of  branches  round 

a  branch-point,  200; 
when  all  the  branch-points  are  simple, 

208; 
in  connection  with  Eiemann's  surface, 
386. 

53 


834 


INDEX 


Algebraic    function   on   a   Eiemann's   surface, 
integrals  of,  436; 

integrals  of,  everywhere  finite,  438 ; 

number  of,  in  a  special  case,  438; 
when  all  branch-points  are  simple,  three 

kinds  of  integrals  of,  439 ; 
infinities  of  integrals  of,  440,  443; 
branch-points  of  integrals  of,  443. 
Algebraic  functions  on  a  Eiemann's  surface, 
constructed  from  normal  elementary  func- 
tions of  second  kind,  520; 

smallest  number  of  arbitrary  infinities 
to  render  this  construction  possible, 
520; 
Kiemann-Roch's  theorem  on,  521 ; 
smallest  number  of  infinities  of,  which, 
f  except  at  them,  is  everywhere  uniform 

and  continuous,  523 ; 
which  arise  as  first  derivatives  of  func- 
tions of  first  kind,  524  ; 

are  infinite  only  at  branch-points, 

525; 
number  of  infinities  of,   and  zeros 

of,  525; 
most  general  form  of,  526 ; 
determined  by  finite  zeros,  526; 
Brill-Nother  law  of  reciprocity  for, 
528; 
determine  a  fundamental  equation  for  a 

given  Eiemann's  surface,  528; 
relations  between  zeros  and  infinities  of, 
535. 
Algebraic  isothermal  curves,  families  of,  707 

et  seq.    (see  isothermal  curves). 
Algebraic  plane  curve  birationally  transformed 
into  another  with  double  points  only,  569-578. 
Algebraic  relation  between  functions  automor- 
phic  for  the  same  infinite  group,  788 ; 
genus  of,  in  general,  789. 
Analytic  function,  monogenic,  67. 
Analytic    function    represented    by    series    of 

polynomials,  69,  134. 
Analytic  function  defined  by  algebraic  equation, 

207. 
Analytical  curve,  459,  478,  658; 

represented  on  a  circle,  478; 
area  bounded  by,  represented  on  a  half- 
plane,  658; 

consecutive  curve  can  be  chosen  at 
will,  659. 
Analytical  test  of  a  branch-point,  186. 
Aiichor-ring  conformally  represented  on  plane, 

612. 
Anharmonic  function,  automorphic  for  the  an- 
harmonic  group,  754. 


Anharmonic  group  of  linear  substitutions,  754. 
Anissimoff,  133. 

Appell,  174,  223,  342,  343,  530,  531,  559,  570. 
Appell's  factorial  functions,  531  (see  factorial 

functions). 
Area,    simply   connected,    can   be   represented 
conformally  upon  a  circle  with  unique  cor- 
respondence of  points,  by  Eiemann's  theorem, 
654; 

form  of  function  for  representation  on  a 

plane,  657,  670;  on  a  circle,  657; 
bounded  by  analytical  curve  represented 

on  half- plane,  658  ; 
bounded  by  cardioid  on  half-plane,  662  ; 
of    convex   rectilinear  polygon,    666   et 

seq.   (see  rectilinear  polygon) ; 
bounded  by  circular  arcs,  679  et  seq.  (see 
curvilinear  polygon). 
Areas,  combination  of,  in  proof  of  existence- 
theorem,  480. 
Argand,  2. 

Argument  (or  amplitude)  of  the  variable,  3. 
Argument  of  function  possessing  an  addition- 
theorem,  forms  of,  for  a  value  of  the  function, 
347  et  seq. 
Argument  and  parameter  of  normal  elementary 

function  of  third  kind,  515. 
Ascoli,  459. 
Automorphic  function,  753 ; 

constructed  for  infinite  group  in  pseudo- 
automorphic   form,   771    et    seq.   (see 
thetafuchsian  functions) ; 
expressed    as    quotient    of    two    theta- 
fuchsian functions,  784; 
its  essential  singularities,  786 ; 
number  of  irreducible  zeros  of,  is  the 
same   as   the   number   of  irreducible 
accidental  singularities,  786; 
different,  for  same  group  are  connected 
by  algebraical  equation,   788;    genus 
of  this  algebraical  equation  in  general, 
789; 
connection  between,  and  general  linear 
differential  equations  of  second  order, 
791; 
modular-functions  as  examples  of,  792. 

Baker,  247,  396,  404,  519,  528,  530,  537,  567, 

579;   a  rule  for  determining  the  genus  of  a 
Eiemann's  surface,  404. 
Barnes,  103. 

Barrier,  impassable,  in  connected  surface,  360 ; 
can  be  used  to  classify  connected  sur- 
faces, 361 ; 
changed  into  a  cut,  361. 


INDEX 


835 


Beltrami,  658,  660,  661. 
Bernoulli's  numbers,  48. 
Bertini,  570. 
Bianchi,  751,  752. 
Bicursal  equations  and  curves,  555. 
Biehler,  342. 
Biermann,  67,  344. 
Bifacial  surfaces,  372,   380. 
Birational  transformation,  415,  587-579 ; 
conserves  genus  of  equation,  542 ; 
conserves  kind  of  function,  543 ; 
conserves  Sp-S  +  p  class-moduli,  545. 
Birational   transformation   of  algebraic   plane 
curves,   569-578:    of   cycles  of  branches  of 
algebraic  function,  571-577. 
Birational     transformation     of     equations     of 
genus  zero,  550 ;    of   genus  unity,  558 ;    of 
genus    greater    than   unity,    566;    of   genus 
greater  than  two,  569. 
Blumenthal,  113. 
Bolza,  716. 
Bonnet,  611. 
Bonola,  714. 
Boole,  585. 
Borchardt,  257. 
Borda,  646. 
Borel,  113,  134,  173. 

Boundary  of  region  of  continuity  of  a  function 
is    composed    of    the    singularities    of    the 
function,   68. 
Boundary,  defined,  369 ; 

assigned  to  every  connected  surface,  361, 

369; 
edges  acquired   by  cross-cut  and  loop- 
cut,  362; 
of  simply  connected  surface  is  a  single 

line,  370; 
effect  of  cross-cut  on,  370; 
and  of  loop-cut  on,  371. 
Boundary   conditions    for   potential   function, 

460  (see  potential  function). 
Boundary,  functions  on  a  Eiemann's  surface 

without,  491. 
Boundary  values  of  potential  function  for  a 
circle,  465; 

may  have  limited  number  of  finite  dis- 
continuities, 470; 
include  all  the  maxima  and  the  minima 
of  a  potential  function,  476. 
Boundaries     of     connected     surface,    relation 
between     number     of,     and     connectivity, 
371. 
Branches  of  a  function,  defined,  16; 

affected  by  branch-points,  180  et  seq. ; 
obtained  by  continuation,  180 ; 


are  uniform  in  continuous  regions  where 

branch-points  do  not  occur,  184; 
which  are  affected  by  a  branch-point, 

can  be  arranged  in  cycles,  185; 
restored  after  number  of  descriptions  of 

circuit  round  branch-point,  186 ; 
analytical  expression  of,  in  vicinity  of 

branch-point,  187; 
number  of,  considered,  188; 
of  an  algebraic  function,  190  (see  alge- 
braic function) ; 
a  function  which  has  a  limited  number 

of,  is  a  root  of  an  algebraic  equation, 

210. 
Branch-lines,  are  mode  of  junction  of  the  sheets 
of  Eiemann's  surfaces,  385 ; 
properties  of,  386  et  seq.; 
free  ends  of,  are  branch-points,  386; 
sequence  along,  how  affected  by  branch- 
points, 387; 
system  of,  for  a  surface,  387; 
special  form  of,  for  two-sheeted  surface, 

391; 
when  all  branch-points  are  simple,  403 ; 
number    of,    when    branch-points    are 

simple,  412. 
Branch-points,  defined,   16,  183 ; 

integral  of  a  function  round  any  curve 

containing  all  the,  42; 
effect  of,  on  branches,  178,  180  et  seq. ; 
analytical  test  of,  186 ; 
expression  of  branches  of  a  function  in 

vicinity  of,  187  ; 
of  algebraic  functions,  191,  197 ; 
simple,  208,  403; 
number  of  simple,  209; 
are  free  ends  of  branch-lines,  886; 
effect   of,   on   sequence   of   interchange 

along  branch-lines,  887 ; 
joined    by    branch-lines    when    simple, 

391; 
deformation    of    circuit    on   Eiemann's 

surface  over,  is  impossible,   396 ; 
circuits  round  two,  are  irreducible,  396 ; 
number  of,  when  simple,  402; 
in    connection    with    loops,     404     (see 

loops) ; 
canonical  arrangement  of,  when  simple, 

411. 
Brill,  404,  415,  528,  530,  570. 
Brill-Nother  law  of  reciprocity,  528. 
Brioschi,  822,  828. 
Briot,  531. 

Briot  and  Bouquet,  vi,  27,  44,  47,  197,  208, 
246,  249,  257,  269,  519. 

53—2 


836 


INDEX 


Bromwich,  6,  21,  292. 

Burnside  (W.),   141,  402,  456,  638,  653,  664, 

689,  716,  754,  774,  789. 
Burnside  (W.  S.)  and  Panton,  440,  584. 

Canonical  form,  of  complete  system  of  simple 
loops,  409; 

of  Kiemann's  surface,  413 ; 
resolved,  414. 
Canonical  resolution  of  Eiemann's  surface,  402. 
Cantor,  176. 

Cardioid,  area  bounded  by,  represented  on  strip 
of  plane,  662 ; 

on  a  circle,  663. 
Carslaw,  21,  23,  652,  714. 
Casorati,  2,  27,  407. 
Categories  of  corners,  725  (see  corners). 
Cathcart,  6,  20,  61. 
Cauchy,  v,  vi,  24,  27,  31,  49,  50,  52,  59,  75, 

82,  207,  214,  359,  585. 
Cauchy's    theorem    on    the    integration    of    a 
holomorphic  function  round  a  simple  curve, 
27; 

and  of  a  meromorphic  function,  30; 
on  the  expansion  of  a  function  in  the 
vicinity  of  an  ordinary  point,  50. 
Cayley,  2,  11,  92,  397,  403,  552,  555,  557,  615, 
622,  623,  629,  657,  658,  661,  665,  679,  705, 
707,  710,  754,  756. 
Cesaro,   113. 
Chessin,   250. 

Christoffel,  652,  666,  670,  679. 
Chrystal,  vi,  2,  6,  199,  218. 
Circle,  areas  of  curves  represented  on  area  of : 
exterior  of  ellipse,  614; 
interior  of  ellipse,  617 ; 
interior  of  rectangle,  615,  674; 
interior  of  square,  615,  674; 
exterior  of  square,  674 ; 
exterior  of  parabola,  618 ; 
interior  of  parabola,  619 ; 
half-plane,  619; 
interior  of  semicircle,  620 ; 
infinitely  long  strip  of  plane,  621 ; 
any   circle,   by   properly   chosen   linear 

substitution,  627; 
any  simply  connected  area,  by  Eiemann's 

theorem,  654; 
interior  of  cardioid,  662 ; 
interior  of  regular  polygon,  678. 
Circle  of  convergence  of  series,   22. 
Circuits,    round    branch-point,    effect    of,    on 
branch  of  a  function,  182,  184; 

restore  initial  branch  after .  number  of 
descriptions,  186; 


on  connected  surface,  374 ; 

reducible,  irreducible,  simple,  multiple, 

compound,  reconcileable,  374; 
represented  algebraically,  375 ; 
drawn  on  a  simply  connected  surface  are 

reducible,  376 ; 
number  in  complete  system  for  multiply 

connected  surface,  377; 
cannot  be  deformed  over  a  branch-point 
on  a  Eiemann's  surface,  397. 
Circular    functions    obtained,    by   integrating 
algebraical  functions,  226; 

on  a  Eiemann's  surface,  430. 
Class-moduli    of    equations    under    birational 

transformation,  544:    number  of,  545. 
Class,  of  transcendental  integral  function  as 
defined  by  its  zeros,  109; 

Laguerre's  criterion  of.  111; 
simple  function  of  given,  112 ; 
essential  singularity,  176 ; 
tertiary-periodic  function,  positive,  335; 

negative,  338; 
(see  genus] . 
Classes   of    doubly-periodic    functions    of    the 

second  order  are  two,  262. 
Clebsch,  208,  247,  403,  407,  408,  411,  415,  453, 

518,  519,  548,  554,  557,  569,  579. 
Clifford,  380,  408. 
Closed  cycles  of  corners  in  normal  polygon  for 

division  of  plane,  730  (see  corners). 
Combination    of    areas,    in    determination    of 

potential  function,  480. 
Complex  variable  defined,  1 ; 

represented  on  a  plane,  2 ; 

and  on  Neumann's  sphere,  4. 
Compound  circuits,  374. 

Conditional  convergence  of  series,  21;  of  pro- 
ducts, 91. 
Conditional  equation  in  Abel's  Theorem,  581. 
Conditions   that   one   complex  variable    be   a 

function  of  another,  7. 
Conduction  of  heat,  application  of  conformal 

representation  to,  649. 
Conformal   representation   applied    to    hydro- 
dynamics,   639 ;    to    electrostatics,    646 ;    to 
conduction  of  heat,  649. 
Conformal  representation  of  planes,  established 
by  functional  relation  between  variables,  11; 
magnification  in,  11 ; 
used   in   Schwarz's  proof  of  existence- 
theorem,  478; 
most  general  form  of  relation  that  secures, 
is  relation  between  complex  variables, 
606; 
examples  of,  614  et  seq. 


INDEX 


837 


Conformal  representation  of  surfaces  is  secured 
by  relation  between  complex  variables  in  the 
most  general  manner,  606 ; 

obtained  by  making  one  a  plane,  607; 
of  surfaces  of  revolution  on  plane,  607 ; 
of  sphere  on  plane,  609 ; 

Mercator's  and  stereographic  projec- 
tion, 609,  610; 
of  oblate  spheroid,  612 ; 
of  ellipsoid,  612; 
of  anchor-ring,  612 ; 
of  surface  of  constant  negative  curvature, 

613; 
Eiemann's  general  theorem  on,  654  ; 
form  of  function  for,  on  a  plane,  657 ; 
on  a  circle,  657. 
Congruent  regions  by  linear  substitutions,  631, 

724. 
Conjugate  edges  of  a  region,  725  (see  edges). 
Conjugate  functions,  9. 

Connected  surface,  supposed  to  have  aboundary, 
360,  368,  375  ; 

to  be  bifacial,  372  ; 

divided  into  polygon  s ,  Lhuilier '  s  theorem 

on,  372  ; 
geometrical  and  physical  deformation  of, 

379; 
can  be  deformed  into  any  other  connected 
surface  of  the  same  connectivity  having 
the  same  number  of  boundaries,  if  both 
be  bifacial,  380  ; 
Klein's  normal  form  of,  381; 
associated  with  irreducible  equation,  392. 
Connection  of  surfaces,  defined,  359  ; 
simple,  360 ; 

definition  of,  362 ; 
,  multiple,  361 ; 

definition  of,  362 ; 
affected  by  cross-cuts,  366 ; 
by  loop-cuts,  367 ; 
and  by  slit,  368. 
Connectivity,  of  surface  defined,  364  ; 
affected  by  cross-cuts,  366  ; 
by  loop-cuts,  367 ; 
by  sUt,  368 ; 
of  spherical  surface  with  holes,  368  ; 
in  relation  to  irreducible  circuits,  376  ; 
of  a  Eiemann's  surface,  with  one  boun- 
dary, 394; 

with  several  boundaries,  396. 
Constant,  uniform  function  is,  everywhere  if 

constant  along  a  line  or  over  an  area,  72. 
Constant   difference   of    integral,   at  opposite 
edges  of  cross-cut,  424  ; 

how  related  for  cross-cuts  that  meet,  425 ; 


for   canonical   cross-cuts,    426  (see 
moduli  of  periodicity). 
Constant  negative  curvature,  surfaces  of,  712. 
Construction  of  rational  function  on  Eiemann's 

surface,  519-524. 
Contiguous  regions,  724. 

Continuation ,  of  function  by  successive  domains, 
67; 

Schwarz's  symmetric,  70 ; 

of  function  with  essential  singularities, 

120; 
of  multiform  function  to  obtain  branches, 
180. 
Continuity  of  a  function,  region  of  (see  region 

of  continuity). 
Continuous  convergence,  22. 
Continuous  group,  718. 
Contour  integration,  43-49. 
Contraction  of  areas  in  conformal  representa- 
tion, 665. 
Convergence  of  products,  kinds  of,  91. 
Convergence,  of  series,  kinds  of,  22 ;  circle  of, 

22  ;  of  products,  91. 
Convex   curve,    area   of,  represented  on  half- 
plane,  deduced  as  the  limit  of  the  representa- 
tion of  a  rectilinear  polygon,  679. 
Convex  normal  polygon  for  division  of  plane, 
in  connection  with  an  infinite  group,  728  ; 
angles  at  corners  of  second  category  and 

of  third  category,  730  ; 
sum  of  angles  at  the  corners  in  a  cycle 
of  the  first  category  is  a  submultiple 
of  four  right  angles,  731 ; 
when  given  leads  to  group,  734  ; 
changed  into  a  closed  surface,  742. 
Corners,  of  regions,  724 ; 

three  categories  of,  for  Fuchsian  group, 

725; 
cycles  of  homologous,  726  ; 
how  obtained,  730 ; 
'  closed,  and  open,  730 ; 

categories  of  cycles,  730  ; 
of    first    category   are  fixed   points   of 

elliptic  substitutions,  734 ; 
of  second  and  third  categories  are  fixed 
points  of  parabolic  substitutions,  734  ; 
sub-categories  of  cycles  of,  741 ; 
open  cycles  of,  do  not  occur  in  Kleinian 
groups,  747. 
Crescent   changed   into   another  of  the  same 
angle  by  a  linear  substitution,  628  ; 
represented  on  a  half- plane,  684. 
Criterion  of  character  of  singularity,  80 ; 

class  of  transcendental  integral  function, 
111. 


838 


INDEX 


Critical  integer,  for  expansion  of  a  function  in 

an  infinite  series  of  functions,  148. 
Cross-cuts,  defined,  361 ; 

effect  of,  on  simply  connected  surface, 
363; 

on  any  surface,  363  ; 
on  connectivity  of  surface,  366  ; 
on  number  of  boundaries,  370 ; 
and  irreducible  circuits,  377  ; 
on  Eiemann's  surface,  398  ; 
chosen  for  resolution  of  Eiemann's  sur- 
face, 399 ; 
in   canonical   resolution   of   Eiemann's 

surface,  401 ; 
in  resolution  of  Eiemann's  surface  in  its 

canonical  form,  413 ; 
difference  of  values  of  integral  at  opposite 

edges  of,  is  constant,  424  ; 
moduli  of  periodicity  for,  426  ; 

number  of  independent  moduli,  428 ; 
introduced   in   proof   of    existence- 
theorem,  487  et  seq. 
Curve,  birational  transformation  of  algebraic 

plane,  569-578. 
Curves,  adjoint,  445. 

Curvilinear  polygon,  bounded  by  circular  arcs, 
represented  on  the  half-plane,  679  et  seq. ; 
function  for  representation  of,  680 ; 
equation  which  secures  the  representa- 
tion of,  683 ; 

connected  with    linear   differential 
equations,  684 ; 
bounded  by  two  arcs,  684  ; 
bounded  by  three  arcs,  685  (see  curvi- 
linear triangles). 
Curvilinear  triangles,  equation  for  representa- 
tion of,  on  half-plane,  685  ; 

connected  with  solution  of  differential 
equation  for  the  hypergeometric  series, 
686; 
when  the  orthogonal  circle  is  real,  688  ; 
any  number  of,  obtained  by  inver- 
sions, lie  within  the  orthogonal 
circle,  689 ; 
equation  is  transcendental,  689  ; 
discrimination  of  cases,  689,  690  ; 
particular  case  when  the  three  arcs 
touch,  691 ; 
when  the  orthogonal  circle  is  imaginary, 
692; 

stereographic  projection  on  sphere 

so   as  to  give  spherical   triangle 

bounded  by  great  circles,  693  ; 

connected  with  division  of  spherical 

surface  by  planes  of  symmetry  of 


inscribed  regular    solids,   694   et 
seq.; 
cases  when  the  relation  is  algebraical 
in  both  variables  and  uniform  in 
one,  694 ; 

equations  which  establish  the 
representation  in  these  cases, 
697  et  seq. ; 
cases  when  the  relation  is  algebraical 
in  both  variables  but  uniform  in 
neither,  704  et  seq. 
Cycles  of  branches  of  algebraic  function,  570: 
birational    transformation    of,    into     linear 
cycles,  571-577. 
Cycles  of  corners,  726  (see  corners). 
Cyclical  interchange  of  branches  of  a  function 
which  are  affected  by  a  branch-point,  185  ; 
when  the  function  is  algebraic,  200. 

Darboux,  23,  53,  70,  83,  379,  613,  666,  679,  712. 
Dedekind,  767,  771. 
Deficiency  of  a  curve,  403  ; 

equal  to  genus  of  associated  Eiemann's 

surface,  403  ; 
determined  by  Baker's  rule,  404  ; 
is  an  invariant  for  rational  transforma- 
tions, 415,  542. 
Deformation,  of  a  circuit  on  a  Eiemann's  surface 
over  branch-point  impossible,  397 ; 

of  connected  surfaces,  geometrical  and 
physical,  379 ; 

can  be  effected  from  one  to  another 
if  they  be  bifacial,  be  of  the  same 
connectivity,  and  have  the  same 
number  of  boundaries,  380 ; 
to  its  canonical  form  of  Eiemann's  sur- 
face with  simple  winding-points,  413  ; 
of  loops,  405  et  seq. ; 
of  path  of  integration,  of  holomorphie 
function  does  not  affect  value  of  the 
integral,  30 ; 

over  pole  of  meromorphic  function 

affects  value  of  the  integral,  39  ; 
of  multiform  function  (see  integral 
of  multiform  function) ; 
form  of,  adopted,  224 ; 
effect  of,  when  there  are  more 
than  two  periods,  247  ; 
on  Eiemann's  surface  (see  path  of 

integration) ; 
of  path  of  variable  for  multiform 

functions,  181  ; 
how  far  it  can  take  place  without 
affecting  the  final  branch,  181- 
184. 


INDEX 


839 


Deformation  of  surfaces  of  constant  negative 

curvature,  712. 
Degree  of  a  function  on  a  Eiemann's  surface, 

420. 
Degree    of    cycle    of    branches    of    algebraic 

function,  570. 
•De  Haan,  47. 
Derivative,  Schwarzian,  657  (see    Schwarzian 

derivative). 
Derivatives,  a  holomorphic  function  possesses 
any  number  of,  at  points  within  its  region, 
36; 

do  not  necessarily  exist  along  the  boun- 
dary of  the  region  of  continuity,  36, 
158; 
superior  limit  for  modulus  of,  38  ; 
of  elliptic  functions  with  regard  to  the 
invariants,  311,  312. 
Description  of  closed  curve,  positive  and  nega- 
tive directions  of,  3. 
De  Sparre,  114. 

Differential  equation  of  first  order,  satisfied  by 
uniform  doubly-periodic  functions,  277 ;. 

in  particular,  by  elliptic  functions,  277, 
278; 
possessing  uniform  integrals,  283; 
satisfied  by  function  which  possesses  an  alge- 
braic addition-theorem,  356. 
Differentiation       of      uniformly      convei-ging 

function-series,  156. 
Dihedral   function,    automorphic   for  dihedral 

group,  765  (see  polyhedral  functions). 
Dihedral  group,  of  rotations,  757  ; 

of  homogeneous  substitutions,  758  ; 
of  linear  substitutions,  759  ; 
function  automorphic  for,  765. 
Dingeldey,  381. 
Dini,  vi. 

Directions  of  description  of  closed  curve,  3. 
Discontinuous,  groups,  717  ; 

properly  and  improperly,  718  ; 
all  finite  groups  are,  719  ; 
division  of  plane  associated  with,  724 
(see  regions). 
Discrete  group,  717. 

Discriminating  circle  for  uniform  function,  133. 
Discrimination  between  accidental  and  essen- 
tial singularities,  61,  80. 
Discrimination  of  branches  of  a  function  ob- 
tained by  various  paths  of  the  variable,  181 
-184. 
Divergence,  of  series,  22  ;  of  products,  91. 
Division   of   surface  into  polygons,  Lhuilier's 

theorem  on,  372. 
Dixon,  139,  589. 


Domain  of  ordinary  point,  60. 
Dominant  function,  39. 
Double  points  of  linear  substitution,  628. 
Double-pyramid,    division   of    surface   of    cir- 
cumscribed sphere  by  planes  of  symmetry, 
694; 

equation  giving  the  conformal  represen- 
tation on  a  half-plane  of  each  triangle 
in  the  stereographic  projection  of  the 
divided  spherical  surface,  698. 
Doubly-infinite  system  of  zeros,  transcendental 

function  having,  104. 
Doubly-periodic  functions,  uniform,  235  ; 
graphical  representation  of,  236  ; 
those  considered  have  only  one  essential 
singularity  which  is  at  infinity,  257, 
267,  281 ; 
fundamental  properties  of  uniform,  258 

et  seq.; 
order  of,  259  ; 
equivalent,  260 ; 
integral    of,    round     parallelogram     of 

periods,  is  zero,  260 ; 
sum  of  residues  of,  for  parallelogram,  is 

zero,  262 ; 
of  first  order  do  not  exist,  262  ; 
of  second  order  consist  of  two  classes, 

262; 
number   of  zeros   equal   to   number  of 

infinities  and  of  level  points,  266 ; 
sum  of  zeros  congruent  with  the  surfi  of 
the  infinities  and  with  the  sum  of  the 
level  points,  267 ; 
of  second  order,  characteristic  equation 
of,  270 ; 

zeros  and  infinities  of  derivative  of, 

271; 
can  be  expressed  in  terms  of  any 
assigned    homoperiodic    function 
of  the  second  order  with  an  ap- 
propriate argument,  273  ; 
of  any  order  with  simple  infinities  can 
be  expressed  in  terms  of  homoperiodic 
functions  of  the  second  order,  274  ; 
are  connected  by  an  algebraical  equation 

if  they  have  the  same  periods,  276  ; 
differential  equation  of  first  order  satis- 
fied by,  276 ; 

in  particular,  by  elliptic  functions, 
277; 
can  be  expressed  rationally  in  terms  of. 
a  homoperiodic  function  of  the  second 
order  and  its  first  derivative,  279  ; 
of  second  order,  properties  of  (see  second 
order) ; 


840 


INDEX 


Liouville's  theorem  as  to,  281 ; 
expressed  in  terms  of  the  f- function,  302; 

and  of  the  (j-function,  305  ; 
possesses  algebraical   addition-theorem, 
344. 
Doubly-periodic  integral  of  differential  equation 

of  first  order,  283. 
Du  Bois-Eeymond,  158. 
Durege,  64,  363,  381. 
Dyck,  381,  716,  718. 

Edges  of  cross-cut,  positive  and  negative,  424, 

499. 
Edges  of  regions  in  division  of   plane  by  an 
infinite  group,  724 ; 

two  kinds  of,  for  real  groups,  725  ; 
congruent,  are  of  the  same  kind,  725  ; 
conjugate,  725 ; 
of  first  kind  are  even  in  number  and  can 

be  arranged  in  conjugate  pairs,  726  ; 
each  pair  of  conjugate,  implies  a  funda- 
mental substitution,  726. 
Eisenstein,  105,  107. 
Electric  force,  electric  intensity,  647. 
Electrostatics,  application  of  conformal  repre- 
sentation to,  646. 
Elementary  function  of  second  kind,  509  (see 

second  kind  of  functions). 
Elementary  functions  of  third  kind,  511  (see 

third  kind  of  functions). 
Elementary  integrals  of  second  kind,  446  ; 

determined  by  an  infinity,  except  as  to 

additive  integral  of  first  kind,  448  ; 
number  of  independent,  449  ; 
connected  with  those  of  third  kind,  453. 
Elementary  integrals  of  third  kind,  452  ; 

connected  with  integrals  of  second  kind, 

453; 
number  of  independent,  with  same  log- 
arithmetic  infinities,  453. 
Elements  of  analytic  function,  67  ; 

can  be  derived  from  any  one  when  the 

function  is  uniform,  68  ; 
any  single  one  of  the,  is  sufficient  for 
the  construction  of  the  function,  68. 
Ellipse,  area  without,  represented  on  a  circle, 
614; 

area  within,  represented  on  a  rectangle, 
616; 

and  on  a  circle,  617. 
•Ellipsoid   conformally   represented   on    plane, 

612. 
Elliptic  equations,  or  curves,  555. 
Elliptic  functions  and  equations  of  genus  unity, 
556. 


Elliptic  functions,  obtained  by  integrating  mul- 
tiform functions,  in  Jacobian  form,  228  ; 

in  Weierstrassian  form,  231,  293  et  seq.; 
■  on  a  Riemann's  surface,  432  et  seq. 
Elliptic  substitutions,  631,  633; 

are  either  periodic  or  infinitesimal,  635; 
occur  in  connection  with  cycles  of  cor- 
ners, 741,  747. 
Enneper,  771. 

Equations,   of  genus   greater  than  two,  566  : 
normal  form  of,  569. 

of  genus  two,  562-565:  variables  in, 
expressible  by  sextic  or  quintic  radical,  563: 
only  limited  number  of  birational  trans- 
formations into  one  another,  566 :  normal 
form  of,  567. 

of  genus  unity,  554-562  :  variables  in, 
expressible  by  quartic  or  cubic  radical,  554, 
and  as  uniform .  elliptic  functions,  556:  bi- 
rationally  transformable  into  one  another 
with  one  arbitrary  parameter,  558  :  normal 
form  of,  567. 

of  genus  zero,  548-554:  variables  in, 
expressible  as  rational  functions,  548 :  bi- 
rationally  transformable  into  one  another 
with  three  arbitrary  parameters,  550  :  sub- 
rational  representation  of  variables  in,  551, 
made  rational,  552  :  normal  form  of,  567. 
Equipotential  lines  in  planar  electrostatics,  647. 
Equivalent  homoperiodic  functions,  260; 

conditions  of  equivalence,  265. 
Essential  singularities,  19,  61 ; 

uniform  function  must  assume  any  value 

at  or  near,  64,  116; 
of   transcendental   integral  function  at 

infinity,  90; 
form  of  function  in  vicinity  of,  118 ; 
continuation  of  function  possessing,  120 ; 
form  of  function  having  finite  number 

of,  as  a  sum,  121 ; 
functions  having  unlimited  number  of, 

Chap.  VII. ; 
line  of,  165 ; 
laeunary  space  of,  166 ; 
classification  of,  into  classes,  175 ; 
into  species,  177 ; 
into  wider  groups,  177; 
of  pseudo-automorphic  functions,  776; 
of  automorphic  fjmctions,  786. 
Essential  singularities  of  groups,  637,  739; 

are  essential  singularities  of    functions 

automorphic  for  the  group,  739  ; 
lie  on  the  fundamental  circle,  739 ; 
may  be  the  whole  of  the   fundamental 
circle,  740. 


INDEX 


841 


Exceptional     values    unattainable     near    an 

essential  singularity,  66. 
Existence  of  functions  on  a  Eiemann's  surface 

without  boundary,  491. 
Existence-theorem    for  functions  on   a   given 
Eiemann's  surface,  Chap.  xvii. ; 
methods  of  proof  of,  459  ; 
abstract  of  Schwarz's  proof  of,  460 ; 
results  of,  relating  to  classes  of  functions 
proved  to  exist  under  conditions,  496. 
Expansion  of  a  function  in  the  vicinity  of  an 
ordinary  point,  by  Cauchy's  theorem,  50 ; 
within  a  ring,  by  Laurent's  theorem,  54. 
Expression  of  uniform  function,  in  vicinity  of 
ordinary  point,  50 ; 

in  vicinity  of  a  zero,  75  ; 

in    vicinity   of    accidental    singularity, 

79; 
in  vicinity  of  essential  singularity,  118 ; 
having  finite  number  of  essential  singu- 
larities, as  a  sum,  122 ; 

as  a  product  when  without  acciden- 
tal singularities  and  zeros,  125, 
126; 
as  a  product,  with  any  number  of 
zeros   and   no   accidental   singu- 
larities, 130; 
as  a  product,  with  any  number  of 
zeros  and  of  accidental  singulari- 
ties, 132; 
in  the  vicinity  of  any  one  of  an  infinite 
number  of  essential  singularities,  135 ; 
having  an  assigned  infinite  number  of 
singularities  over  the  plane,  137 ; 
generalised,  138 ; 
having  infinity  as   its   single   essential 

singularity,  140; 
having  unlimited  singularities   distrib- 
uted over  a  finite  circle,  140. 
Expression  of  multiform  function  in  the  vicinity 
of  branch-point,  187. 

Factor,  generalising,  of  transcendental  integral 
function,  99; 

primary,  101 ; 

fractional,   for  potential-function,   476 ; 
major  and  minor,  477. 
Factorial  functions,  pseudo-periodic  on  a  Eie- 
mann's surface,  531 ; 

their  argument,  531; 
constant  factors  (or  multipliers)  for  cross- 
cuts of,  532 ; 

forms  of,  when  cross-cuts  are  canon- 
ical, 532; 
general  form  of,  532 ; 


expression  of,  in  terms  of  normal  ele- 
mentary functions  of  the  third  kind, 
533  et  seq. ; 
zeros  and  infinities  of,  535  ; 
cross-cut   multipliers    and   an  assigned 
number     of    infinities    determine     a 
limited  number  of  independent,  537. 
Factorial  periodicity,  719. 

Factors  (or  miTltipliers)  of  factorial  functions 
at  cross-cuts,  532; 

forms  of,  when  cross-cuts  are  canonical, 
532. 
Falk,  239. 
Famihes  of  groups,  seven,  740  ; 

for  one  set,  the  whole  line  conserved  by 
the  group  is  a  line  of  essential  singu- 
larity ;   for  the  other  set,  only  parts 
of    the    conserved  line   are   lines  of 
essential  singularity,  741. 
Finite  groups  of  linear  substitutions,  719,  754 ; 
containing  a  single  fundamental  substi- 
tution, 719; 
anharmonic,     containing    two     elliptic 
fundamental  substitutions,  720. 
Finite  number  of  essential  singularities,  func- 
tion having,  expressed  as  a  sum,  122. 
First  kind  of  pseudo-periodic  function,  320. 
First  kind  of  functions  on  a  Eiemann's  surface, 
498; 

moduli  of  periodicity  of  functions  of, 
500  et  seq. ;   ' 

relation   between,   and   those  of  a 

function  of  second  kind,  503; 
when  the  functions  are  normal,  508 ; 
number  of  linearly  independent  functions 

of,  505 ; 
normal  functions  of,  508 ; 
inversion  of,  leading  to  multiply  periodic 

functions,  515 ; 
derivatives  of,  as  algebraical  functions, 
524; 

infinities  and  zeros  of,  525 ; 
conserved  under  birational  transforma- 
tion, 542. 
First  kind  of  integrals  on  Eiemann's  surface, 
444; 

number  of,  linearly  independent  in  par- 
ticular case,  445 ; 
are  not  uniform  functions,  445  ; 
general  value  of,  446  (see  first  kind  of 

functions) ; 
sum  of,  expressed  by  Abel's  Theorem, 
590. 
Fixed  circle  of  elliptic  Kleinian  substitution, 
when  the  equation  is  generalised,  747. 


842 


INDEX 


Fixed  points  of  linear  substitution,  628. 

Floquet,  329. 

Form  of  argument  for  given  value  of  function 

possessing  an  addition-theorem,  347  et  seq. 
Fourier,  651. 
Fractional  factor  for  potential  function,  476 ; 

major,  minor,  477. 
Fractional   part   of  doubly-periodic   function, 

259. 
Fredholm,  54. 
Fresnel's  integrals,  44. 

Fricke,  vii,  153,  453,  523,  526, 530,  625, 704,  754. 
Frobenius,  312,  322,  328. 
Fuchs,  133,  771. 
Fuchsian    functions,     753    (see    automorpMc 

functions). 
Fuchsian  group,  723,  740 ; 

if  real,  conserves  axis  of  real  quantities, 

723; 
when  real,   it   is   transformed   by   one 
complex  substitution  and  then  con- 
serves a  circle,  737; 
division    of    plane    into    two    portions 
within  and  without  the  fundamental 
circle,  737; 
families  of,  740; 
genus  of,  742. 
Function    defined    by    algebraic    equation    is 

analytic,  207. 
Function  on  Eiemann's  surface,  construction 

of  rational,  523  ;  special,  526. 
Function,  Eiemann's  general  definition  of,  8 ; 
relations  between  real   and  imaginary 

parts  of,  9 ; 
equations  satisfied  by  real  and  imaginary 

parts  of,  12 ; 
monogenic,  defined,  15 ; 
uniform,  multiform,  defined,  16; 
branch,  and  branch-point,  defined,  16 ; 
holomorphic,  defined,  17; 
meromorphic,  defined,  17; 
continuation  of  a,  67; 
region  of  continuity  of,  67 ; 
element  of,  67 ; 

monogenic  analytic,  definition  of,  67 ; 
constant  along  a  line  or  area,  if  uniform, 

is  constant  everywhere,  73 ; 
properties  of  uniform,  without  essential 

singularities,  Chap.  iv. ; 
rational  integral,  84; 
transcendental,  84; 

having  a  finite  number  of  branches  is 
a   root    of    an    algebraical    equation, 
210; 
potential,  457  (see  potential  function). 


Function  possessing  an  algebraic  addition- 
theorem,  is  either  algebraic,  or  algebraic 
simply-periodic,  or  algebraic  doubly-periodic, 
347; 

has  only  a  finite  number  of  values  for 

one  value  of  the  argument,  355 ; 
if   uniform,    then    either    rational,    or 
simply-periodic,     or    doubly-periodic, 
355  ; 
satisfies  a  differential  equation  between 
itself  and  its  first  derivative,  356. 
Functional  dependence  of   complex  variables, 
form  of,  adopted,  7; 

analytical  conditions  for,  7; 
establishes     conformal    representation, 
11. 
Functionality,  monogenic,  not  coextensive  with 

arithmetical  expression,  164. 
Functions,  expression  in  series  of  (see  series  of 

functions). 
Functions     of     two     variables,     Weierstrass's 

theorem  on  regular,  203-6. 
Fundamental  circle  of  Fuchsian  group,  737 ; 

divides  plane  into  two  parts  which  are 
inverses  of  each  other  with  regard  to 
the  circle,  738; 
essential  singularities  of  the  group  lie 
on,  740. 
Fundamental  equation  for  a  Biemann's  surface 
is  determined  by  algebraical  functions  that 
exist  on  the  surface,  529. 
Fundamental  parallelogram  for  double  period- 
icity, 237,  244; 

is  not  unique,  244. 
Fundamental  region  (or  polygon)  for  division 
of    plane   associated   with   a   discontinuous 
group,  724; 

can  be  taken  so  as  to  have  edges  of  the 
first  kind  cutting  the  conserved  line 
orthogonally,  728,  738; 

in  this  case,  called  a  normal  polygon, 
727; 
which  can  be  taken  as  convex, 
728; 
angles  of,  730  (see  convex  normal 
polygon) ; 
characteristics  of,  732. 
Fundamental  set  of  loops,  407. 
Fundamental  substitutions  of  a  group,  716; 
relations  between,  717,  726,  732; 
one  for  each  pair  of  conjugate  edges  of 
region,  726. 
Fundamental  systems  of  isothermal  curves, 712 ; 
given  by  a  uniform  algebraic  function, 
or  a  uniform  simply -periodic  function, 


INDEX 


843 


or  a  uniform  doubly-periodic  function, 
712; 
all  families  of  algebraic  isothermal  curves 
are  derived  from,  by  algebraic  equa- 
tions, 713. 

Galois,  715. 

Gauss,  2,   11,    103,   458,   602,   607,   611,  712, 

714. 
General  conditions  for  potential  function,  460 

(see  potential  function). 
Generalised  equations  of  Kleinian  group,  745 
(see  Kleinian  group) ; 

polyhedral  division  of  space  in  connec- 
tion with,  747. 
Generalising  factor  of  transcendental  integral 

function,  99. 
Genus  of,  algebraic  equation  associated  with  a 
Kiemann's  surface,  395 ; 

between  automorphic  functions,  789 ; 
connected  surface,  371 ;  conserved  under 

birational  transformation,  542; 
Fuchsian  group,  742 ; 
Eiemann's  surface,  395; 
of  Eiemann's  surface  equal  to  deficiency 
of  associated  curve,  403  ; 

determined  by  Baker's  rule,  404. 
Genus  zero,  equations  of,  548-554; 
unity,  equations  of,  554-562; 
two,  equations  of,  562-565 ; 

curve  of,  transformable  into  a  quar- 
tic,  565. 
Gordan,  208,  247,  407,  415,  453,  518,  519,  569, 

579,  719. 
Goursat,  82,  105,  172,  222,  223,  243,  342,  530, 

676,  679,  752. 
Graphical  determination  of,  order  of  infinity  of 
an  algebraic  function,  194  ; 

the   leading  term  of    a   branch   in   the 
vicinity  of  an  ordinary  point  of  the 
coefficients  of  the  equation,  196; 
the  branches  of  an  algebraic  function  in 
the  vicinity  of  a  branch-point,  199. 
Graphical  representation  of  periodicity  of  func- 
tions, 236,  237. 
Green,  458. 
GreenhiU,  227. 
Group  of  linear  substitutions,  715 ; 

fundamental  substitutions  of,  716 ; 

relations  between,  717 ; 
continuous,  and  discontinuous  (or  dis- 
crete), 717; 
properly  and  improperly  discontinuous, 

718; 
finite,  719  (see  finite  groups) ; 


modular,    with   two    fiindamental   sub- 
stitutions, 720 ; 

division    of    plane    into    polygons 
associated  with,  721  et  seq. ; 
relation    between    the    funda- 
mental substitutions,  723 ; 
division  of  plane  for  any  discontinuous 
group,  724  (see  region) ; 

fundamental  region  for,  724 ; 
Fuchsian,  724, 740  (see  Fuchsian  group) ; 
when  real,  conserves  axis  of  real  quanti- 
ties, 724; 
fundamental  substitutions  of,  connected 
with  the  pairs  of  conjugate  edges  of  a 
region,  726 ; 
seven  families  of,  740 ; 
conserved  line  in  relation  to  the  essential 

singularities,  741 ; 
Kleinian,  743  (see  Kleinian  group) ; 
dihedral,  757; 
tetrahedral,  759. 
Grouping  of  branches  of  algebraical  function 

at  a  branch-point,  200. 
Giinther,  530. 

Guichard,  126,  176,  177,  256,  257. 
Gutzmer,  53. 
Gyld^n,  150. 

Hadamard,  54,  113,  803. 
Half-plane  represented  on  a  circle,  619; 
on  a  semicircle,  620; 
on  an  infinitely  long  strip,  621 ; 
on  a  sector,  622; 
on  a  rectilinear  polygon,  665  et  seq.  (see 

rectilinear  polygon) ; 
on  a  curvilinear  polygon,  bounded  by 
circular  arcs,  79  et  seq.  (see  curvilinear 
polygon,  curvilinear  triangle). 
Halphen,  105,  309,  312,  322,  332,  342,  343,  572. 
Halphen's  birational  transformation  of  plane 
curves,    572 ;    used   to    transform   cycle    of 
branches  of  algebraic  function,  572-577. 
Hankel,  153,  223. 
Hardcastle,  F.,  381. 
Hardy,  6. 

Harnack,  6,  10,  20,  61,  459. 
Heine,  223. 
Helmholtz,  642,  646. 
Henrici,  459. 

Hermite,  vii,  23,  48,  95,   103,  113,  134,  165, 
219,  220,  222,  302,  322,  324,  326,  333,  342, 
518,  531,  541,  547,  767,  771. 
Hermite's    sections    for  integrals  of  uniform 

functions,  220. 
Herz,  611. 


844 


INDEX 


Hexagon,    symmetrical    about   one   diagonal, 

area  of,  represented  on  half-plane,  678. 
Hilbert,  134. 
Hill,  M.  J.  M.,  69. 
Hobson,  6,  21,  102. 
Hodgkinson,  690. 
Hofmann,  414. 
Holder,  64. 

Hole  in  surface,  effect  of  making,  on  connec- 
tivity, 367. 
Holomorphie  function,  defined,  17; 

integral  of,  round  a  simple  curve,  27 ; 
along  a  line,  28; 
when  line  is  deformed,  29 ; 
when  simple  curve  is  deformed,  30 ; 
has  a  derivative  for  points  within,  but 
not  necessarily  on  the  boundary  of, 
its  region,  36; 
superior  limit  for  modulus  of  derivatives 

of,  38 ; 
expansion  of,  in  the  domain  of  an  ordi- 
nary point,  50,  60; 

within   a  -ring   of  convergence  by 
Laurent's  theorem,  55. 
Holzmiiller,  2,  391,  615,  625. 
Homen,  172. 

Homogeneous  form  of  linear  substitutions,  756. 
Homogeneous  substitutions,  756; 

two  derived  from  each  linear  substitu- 
tion, 756; 
dihedral  group  of,  758. 
nomographic  substitution  connected  with  sphe- 
rical rotation,  755. 
nomographic  transformation,  or  substitution, 

625  (see  linear  substitution). 
Homologous  points,  237,  724. 
Homoperiodic  functions,  263; 

when  in  a  constant  ratio,  263 ; 

are  connected  by  an  algebraical  equation, 

263. 
when  equivalent,  265; 
Hotiel,  2.      . 
Humbert,  519,  530. 
Hurwitz,  456,  566,  718,  721,  771,  792. 
Hydrodynamics,  application  of  conformal  repre- 
sentation to,  639. 
Hyperbolic  substitutions,  631,  633; 

neither  periodic  nor  infinitesimal,  636; 
do  not  occur  in  connection  with  cycles 
of  corners,  741,  748. 
Hyperelliptic  equations  or  curves,  565. 
Hypergeometric  series,  solution  of  differential 
equation  for,  connected  with  conformal  repre- 
sentation of  curvilinear  triangle,  685  et  seq. ; 
cases  of  algebraical  solution,697  et  seq. 


Icosahedral  (and  dodecahedral)  division  of  sur- 
face of  circumscribed  sphere,  696; 

equation  giving  the  conformal  represent- 
ation on  a  half-plane  of  each  triangle 
in  the  stereographic  projection  of  the 
divided  surface,  704. 
Identical  substitution,  716. 
Imaginary  parts  of  functions,  how  related  to 
real  parts,  9 ; 

equations  satisfied  by  real  and,  12. 
Improperly  discontinuous  groups,  718 ; 

example  of,  749  et  seq. 
Index  of  a  composite  substitution,  716; 

not  entirely  determinate,  717. 
Infinite  circle,  integral  of  any  function  round, 

41. 
Infinite  class  of  integral  function,  113. 
Infinitesimal  curve,  integral  of  any  function 

round,  40. 
Infinitesimal  substitution,  717. 
Infinities,  of  a  function  defined,  17; 

of  algebraic  function,  192. 
Infinities   of  doubly-periodic    functions,    irre- 
ducible, are  in  number  equal  to  the  irreducible 
zeros,  266; 

and,  in  sum,   are  congruent  with  their 

sums,  267; 
of  pseudo-periodic  functions  (see  second 
kind,  third  kind). 
Infinities  of  potential  function  on  a  Riemann's 

surface,  495. 
Integral  function,  52; 

of  infinite  class,  113. 
Integral  with  complex  variables,  defined,  20; 
elementary  properties  of,  22,  23 ; 
over  area  changed  into  integral  round 
boundary,  by  Riemann's  fundamental 
lemma,  25; 
of  holomorphie  function  round  simple 

curve  is  zero,  28 ; 
of  holomorphie  function  along  a  line  is 

holomorphie,  29; 
of  meromorphic  function  round  simple 
curve  containing  one  simple  pole,  31; 
round  simple  curve,  containing  seve- 
ral simple  poles,  33 ; 
round    curve    containing    multiple 
pole,  37; 
of  any  function  round  infinitesimal  circle, 
40; 

round  infinitely  great  circle,  41; 
round  any  curve  enclosing  all  the 
branch-points,  42. 
Integral  of  multiform  function,   between  two 
points  is  unaltered  for  deformation  of  path 


INDEX 


845 


not  crossing  a  branch-point  or  an  infinity, 
215; 

round  a  curve  containing  branch-points 
and  infinities  is   unaltered  when  the 
curve  is  deformed  to  loops,  216 ; 
also  when  the  curve  is  otherwise  deformed 

under  conditions,  217; 
round  a  small  curve  enclosing  a  branch- 
point, 217; 
round  a  loop,  224; 
deformed  path  adopted  for,  225 ; 
with   more   periods   than   two,   can   be 
made  to  assume  any  value  by  modi- 
fying the  path  of  integration  between 
the  limits,  246. 
Integral  of  uniform  function  round  parallelo- 
gram of  periods,  is  zero   when  function  is 
doubly-periodic,  260; 

general  expression  for,  261. 
Integrals,  at  opposite  edges  of  cross-cut,  values 
of,  differ  by  a  constant,  424; 

when  cross-cuts  are  canonical,  426 ; 
discontinuities  of,   excluded  on  a  Kie- 

mann's  surface,  427 ; 
general  value  of,  on  a  Eieraann's  surface, 

428; 
of  algebraic  functions,  436 ; 

when  branch-points  are  simple,  438 ; 
infinities    of,    of    algebraic    functions, 

439; 
first  kind  of,  444 ; 

number  of  independent,  of  first  kind, 

445; 
arenot  uniform  functionsof  position, 

445; 
general  value  of,  446; 
second  kind  of,  446  (see  second  kind) ; 
elementary,  of  second  kind,  446  (see 
elementary  integrals) ; 
third  kind  of,  450  (see  third  kind) ; 

elementary,  of  third  kind,  452  (see 

elementary  integral) ; 
connected  with  integrals  of  second 
kind,  453. 
Integration,  Eiemann's  fundamental  lemma  in, 

24. 
Interchange,  cyclical,  of  branches  of  a  function 
affected  by  a  branch-point,  185; 
of  algebraical  function,   210. 
Interchange   of  argument    and  parameter   in 
normal    elementary  function    of  the   third 
kind,  515. 
Interchange,  sequence  of,  along  branch-lines 

determined,  387. 
Interchangeable  substitutions,  719. 


Invariants,  derivatives  of  elliptic  functions  with 
regard  to  the,  312 ; 

as  automorphic  functions,  785. 
Inversion-problem,  517; 

of  functions  of  the  first  kind  with  several 
variables  leading  to  multiply  periodic 
functions,  517  et  seq. 
Inversions  at  circles,  even  number  of,  lead  to 
lineo-linear  relation  between  initial  and  final 
points,  638. 
Irreducible  circuits,  374; 

complete  system  contains  same  number 

of,  375; 
cannot  be  drawn  on  a  simply  connected 

surface,  376; 
round  two  branch-points,  398. 
Irreducible   equation   and   singleness   of   con- 
nected surface,  392. 
Irreducible,  points,  236,  237,  724,  772; 

zeros  of  doubly-periodic  function  are  the 
same  in  number  as  irreducible  infini- 
ties, 266; 
hkewise    the    number    of    level-points, 

266; 
also  of  automorphic  functions,  787; 
sum  of  irreducible  points  is  independent 
of   the   value    of  the   doubly-periodic 
function,  267. 
Isothermal  curves,  families  of  plane  algebraical, 
707; 

form  of  equation  that  gives  such  families 
as   the   conformal    representation    of 
parallel  straight  lines,  710; 
three  fundamental  systems  of,  710 ; 
all,    are    conformal    representations    of 
fundamental   systems   by   algebraical 
equations,  711; 
isolated,   may  be   algebraical  by   other 
relations,  711. 
Isothermal  lines  in  conduction  of  heat,  650. 

Jacobi,   108,   223,  228,  239  et  seq.,  278,  518, 

592,  611,  612. 
Jacobi's  theorem  in  algebraic  equations  used  to 

deduce  Abel's  Theorem,  592-594. 
Jeans,  649. 
Jordan,  40,  87,  222,  570,  716. 

Kapteyn,  740. 

Kinds  of  edges  in  region  for  Fnchsian  group, 
725  (see  edges). 

Kinds  of  pseudo-periodic  functions,  three  prin- 
cipal, 320,  321 ; 

examples  of  other,  342. 

Kirchhoff,    628,  641.       . 


846 


INDEX 


Klein,  vii,  153,  381,  417,  453,  458,  518,  523, 
526,  530,  545,  566,  579,  612,  625,  631,  679, 
704,  716  et  seq.,  753  et  seq. 
Kleinian     functions,     753     (see     automorphic 

functions). 
Kleinian  group,  748; 

conserves  no  fundamental  line,  743; 
generalised  equations  of,  applied  to  space, 
745; 

conserve  the  plane  of  the  complex 

variable,  745 ; 
double    (or  fised)   circle  of  elliptic 
substitution  of,  746 ; 
polygonal  division  of  plane  by,  746; 
polyhedral  division  of  space  in  connec- 
tion   with   generalised    equations    of, 
747; 
relation  between  polygonal  division  of 
plane  and  polyhedral  division  of  space 
associated  with,  748. 
Konigsberger,  269,  519. 
Kopeke,  161. 
Korkine,  608,  611. 
Krause,  342. 
Krazer,  519. 
Kronecker,   153. 

Lachlan,  688,  692. 

Lacunary  functions,  166. 

Lagrange,  608,  611. 

Laguerre,  109,  HI,   112. 

Laguerre's  criterion  of  class  of  transcendental 
integral  function.  111. 

Lamb,  646. 

Lame,  328,  707. 

Lamp's  differential  equation,  328 ; 

can  be  integrated  by  secondary  periodic 

functions,  330; 
general  solution  for  integer  value  of  n, 
331; 

special  cases,  332. 

Laurent,  50,  54,  57,  58,  82,  252,  253. 

Laurent's  theorem  on  the  expansion  of  a  func- 
tion which  converges  within  a  ring,  54. 

Law  of  reciprocity,  Brill-Nother's,  528. 

Leading  term  of  a  branch  in  vicinity  of  an 
ordinary  point  of  the  coefficients  of  the 
equation  determined,   196. 

Leathern,  646. 

Legendre,  228. 

Lerch,  161. 

Level  places  are  isolated  points,  74. 

Lhuilier,  372. 

Lhuilier's  theorem  on  division  of  connected 
surface  into  polygons,  372. 


Limit,  natural,  of  a  power-series,  153. 
Lindelof,  49. 
Lindemann,  403,  530. 

Linear  cycles  of  branches  of  algebraic  func- 
tions, 570 ;  all  cycles  can  be  birationally 
transformed  into,  577. 
Linear  differential  equations  of  the  second 
order,  connected  with  automorphic  functions, 
791. 
Linear  substitution,  625; 

equivalent  to  two  translations,  a  reflexion 

and  an  inversion,  626; 
changes  straight  lines  and  circles  into 

circles  in  general,  627 ; 
can  be  chosen  so  as  to  transform  any 

circle  into  any  other  circle,  628; 
changes  a  plane  crescent  into  another  of 

the  same  angle,  628; 
fixed  points  of,  628; 
multiplier  of,  628 ; 
condition  of  periodicity,  629; 
parabolic,  631 ; 

and  real,   632; 
elliptic,  631; 

and  real,   633 ; 

is  either  periodic  or  infinitesimal, 
635; 
hyperbolic,  631 ; 

and  real,  633 ; 
loxodromie,  631,  635 ; 
can  be  obtained  by  any  number  of  pairs 

of  inversions  at  circles,  637; 
group  of,  715  et  seq.  (see  group) ; 
normal  form  of,  715  ; 
identical,  716 ; 

algebraical  symbols  to  represent,  716  ; 
index  of  composite,  716 ; 
infinitesimal,  717; 
interchangeable,  719; 
in  homogeneous  form,  756. 
Lines   of    flow   in    conduction    of    heat,    649, 

650. 
Liouville,  190,  249,  257,  269. 
Liouville's  theorem   on   doubly-periodic   func- 
tions, 281. 
Lippich,  363,  381. 
Logarithmic     differentiation      of     converging 

products  is  possible,  92. 
Logarithmic  infinities,  integral  of  third  kind 
on    a   Eiemann's    surface   must   possess   at 
least  two,  452. 
Loop-cuts,  defined,  362 ; 

changed  into  a  cross-cut,  367; 
effect  of,  on  connectivity,  367; 

on  number  of  boundaries,  371. 


INDEX 


847 


Loops,  defined,  182; 

effect  of  a  loop  on  a  branch,  is  unique, 

184; 
symbol  to  represent  effect  of,  405; 

change  of,  when  loop  is  deformed, 
406; 
fundamental  set  of,  407 ;  ' 
simple  cycle  of,  408; 
canonical  form  of  complete  system  of 
simple,  409. 
Love,  672. 
Loxodromic  substitutions,  631,  635; 

neither  periodic  nor  infinitesimal,  637 ; 
do  not  occur  in  connection  with  cycles 
of  corners,  747. 
Liiroth,  407,  408,  551,  554. 

Magnification  in  conformal  representation,  11, 
603; 

in  star-maps,   611. 
Mair,  381. 

Major  fractional  factor,  477. 
Maps,  611. 
Mathews,  751. 
Mathieu,  652. 

Maximum  and  minimum  values  of  potential 
function  for  a  region  he  on  its  boundary, 
476. 
Maxwell,  458. 

Mercator's  projection  of  sphere,  610. 
Meromorphic  function,  defined,   17; 

integral  unchanged  by  deformation   of 
simple  curve  in  part  of  plane  where 
function  is  uniform,  31; 
integral  round  a  simple  curve,  containing 
one  simple  pole,  31 ; 

round  a   curve   containing   several 

simple  poles,  33 ; 
round  a  curve  containing  multiple 
pole,  36; 
cannot,    without   change,   be   deformed 

across  pole,  39; 
is  form    of    uniform    function   with    a 
limited  number  of  accidental   singu- 
larities, 85 ; 
all   singularities   of  rational,  are   acci- 
dental, 87. 
Meyer,  707. 
Michell,  670. 
Minding,   712. 

Minimum   number  of    integrals   in   terms   of 
which  any  number  is  expressible  by  Abel's 
Theorem,  599 ;  the  same  as  genus  of  equa- 
tion, 599. 
Minor  fractional  factor,  477. 


Mittag-Leffler,    vi,    68,    69,    70,    134   et   seq., 

176,  322,  324,  326.     ' 
Mittag-Leffler' s  theorem  on  the  expression  of  a 
uniform  function  over  its  whole  region  of 
existence,  69. 
Mittag-Leffler's  theorems  on  functions  having 
an  unlimited  number  of  singularities,    dis- 
tributed over  the  whole  plane,  134; 
distributed  over  a  finite  circle,  183. 
Mobius,  372,  623,  704. 
Modular-function  defined,  767; 

connected  with  elliptic  quarter-periods, 

767; 
(see  modular  group) ; 
as  automorphic  function,  792. 
Modular  group  of  substitutions,  719 ; 

is    improperly    discontinuous    for    real 

variables,  718  ; 
division  of  plane  into  polygons,   asso- 
ciated with,  720  et  seq. ; 
relation  between  the  fundamental  sub- 
stitutions of,  723; 
for  modulus  of  elliptic  integral,  768; 
for  the  absolute  invariant  of  an  elliptic 
function,  770. 
Moduh  of  periodicity,  for  cross-cuts,  427 ; 

values  of,  for  canonical  cross-cuts,  427  ; 
namber   of  linearly  independent   on   a 

surface,  428; 
examples  of,  429  et  seq. ; 
introduced  in  proof  of  existence-theorem , 

487  et  seq. ; 
of  function  of  first  kind  on  a  Riemann's 

surface,  498  et  seq. ; 
relation  between,  of  a  function  of  first 
kind  and  a  function  of  second  kind,  503 ; 
properties   of,    for   normal   function   of 

first  kind,  508; 
of  normal  elementary  function  of  second 
kind   are   algebraic    functions    of    its 
infinity,  510; 
of  normal  elementary  function  of  third 
kind  are  expressed  as  normal  functions 
of  first  kind  of  its  two  infinities,  513. 
Modulus  of  variable,  3. 
Monogenic,  defined,  15; 

function  has  any  number  of  derivatives, 

36; 
analytic  function,  67. 
Monogenic  functionality  not  coextensive  with 

arithmetical  expression,  164. 
Multiform  function,  defined,  16; 

elements  of,  in  continuation,  68; 
expression  of,  in  vicinity  of  a  branch- 
point, 187; 


848 


INDEX 


defined  by  algebraic  equation,  190  (see 

algebraic  function) ; 
integral  of  (see  integral  of  multiform 

function) ; 
is  uniform  on  Eiemann's  surface,  384, 
390. 
Multiple  circuits,  374. 
Multiple  periodicity,  247 ; 

of  uniform  function  of  several  variables, 
248. 
Multiplication-theorem,  344. 
Multiplicity  of  zero,  75 ; 
of  pole,  80; 

of  a  function  on  a  Eiemann's  surface, 
421. 
Multiplier  of  linear  substitution,  628. 
Multipliers  of  factorial  functions  at  cross-cuts, 
532; 

forms  of,  when  cross-cuts  are  canonical, 
532.- 
Multiply  connected  surface, 'SSO; 
defined,  860; 

connectivity  modified  by  cross-cuts,  364 ; 
by  loop-cuts,  367; 
and  by  slit,  368; 
boundaries  of,  affected  by  cross-cuts,  370 ; 
relation  between  boundaries  of,  and  con- 
nectivity, 371 ; 
divided  into  polygons,  Lhuilier's  theorem 

on,  372; 
number  of  circuits  in  complete  system 
of  circuits  on,  377. 
Multiply-periodic  uniform  functions  of  n  vari- 
ables,  cannot  have  more  than  2n  periods. 
248; 

obtained   by  inversion    of  functions  of 
first  kind,  515  et  seq. 

Natural  limit,  of  a  power-series,  153; 
of  part  of  plane,  689 ; 
for   pseudo-automorphic   function   with 
certain  families  of  groups,  777. 
Negative  curvature,  surfaces  of  constant,  712. 
Negative  edge  of  cross-cut,  424,  499. 
Nekrassoff,  133. 
Netto,  716. 
Neumann,  vii,  5,  6,  42,  182,  190,  363  et  seq., 

384,  401,  458,  459,  518,  531,  535,  586. 
Neumann's  sphere  used  to  represent  the  vari- 
able, 4; 

used  for  multiform  functions,  182. 
Normal  elementary  function  of  second  kind, 

509  (see  second  kind  of  functions). 
Normal  elementary  function  of  third  kind,  510 
(see  third  kind  of  functions). 


Normal  form  of  equations  subject  to  birational 

transformation,  567-569. 
Normal  form  of  linear  substitution,  715. 
Normal  functions  of  first  kind,  508  (see  first 

kind  of  functions). 
Normal  polygon  for  division  of  plane,  728; 

can  be  taken  convex,   728  (see  convex 
normal  polygon). 
Normal  surface,  Klein's,  as  a  surface  of  refer- 
ence of  given  connectivity  and  number   of 
boundaries,  381,  413. 
Nother,  404,  528,  530,  570. 
Number  of  zeros  of  uniform  function  in  any 
area,  75,  77,  82,  86; 

of  periodic  functions  (see  doubly-periodic 

functions,  second  kind,  third  kind) ; 
of   pseudo-automorphic    functions    (see 
pseudo-automorphic  functions). 

Octahedral  (and  cubic)  division  of  surface  of 
circumscribed  sphere,  695 ; 

equation    giving    the   conformal   repre- 
sentation   on    a    half-plane    of   each 
triangle  in  the  stereographic  projec- 
tion of  the  divided  surface,  701. 
Open  cycles  of  corners  in  normal  polygon  for 
division   of  plane  by  Fuchsian  group,   710 
(see  corners); 

do  not  occur  in   division   of  plane   by 
Kleinian  group,  747. 
Order  (Borel's)  of  integral  function,  113. 
Order  of  a  function  on  a  Eiemann's  surface, 

420. 
Order  of  doubly-periodic  function,  259. 
Order  of  infinity  of  a  multiform  function  deter- 
mined, 193. 
Ordinary  point  of  a  function,  60; 

domain  of,  60. 
Origin    of    cycle    of    branches    of    algebraic 

function,  570. 
Oscillating  series,  21. 

Painleve,  70,  134,  165. 

Parabola,  area  without,  represented  on  a  circle, 
618; 

area  within,  represented  on  a  circle,  619. 
Parabolic  substitutions,  631,  632; 

neither  periodic  nor  infinitesimal,  636 ; 
occur  in  connection  with  cycles  of  cor- 
ners, 741,  747. 
Parallelogram   for   double   periodicity,   funda- 
mental, 238,  243; 

edges  and  corners  in  relation  to  zeros 
and  to  accidental  singularities  of  func- 
tions, 258. 


,    INDEX 


849 


Parametric  integer  of  thetafuchsian  functions, 

784. 
Path  of  integration,  20 ; 

can  be  deformed  in  region  of  holomor- 
phic  function  without  affecting  the 
value  of  the  integral,  30 ; 
on  a  Eiemann's  surface,  can  be  de- 
formed except  over  a  discontinuity, 
422. 
Periodic  hnear  substitutions,  629 ; 

are  elliptic,  633. 
Periodicity  of  uniform  functions,  of  one  variable, 
235  et  seq. ; 

of  several  variables,  247. 
Periodicity,  modulus   of,   427   (see   moduli   of 

periodicity). 
Periods  of  a  function  of  one  variable,  235 ; 

cannot  have  a  real  ratio  when  the  func- 
tion is  uniform,  237; 
cannot  exceed  two  in  number  indepen- 
dent  of   one  another  if  function  be 
uniform,  242. 
Permanent  equation  in  Abel's  Theorem,  581. 
Phragmen,  134,  444,  457,  666. 
Picard,  64,  66,  166,  329,  343,  491,  530,  560, 

566,  569,  751. 
Pincherle,   174,  719. 
Plane  used  to  represent  variation  of  complex 

variable,  2. 
Pochhammer,  222. 

Poincare,  vii,  39,  113,  114,  166,  172,  342,  344, 
566,  623,  632  et  seq.,  637,  716  et  seq.,  740, 
752  et  seq. 
Poisson,  458. 

Poles  of  a  function  defined,  17,  61. 
Polyhedral  division  of  space  in  connection  with 
generalised  equations  of  group  of  Kleinian 
substitutions,  748. 
Polyhedral  functions,  connected  with  conformal 
representation,  696  et  seq. ; 

for  double-pyramid,  697,  766  ; 
for  tetrahedron,  698-764 ; 
for  octahedron  and  cube,  700 ; 
for  icosahedron  and  dodecahedron,  703. 
Polynomials,  adjoint,  445. 
Polynomials,  analytic  function  represented  by 

series  of,  70,  134. 
Polynomials  on  a  Eiemann's  surface,  adjoint, 

lead  to  special  functions,  527. 
Position  on  Eiemann's  surface,  most  general 
uniform  function  of,  417; 

their  algebraical  expression,  419 ; 
has  as  many  zeros  as  infinities,  420. 
Positive  edge  of  cross-cut,  424,  459. 
Potential  function,  defined,  457; 

F.  F. 


conditions    satisfied    by,    when    derived 
from  a  function  of  position  on  a  Eie- 
mann's surface,  467; 
general  conditions  assigned  to,  460; 
boundary  conditions  assigned  to,  460; 
Green's     integral- theorems     connected 

with,  461  et  seq. ; 
is  uniquely  determined  for  a  circle  by 
general    conditions    and    continuous 
finite  boundary  values,  463 ; 

integral    expression    obtained    for, 

satisfies  the  conditions,  467  ; 
the  boundary  values  for  circle  may 
have   finite   discontinuities   at   a 
limited  number  of  isolated  points, 
470; 
properties  of,  for  a  circle,  475 ; 
maximum  and  minimum  values  of,  in  a 

region,  lie  on  the  boundary,  476  ; 
is  determined  by  general  conditions  and 
boundary  values,  for  area  conformally 
representable  on  area  of  a  circle,  478 ; 
for  combination  of  areas  when  it 
can  be  obtained  for  each   sepa- 
rately, 480; 
for  area  containing  a  winding-point, 

485; 
for  any  simply  connected   surface, 
486; 
introduction  of  cross-cut  moduli  for,  on 
a  doubly  connected  surface,  487  ; 
on  a  triply  connected  surface,  490  ; 
on  any  multiply  connected  surface, 
491; 
number  of  linearly  independent,  every- 
where finite,  495,  505 ; 
introduction  of  assigned  infinities,  495  ; 
classes  of,  determined,  496  ; 
classes  of  complex  functions  derived  from, 
with  the  respective  conditions,  496. 
Power-series,    as    elements    of   an    analytical 
function,  67  et  seq.,  152  et  seq.; 

region  of  continuity  of,  consists  of  one 
connected  part,  152 ; 

may  have  a  natural  limit,  153. 
Primary  factor,  101. 
Primitive  parallelogram  of  periods,  244. 
Pringsheim,  21,  91,  162,  289. 
Product-form  of  transcendental  integral  func- 
tion with  infinite  number  of  zeros  over  whole 
plane,  99. 
Products,  convergence  of,  91. 
Prym,  400,  401,  417,  459,  519,  531. 
Pseudo-automorphic  functions,  777  (see  theta- 
fuchsian functions). 

54 


850 


INDEX 


Pseudo-periodic  functions,  Chap,  xii.; 
of  the  first  kind,  320  ; 
of  the  second  kind,  321 ; 

properties  of  (see  second  kind) ; 
of  the  third  kind,  321 ; 

properties  of  (see  third  kind) ; 
on   a   Kiemann's   surface   (see   factorial 
functions) . 
Pseudo-periodicity  of  the  ^-function,  301 ; 

of  the  (T-function,  305. 
Puiseux,   197. 

Quadrilateral,    area   of,    represented   on   half- 
plane,  676; 

determination  of  fourth  angular  point, 
three  being  arbitrarily  assigned,  678. 
Quartic  transformable  into  sextic  curve,  546 ; 
into  another  quartic,  547. 

EafEy,  896. 

Eamification  of  a  Eiemann's  surface,  395. 

Eatio  of  periods  of  uniform  periodic  function 

cannot  be  real,  238. 
Eational  function  on  Eiemann's  surface,  how 

to  construct,  523. 
Eational  integral   of  differential   equation    of 

first  order,  283. 
Eational  representation  of  variables  in  equation 

of  genus  zero,  548. 
Eational  transformation,  537,  579. 
Eauseuberger,  342,  719. 

Eeal  and  imaginary  parts  of  functions,  how 
related,  9; 

equations  satisfied  by,  12  ; 
each  can  be  deduced  from  the  other,  12. 
Eeal    potential    function,    457    (see    potential 

function). 
Eeal  substitutions,  723  (see  Fuchsian  group). 
Eeciprocity,  Brill-Nother's  law  of,  528. 
Eeconcileable  circuits,  374. 
Eectangle,  area  within,  represented  on  a  circle, 
613; 

and  on  an  ellipse,  615 ; 
on  a  half-plane,  674,  675. 
Eectilinear    polygon,    convex,    represented   on 
half- plane,  666  et  seq.; 

function  for  representation  of,  668 ; 
equation  which  secures  the  representa- 
tion of,  668 ; 
three  angular  points  (but  not  more)  may 
be  arbitrarily  assigned  in   the  repre- 
sentation, 670 ; 

determination  of  fourth  for  quadri- 
lateral, 677; 
three  sides,  673  (see  triangle) ; 


four  sides,   674  (see  rectangle,    squa,re 

quadrilateral) ; 
limit  in  the  form  of  a  convex  curve,  678. 
Eeducible  circuits,  374. 
Eeducible  points,  236,  237. 
Eegion  of  continuity,  of  a  uniform  function, 
67,  150; 

bounded  by  the  singularities,  68 ; 
of  a  power-series  consists  of  one  con- 
nected part,  152; 

may  have  a  natural  limit,  153 ; 
of  a  series  of  uniform  functions,  153  et 

seq.; 
of  multiform  function,  179. 
Eegions  in  division  of  plane  associated  with 
discontinuous  group: 
fundamental,  724; 

uniform  correspondence  between,  724 ; 
contiguous,  724; 
edges  of,  724  (see  edges) ; 
corners  of,  724  (see  corners). 
Eegular   functions    of   two    variables,   Weier- 

strass's  theorem  on,  204-6. 
Eegular  in  vicinity  of  ordinary  point,  function 

is,  60. 
Eegular  polygon,  area  of,  conformally  repre- 
sented on  a  circle,  678. 
Eegular  singularities  of  algebraical  functions, 

192. 
Eegular  solids,  planes  of  symmetry  of,  dividing 
the  surface  of  the  circumscribed  sphere,  694 
et  seq. 
Eepresentation,  conformal,  11  (see  conformal 

representation). 
Eepresentation  of  complex  variable  on  a  plane, 
2; 

and  on  Neumann's  sphere,  4. 
Eesidue  of  function,  defined,  48; 

when  the  function  is  doubly-periodic,  the 
sum  of  its  residues  is  zero,  261. 
Eesidues  (Cauchy's)  in  Abel's  Theorem,  585. 
Eesolution  of  Eiemann's  surface,  398  et  seq.; 
how  to  choose  cross-cuts  for,  399 ; 
canonical,  402; 

when  in  its  canonical  form,  413. 
Eevolution,  surface  of,  conformally  represented 

on  a  plane,  608. 
Eiemann,  v,  vi,  vii,  8,  10,  15,  24,  158,  214,  220, 
359  et  seq.,  372  et  seq.,   416  et  seq.,  421, 
453,  458,  459,  509,  518,  521,  527,  530,  543, 
545,  548,  567,  611,  654,  792. 
Eiemann,  J.,  459. 

Eiemann-Eoch's  theorem  on  algebraic  functions 
having  assigned  infinities,  521;  comple- 
mented by  Brill-Nother  law,  528. 


INDEX 


851 


Riemann's  definition  of  function,  8. 
Riemann's  fundamental  lemma  in  integration, 

24. 
Riemann's  surface,  aggregate  of  plane  sheets, 
382; 

used  to  represent  algebraic  functions,  384 ; 
sheets  of,  joined  along  branch-lines,  385 ; 
can  be  taken  in  spherical  form,  393; 
connectivity  of,  with  one  boundary,  394 ; 

with  several  boundaries,  396 ; 
genus  of,  395 ; 
ramification  of,  395; 
irreducible  circuits  on,  397; 
resolution  of,  by  cross-cuts  into  a  simply 

connected  surface,  398  at  seq. ; 
canonical  resolution  of,  402; 
form  of,  when  branch-points  are  simple, 
411; 
*  deformation  to  canonical  form  of, 

412; 
resolution  of,  in  canonical  form,  414; 
uniform  functions  of  position  on,  417 ; 
their  expression  and  the  equation 

satisfied  by  them,  419; 
have  as  many  zeros  as  infinities, 
420; 
integrals  of  algebraic  functions  on  a,  423 

et  seq.; 
existence-theorem   for   functions    on    a 

given,  455 ; 
functions  on  (see  first  kind,  second  kind, 
third    kind    of    functions,    algebraic 
functions). 
Riemann's  theorem  on  conformalrepresentation 
of  any  plane  area,  simply  connected,  on  area 
of  a  circle,  654. 
Ritter,  754. 
Roch,  521,  530. 
Roots  of  a  function,  defined,  17 ;  of  an  algebraic 

equation,  88. 
Rotations,  connected  with  linear  substitutions, 
754; 

groups  of  for  regular  solids,  757 ; 
dihedral  group  of,  757 ; 
tetrahedral  group  of,  759. 
Rouche,  53. 
Rowe,  580. 
Runge,  134. 

Salmon,  403,  415,  524. 
Schlafli,  666. 
Schlesinger,  754. 
Schlomilch,  2. 
Schonflies,  679,  752. 
Schottky,  653,  753. 


Schroder,  162. 

Schwarz,  vii,  13,  70,  161,  344,  455  et  seq.,  491, 

566,  617,  619,  654  et  seq. 
Schwarz- Christoff el  transformation,  670. 
Schwarz's  symmetric  continuation,  70. 
Schwarzian  derivative,  used  in  conformal  re- 
presentation, 657,  680  et  seq. 
Scott,  C.  A.,  578. 

Second  kind  of  pseudo-periodic  function,  321 ; 
Hermite's  expression  for,  324,  326; 

limiting  form  of,  when  function  is 
periodic    of   the   first   kind,   325, 
327; 
Mittag-Leffler's  expression  for,  in  inter- 
mediate case,  325,  327 ; 
number   of  irreducible    infinities   same 
as  the  number   of  irreducible  zeros, 
327; 
difi'erence  between  the  sum  of  irreducible 
infinities  and  sum  of  irreducible  zeros, 
328; 
expressed   in    terms  of  the  cr-fnnction, 

328; 
used  to  solve  Lamp's  differential  equa- 
tion, 328. 
Second  order  of  doubly-periodic  functions  (see 
also  doubly-periodic  functions),  properties  of. 
Chap.  XI. ; 

of  second  class  and  odd,  286 ; 

connected    with    Jacobian    elliptic 

functions,  289; 
addition-theorem  for,  290 ; 
of  first   class  and  even,  illustrated  by 
Weierstrassian  elliptic  functions,  293 
et  seq. ; 
of  second  class  and  even,  -313  et  seq. 
Second  kind,  of  functions  on  a  Riemann's  sur- 
face, 498 ; 

relation  between  moduh  of  periodicity  of 
functions  of,  and  those  of  a  function 
of  first  kind,  503; 
elementary  function  of,  is  determined  by 

its  infinity  and  moduli,  509 ; 
normal  elementary  function  of,  509  ; 
moduli  of  periodicity  of,  510 ; 
used  to  construct  algebraic  functions 
on  a  Riemann's  surface,  520. 
Second  kind,  of  integrals  on  a  Riemann's  sur- 
face, 446; 

elementary  integrals  of,  446 ; 
general  value  of,  448 ; 
elementary  integrals  of,  determined  by 
an   infinity  except   as   to    integral  of 
first  kind,  448 ; 
number  of,  449 ; 


852 


INDEX 


(see  second  kind  of  functions) ; 
sum  of,  expressed  by  Abel's  Theorem,  594. 
Secondary  periodic  function,  322  (see  second 

kind  of  pseudo-periodic  function). 
Sections  for  integrals   of   uniform   functions, 

Hermite's,  69,  165,  220. 
Sector  on  a  half-plane,  622. 
Seidel,  162. 
Semicircle  represented  on  a  half-plane,  620; 

on  a  circle,  620. 
Sequence   of    interchange    along  branch-lines 

determined,  387. 
Series,  convergence  of,  21. 
Series  of  functions,  expansion  in,  185  et  seq. ; 
region  of  continuity  of,  156 ; 
represents  the  same  function  throughout 
any  connected  part  of  its  region  of 
continuity,  157; 
may  represent  different  functions  in  dis- 
tinct parts  of  its  region  of  continuity, 
162. 
Series    of   polynomials  representing   analytic 

function,  69,  134. 
Series  of  powers,  expansion  in,  50  et  seq. ; 

function    determined   by,    is   the    same 
throughout  its  region  of  continuity, 
152; 
natural  limit  of,  153. 
Serret,  716. 
Sextic  hyperelliptic  curve  transformable  into 

quartic,  conditions,  546. 
Sheets  of  a  Eiemann's  surface,  382 ; 

relation  between  variable  and,  384 ; 
joined  along  branch-lines,  385. 
Siebeck,  615,  710. 

Simple  branch-points  for  algebraic  function, 
208; 

number  of,  209,  403  ; 
in  connection  with  loops,  404; 
canonical  arrangement  of,  411. 
Simple  circuit,  374. 
Simple  curve,  defined,  24 ; 

used  as  boundary,  369. 
Simple  cycles  of  loops,  408  ; 

number  of  independent,  409. 
Simple  element  for  tertiary  periodic  functions, 
of  positive  class,  338  ; 

of  negative  class,  340. 
Simply  connected  surface,  360 ; 
defined,  362; 
effect  of  cross-cut  on,  363 ; 

and  of  loop-cut  on,  367; 
circuits  drawn  on,  are  reducible,  376 ; 
winding-surface  containing  one  winding- 
poiilt  is,  395. 


Simply  infinite  system  of  zeros,  function  having, 

101. 
Simply-periodic  functions,  237 ; 

graphical  representation,  237,  251; 
properties  of,  with  an  essential  singu- 
larity at  infinity,  252  et  seq. ; 
when  uniform,  can  be  expressed  as  series 

of  powers  of  an  exponential,  253; 
of  most  elementary  form,  255 ; 
limited  class  of,  considered,  257; 
possess     algebraical     addition-theorem, 
345. 
Simply-periodic  integral  of   differential  equa- 
tion of  first  order,  283. 
Single  connected  surface  associated  with  irre- 
ducible    equation     or     with    repetition     of 
irreducible  equation,  393. 
Singular  line,  165. 

Singular  points,  17.  * 

Singularities,    accidental,    17    (see    accidental 
singularity) ; 

essential,  19  (see  essential  singularity) ; 
discrimination  between,  61,  80; 
•     bound  the  region  of  continuity  of  the 
function,  67; 
must  be  possessed  by  uniform  functions, 

78; 
of  algebraical  functions,  regular,  192. 
Singularity  of  a  coefficient  of  an  algebraic  equa- 
tion is  an  infinity  of  a  branch  of  the  function, 
193. 
Slit,  effect  of,  on  connectivity  of  surface,  368. 
Special  function  on  Eiemann's  surface,  508 ; 
is   quotient  of  one   adjoint  polynomial  by 
another,  517. 
Species  of  essential  singularity,  177. 
Sphere   conformally  represented   on  a  plane, 
609; 

Mercator's  projection,  609; 
stereographic  projection,  610. 
Spherical  form  of  Eiemann's  surface,  393 ; 

related  to  plane  form,  393. 
Spherical  surface  with  holes,  connectivity  of, 

368. 
Spheroid,   oblate,   conformally  represented  on 

plane,  612. 
Square,  area  within,  represented  on  a  circle, 
615,  674; 

on  a  half-plane,  673,  675; 
area  without,  represented  on  a  circle,  674. 
Stahl,  519,  753,  788. 
Star-shaped  region  of  continuity,  constructed 

by  Mittag-Leffler,  69. 
Stereographic  projection  of  sphere  on  plane  as 
a  conformal  representation,  610; 


INDEX 


853 


of  curvilinear  triangle  on  the  surface  of 
a  sphere,  693. 
Stickelberger,  312,  519. 
Stieltjes,  172,  173. 
Stokes,  458. 
Stolz,  vi. 
Straight  line  changed  into  a  circle  by  a  linear 

substitution,  626. 
Stream-lines,  640,  641  et  seq. 
Strip  of  plane,  infinitely  long,  represented  on 
half -plane,  621; 

on  a  circle,  619; 
on  a  cardioid,  662. 
Subcategories  of  cycles  of  corners,  741. 
Sub-rational    representation    of    variables    in 

equation  of  genus  zero,  551. 
Substitution,  linear  or  homographic,  625  (see 

linear  substitution). 
Sum  of  residues  of  doubly-periodic  function, 
relative  to  a  fundamental  parallelogram,  is 
zero,  261. 
Sum  of  transcendental  integrals,  Abel's  expres- 
sion for,  583 :  examples  of,  587-590 :  of  first 
kind,  590:  of  second  kind,  594 :  of  third  kind, 
597 :  minimum  number  equivalent  to,  599. 
Surface,  connected,  359 ; 

has  a  boundary  assigned,  360,  368,  375; 
effect  of  any  number  of  cross-cuts  on,  363; 
connectivity  of,  364; 

affected  by  cross-cuts,  366; 
by  loop-cuts,  367 ; 
and  by  slit,  368  ; 
genus  of,  371 ; 

of  constant  negative  curvature   repre- 
sented on  a  plane,  613,  712 ; 
supposed  bifacial,  not  unifacial,  872; 
Lhuilier's  theorem  on  division  of,  into 

polygons,  372; 
Biemann's  (see  Eiemann's  surface). 
Symbol  for  loop,  405 ; 

change  of,  when  loop  is  deformed,  406. 
Symmetric  continuation,  Schwarz's,  70. 
System  of  branch-lines  for  a  Eiemann's  surface, 

387. 
System  of  zeros  for  transcendental  function, 
simply-infinite,  101; 

doubly-infinite,  104; 

cannot    be    triply-infinite   arithmetical 

series,  108; 
used  to  define  its  class,  109. 

Tannery,  vi,  162. 

Tannery's  series  of  functions  representing  dif- 
ferent functions  in  distinct  parts  of  its  region 
of  continuity,  162. 


Teixeira,  174. 

Tertiary  periodic  functions,  322  (see  third  kind). 
Test,  analytical,  of  a  branch-point,  186. 
Tetrahedral  division  of  surface  of  circumscribed 
sphere,  695; 

equation  giving  the  conformal  represent- 
ation on  a  half-plane  of  each  triangle 
in  the  stereographic  projection  of  the 
divided  surface,  699. 
Tetrahedral   function,  automorphie  for  tetra- 
hedral group,  764  (see  polyhedral  functions). 
Tetrahedral  group,  of  rotations,  759 ; 
of  substitutions,  761 ; 

in  another  form,  762 ; 
function  automorphie  for,  764. 
Thetafuchsian  functions,  776; 

their  essential  singularities,  776  ; 
exist  either  only  within  the  fundamen- 
tal circle,  or  over  whole  plane,  accord- 
ing to  family  of  group,  777; 
pseudo-automorphic  for  infinite  group, 

778; 
number  of  irreducible  accidental  singu- 
larities of,  778 ; 
number  of  irreducible  zeros  of,  782 ; 
parametric  integer  for,  784; 
quotient  of   two  with  same  parametric 
integer  is  automorphie,  785. 
Third  kind,  of  functions  on  a  Eiemann's  sur- 
face, 498; 

normal  elementary  function  of,  511 ; 

moduli  of  periodicity  of,  512 ; 
elementary  functions  of,  511 ; 

interchange  of  argument  and  para- 
meter in,  513; 
used  to  construct  Appell's  factorial 
functions,  533  et  seq. 
Third  kind,  of  integrals  on  a  Eiemann's  surface, 
450; 

sum  of  logarithmic  periods  of,  is  zero, 

451; 
must  have  two  logarithmic  infinities  at 

least,  452; 
elementary  integrals  of,  452  (see  third 

kind  of  functions); 
sum  of,  expressed  by  Abel's  Theorem,  597. 
Third  kind  of  pseudo-periodic  function,  321 ; 

canonical  form   of  characteristic  equa- 
tions, 322 ; 
relation  between  number  of  irreducible 
zeros  and  number  of  irreducible  infini- 
ties, 333; 
relation  between  sum  of  irreducible  zeros 
and  sum  of  irreducible  infinities,  334 ; 
expression  in  terms  of  cr-function,  335 ; 


854 


INDEX 


of  positive  class,  335  ; 

expressed  in  terms   of  simple   ele- 
ments, 337; 
of  negative  class,  338 ; 

expressed  in  terms  of  Appell's  ele- 
ment, 340; 
expansion  in  trigonometrical  series, 
340. 
Thomas,  580. 

Thomson  (Lord  Kelvin),  458. 
Thomson,  Sir  J.  J.,  649. 

Three    principal    classes    of    functions   on    a 
Biemann's  surface,  498  (see  first  kind,  second 
kind,  third  kind,  of  functions). 
Tractrix    and    surface    of    constant    negative 

curvature,  612,  672. 
Transcendental  function,  84; 

it    has    2^00    for   an   essential    singu- 
larity, 90; 
with  unlimited  number  of  zeros  over  the 
whole  plane,  in  form  of  a  product, 
92  et  seq. ; 
most  general  form  of,  99 ; 
having  simply-infinite  system  of  zeros, 

102; 
having  doubly-infinite  system  of  zeros, 
104; 
Weierstrass's  product  form  of,  107 ; 
cannot  have  triply-infinite  arithmetical 

series  of  zeros,  108  ; 
class  of,  determined  by  zeros,  109 ; 
simple,  of  given  class,  112. 
Transcendental  integrals,  Abel's  expression  for 
sum  of,  585 :  examples  of,  587-590 :  of  first 
kind,  590 :   of  second   kind,   594 :   of  third 
kind,  597. 
Transcendents,    Abel's  Theorems  relating  to, 

579-601. 
Transformation,  birational,  537-579;  effect  of, 

on  irreducible  equation,  539. 
Transformation,  homographic,  625  (see  linear 

substitution). 
Transformation,  rational,  537,  579;  effect  of, 

on  irreducible  equation,  540. 
Transformation,  uniform,  415 ;  birational  (see 

birational  transformation). 
Triangle,  rectilinear,   represented   on   a   half- 
plane,  671 ;  with  special  cases,  672 ; 

separate  cases  in  which  representation  is 
complete  and  uniform,  672; 

curvilinear,  represented  on  a  half- 
plane,  685   (see    curvilinear   tri- 
angle). 
Trigonometrical  series,  expansion  of  tertiary 
periodic  functions  in,  340. 


Triply-infinite  arithmetical  system  of  zeros  can- 
not be  possessed  by  transcendental  function, 
108. 

Triply-periodic  uniform  functions  of  a  single 
variable  do  not  exist,  243 ; 

example  of  this  proposition,  435. 

Two  equations  of  genus,  562-565. 

Two-sheeted  surface,  special  form  of  branch- 
lines  for,  390. 

Two  variables,  Weierstrass's  theorem  on 
regular  functions  of,  204-6. 

Unconditional   convergence   of  series,   21 ;   of 

products,  91. 
Unicursal  equations  or  curves,  548. 
Unifacial  surfaces,  372,  380. 
Uniform  convergence   of  series,    21 ;    of  pro- 
ducts, 91. 
Uniform  function,  defined,  16. 
Uniform  function,  must  assume  any  value  at 
an  essential  singularity,  61,  64,  115 ; 

has  a  unique  set  of  elements  in  continua- 
tion, 68; 
is  constant  everywhere  in  its  region  if 

constant  over  a  line  or  area,  72 ; 
number  of  zeros  of,  in  an  area,  77 ; 
must  assume  any  assigned  value,  78 ; 
must  have  at  least  one  singularity,  78 ; 
is    polynomial    if    only   singularity    be 

accidental  and  at  infinity,  83 ; 
is  rational  and  meromorphic  if  there  be 
no  essential   singularity  and  a  finite 
number   of    accidental    singularities, 
85; 
transcendental  (see  transcendental  func- 
tion) ; 
Hermite's  sections  for  integrals  of,  220 ; 
of  one  variable,  that  are  periodic,  238  et 

seq.; 
of  several  variables  that  are.  periodic, 

247; 
simply-periodic  (see  simply-periodic  uni- 
form functions) ; 
doubly-periodic  (see  doubly-periodic  uni- 
form functions). 
Uniform  function  of  position  on  a  Eiemann's 
surface,   multiform   function   becomes,  383, 
390; 

most  general,  418; 

algebraic  equation  determining,  419 ; 
has  as  many  zeros  as  infinities,  420. 
Uniform  integrals  of  differential  equations  of 
the  first  order,  and  their  characters :    con- 
ditions for,  283. 
Uniformity  of  elliptic  functions,  233-235. 


INDEX 


855 


Uniformly  converging  function-series  can   be 

differentiated,  156. 
Unity,  equations  of  genus,  554-562. 
Unlimited  number  of   essential   singularities, 
functions  possessing.  Chap.  vii. ; 

distributed  over  the  plane,  134  ; 
over  a  finite  circle,  140. 

Velocity,    and    velocity   potential,    in    hydro- 
dynamics, 639  et  seq. 
Vivanti,  113. 

Von  der  Miihll,  611,  707. 
Von  Mangoldt,  753,  788. 
Voss,  606. 

Watson,  103. 

Weber,  223,  625,  753,  767,  771. 

Weierstrass,  v,  vi,  vii,  15,  51,  61  et  seq.,  64,  67, 

68,  92  et  seq.,  118  et  seq.,  134  et  seq.,  166, 

173,    204,    277,    299,    344,    358,    518,    519, 

528. 

Weierstrass's  il/-test  for  uniform  convergence, 

292. 
Weierstrass's  ^-function,  296; 
is  doubly-periodic,  297 ; 
is  of  the  second  order  and  the  first  class, 

298; 
its  differential  equation,  299 ; 
its  addition-theorem,  307 ; 
derivatives  with  regard  to  the  invariants 
and  the  periods,  311. 
Weierstrass's  (7-f unction,  293; 

its  pseudo-periodicity,  305 ; 

periodic  functions  expressed  in  terms  of, 

306; 
its  quasi-addition-theorem,  307 ; 
differential  equation  satisfied  by,  312  ; 
used    to    construct   secondary   periodic 
functions,  328 ; 

and  tertiary  periodic  functions,  335. 


Weierstrass's  f- function,  292; 

its  pseudo-periodicity,  301; 

periodic  functions  expressed  in  terms  of, 

302; 
relation    between    its    parameters   and 

periods,  303; 
its  quasi-addition-theorem,  307. 
Weierstrass's  product-form  for  transcendental 
integral  function,  with   infinite   number   of 
zeros  over  the  plane,  92  et  seq. ; 

with  doubly-infinite  arithmetic  series  of 
zeros,  107. 
Weierstrass's  theorem  on  regular  functions  of 

two  variables,  204-6. 
Weyr,  103. 
Whittaker,  103. 
Wiener,  161. 
Winding-point,  392. 
Winding-surface,  defined,  392; 

portion  of,  that  contains  one  winding- 
point  is  simply  connected,  395. 
Witting,  114. 

Zero,  equations  of  genus,  548-554. 

Zero  of  a  function  on  Eiemann's  surface,  how 

estimated  in  multiplicity,  421. 
Zeros  of  doubly -periodic  function,  irreducible, 
are  in  number  equal  to  the  irreducible  infini- 
ties and  the  irreducible  level  points,  266 ; 
and   in  sum  are  congruent  with  their 
sum,  267. 
Zeros  of  uniform  function  are  isolated  points,  74 ; 
form  of  function  in  vicinity  of,  75 ; 
in  an  area,  number  of,  75,  77,  82,  86 ; 
of  transcendental  function,  when  simply- 
infinite,  102; 

when  doubly-infinite,  104; 
cannot    form    triply-infinite    arith- 
metical series,  108. 
Zuhlke,  690. 


CAMBRIDGE  :    PRINTED  BY 

J.  B.  PEACE,  M.A., 
AT  THE  UNIVERSITY  PRESS 


521  I       2 


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